Free Math Worksheets

Find the slope of the tangent line to \(f(x) = 0.8x-5\) at each x-value. a) \(x = 4\) b) \(x = a\)

A rocket’s distance traveled, \(s\) (in \(\text{km}\)), is modeled by \(s = 3t^2\), where \(t\) is time in minutes. Find the rocket’s average speed from \(t = 4\,\text{min}\) to \(t = 6\,\text{min}\).

Let \(f(x) = \frac{1}{2}x^3 - 2x\). Find the average rate of change of \(f\) over each interval: 1. \([-2, 0]\) 2. \([1, 4]\)

Let \(h(x) = -x^2 + 6x - 5\). Find the average rate of change of \(h\) over \([0, 2]\) and \([3, 5]\).

Let \(f(x) = 2x^2 - 4x\). Find the average rate of change of \(f\) over each interval: a) \([2, 4]\) b) \([2, 2.5]\)

Let \(g(x) = \frac{4}{x}\), where \(x \ne 0\). Find the average rate of change of \(g\) over \([1, 4]\) and \([1, 1.6]\).

Temperatures were recorded in a city during one day. The table shows the measurements at different times, where \(t\) is the number of hours after midnight. <table> <tr> <td>Time \(t\) (in \(\text{h}\))</td> <td>\(0\)</td> <td>\(4\)</td> <td>\(8\)</td> <td>\(12\)</td> <td>\(16\)</td> <td>\(20\)</td> <td>\(24\)</td> </tr> <tr> <td>Temperature \(T\) (in \(^\circ\text{C}\))</td> <td>\(5\)</td> <td>\(2\)</td> <td>\(6\)</td> <td>\(15\)</td> <td>\(19\)</td> <td>\(13\)</td> <td>\(7\)</td> </tr> </table> Find the average rate of change of the temperature over each interval: a) The entire 24-hour period, from \(t = 0\) to \(t = 24\). b) The early morning, from \(t = 4\) to \(t = 8\). c) From \(t = 8\) to \(t = 16\). d) The evening, from \(t = 16\) to \(t = 24\).

A remote-controlled car travels along a straight test track. Its distance \(s\) from the starting point was recorded as a function of time \(t\). <table> <tr> <td>Time \(t\) (in \(\text{s}\))</td> <td>\(0\)</td> <td>\(2\)</td> <td>\(4\)</td> <td>\(6\)</td> <td>\(8\)</td> <td>\(10\)</td> </tr> <tr> <td>Distance \(s\) (in \(\text{m}\))</td> <td>\(0\)</td> <td>\(3.2\)</td> <td>\(12.8\)</td> <td>\(28.8\)</td> <td>\(51.2\)</td> <td>\(80.0\)</td> </tr> </table> Find the average speed for: a) The first 4 seconds. b) The interval from 4 to 8 seconds. c) The final 2 seconds of the measurement. d) The entire observed time interval.

A hot drink cools in a room. Its temperature \(T\) was measured at several times \(t\). <table border="1"> <tr> <td>Time \(t\) (in \(\text{min}\))</td> <td>\(0\)</td> <td>\(4\)</td> <td>\(10\)</td> <td>\(20\)</td> </tr> <tr> <td>Temperature \(T\) (in \(^\circ\text{C}\))</td> <td>\(92\)</td> <td>\(76\)</td> <td>\(58\)</td> <td>\(38\)</td> </tr> </table> 1. Find the average rate of change of the temperature over \([0, 4]\) and over \([4, 20]\). 2. Interpret the sign of each result in context. 3. Give a common name for the quantity you calculated.

Match each quantity \(G\) with its correct unit, the meaning of its instantaneous rate of change \(r\), and the unit of \(r\). The entries in the last three columns are choices; they are not intended to match row by row. For \(E(t)\), \(t\) is measured in seconds; for \(h(t)\), \(t\) is measured in minutes; \(s\) is measured in feet; and \(q\) counts items. <table> <thead> <tr> <th>Quantity \(G\)</th> <th>Unit of \(G\)</th> <th>Meaning of \(r\)</th> <th>Unit of \(r\)</th> </tr> </thead> <tbody> <tr> <td>Energy \(E(t)\)</td> <td>\(\text{lb}\)</td> <td>marginal profit</td> <td>\(\frac{\text{in.}}{\text{min}}\)</td> </tr> <tr> <td>Fill height \(h(t)\)</td> <td>\(\text{J}\)</td> <td>power</td> <td>\(\frac{\$}{\text{item}}\)</td> </tr> <tr> <td>Mass \(m(s)\) of a rope</td> <td>\(\text{in.}\)</td> <td>rising speed</td> <td>\(\text{W}\), or \(\frac{\text{J}}{\text{s}}\)</td> </tr> <tr> <td>Profit \(P(q)\)</td> <td>\(\$\)</td> <td>linear mass density</td> <td>\(\frac{\text{lb}}{\text{ft}}\)</td> </tr> </tbody> </table>

A function \(A\) gives the area of a lake covered by algae, where \(A(t)\) is measured in square meters and \(t\) is the number of days after the first measurement. Interpret each statement in context. a) \(A(10) = 50\) b) \(\frac{A(10)-A(0)}{10} = 4\) c) \(A'(10) = 6\)

A function \(s\) gives a vehicle’s distance traveled during acceleration, where \(s(t)\) is measured in meters and \(t\) in seconds after the start. Interpret each statement in context. a) \(s(4) = 40\) b) \(\frac{s(10)-s(0)}{10} = 25\) c) \(s'(10) = 45\)

Let \(f(x) = \frac{1}{2}x^3 - 2x\). First predict whether the average rate of change of \(f\) over \([1, 3]\) is positive or negative. Then verify your prediction by calculating the difference quotient.

Let \(g(x) = \frac{8}{x^2}\) on the interval \([1, 4]\). First decide whether the average rate of change over this interval is positive or negative, and briefly justify your prediction. Then calculate its exact value.

The function \(h(t)\) gives the height of a sunflower in inches \(t\) days after it sprouts. Interpret each expression in context and give its units. a) \(h(t+7)-h(t)\) b) \(\frac{h(t+\Delta t)-h(t)}{\Delta t}\) c) \(h'(t)\)

A test vehicle moves along a straight track. The function \(s(t)\) gives the vehicle's position in meters at time \(t\) seconds. Explain the physical meaning of each expression. a) \(\frac{s(t_2)-s(t_1)}{t_2-t_1}\) b) \(s'(t)\)

A hot-air balloon rises after takeoff. The function \(h(t)\) gives its height in feet at time \(t\) minutes. Interpret each expression in context and give its units. a) \(h(t)-h(t_0)\) b) \(\frac{h(t)-h(t_0)}{t-t_0}\) c) \(h'(t_0)\)

The derivative of a function \(f\) is constant: \(f'(x)=c\) for every real number \(x\). The graph of \(f\) passes through \(P(0, 1.5)\) and \(Q(5, 9)\). 1. Find \(c\). 2. Find an equation for \(f\). 3. Describe how the value of \(f\) changes whenever \(x\) increases by exactly \(1\).

Use the graph of \(f\) to find the average rate of change over each interval: a) \([-1, 1]\) b) \([0, 2]\)

The graph shows the height \(h(t)\), in meters, of a research drone during a test flight, where \(t\) is measured in minutes. a) Find the average rate of change of the height over \([1, 5]\). Include units. b) Find the average rate of change over \([6, 8]\) and explain the meaning of its sign in context.

The three panels show the graphs of \(f\), \(g\), and \(h\), together with their tangent lines at \(x = 2\). Use the tangent lines to find each derivative value. a) \(f'(2)\) b) \(g'(2)\) c) \(h'(2)\)

Let \(f(x) = -0.5x^3 + 1.5x^2 + 2x\). The graph shows \(f\), the tangent line \(t_0\) at \(x = 0\), and the tangent line \(t_2\) at \(x = 2\). Use the graph to find: a) the average rate of change of \(f\) over \([-1, 1]\), b) the average rate of change of \(f\) over \([2, 4]\), c) the instantaneous rates of change \(f'(0)\) and \(f'(2)\).

The graph of \(f\) is shown with tangent lines labeled \(t_{-2}\), \(t_0\), and \(t_2\) at \(x = -2\), \(x = 0\), and \(x = 2\), respectively. Use the marked points on the tangent lines to find each derivative value. a) \(f'(-2)\) b) \(f'(0)\) c) \(f'(2)\)

For each panel, find the instantaneous rate of change at the marked point \(P\) by using the drawn tangent line. a) Graph of \(f\) at \(P(2, 0)\) b) Graph of \(g\) at \(P(2, 1)\) c) Graph of \(h\) at \(P(1, 1)\)

The graph shows the tidal water level \(h(t)\), in meters, in a harbor over a 12-hour period, where \(t\) is measured in hours. a) Read \(h(0)\) and \(h(6)\) from the graph, and find the average rate of change over \([0, 6]\). b) Read \(h(2)\) and \(h(8)\) from the graph, and find the average rate of change over \([2, 8]\). c) Interpret the value from part a in context.

Use the graph of \(f\). a) Find the average rate of change of \(f\) over \([-2, -1]\) and over \([-1, 1]\). b) Give one interval over which the average rate of change is \(0\).

A large water tank is being filled. The graph shows the water volume \(V\), in cubic meters, as a function of time \(t\), in hours. Find the average inflow rate over each interval. Include units. a) \(I_1 = [0, 4]\) b) \(I_2 = [4, 8]\)

The two panels show the graphs of \(f\) and \(g\). Find the average rate of change of each function over each interval: a) \(I_1 = [2, 4]\) b) \(I_2 = [3, 5]\) c) \(I_3 = [4, 6]\) d) \(I_4 = [2, 6]\)

The graph of \(f\) and its tangent line \(t\) at \(x = 1\) are shown. Use the tangent line to find \(f'(1)\).

The graph of \(f\) has four marked points, \(A\), \(B\), \(C\), and \(D\). a) At which marked points is the slope of the graph negative? b) Order the points from the smallest slope to the largest slope.

The graph of \(f\) includes tangent lines at \(x = 0\), \(x = 2\), and \(x = 4\). a) Use the tangent lines to find \(f'(0)\), \(f'(2)\), and \(f'(4)\). b) Determine whether the derivative is positive, negative, or zero at \(x = -1\), \(x = 1\), and \(x = 5\).

The graph shows the water level \(h(t)\), in meters, in a hydroelectric reservoir, where \(t\) is the number of hours after midnight. a) Find the average rate of change of the water level over each interval. Include units. 1. \([0, 2]\) 2. \([2, 6]\) 3. \([10, 12]\) b) Find the average rate of change over the entire 12-hour period, \([0, 12]\), and interpret the result. c) At what time is the water level highest, and what is the maximum level?

Four function graphs, labeled \(A\) through \(D\), are shown. Match each graph with the correct derivative value at \(x = 1\). (1) \(f'(1) = 0\) (2) \(f'(1) = 2\) (3) \(f'(1) = -2\) (4) \(f'(1) = 1\)

The graph of \(f\) has marked points at \(x_1 = -2\), \(x_2 = 0\), and \(x_3 = 2\). Use the shown tangent lines to find the derivative at each x-value.

The graph of a function \(f\) is shown. Find the average rate of change of \(f\) on the intervals \([-5, 0]\) and \([0, 5]\). What do you notice when you compare the two results?

The figure shows the graph of a quadratic function \(f\). a) Find the average rate of change on \([-4, 4]\). b) Give another interval \([a, b]\), where \(a < b\), on which the average rate of change is also \(0\). Explain your choice.

The graph shows the water level in a rainwater collection basin over \(10\) hours. Find the average rate of change of the water level, in meters per hour, from \(t = 2\,\text{h}\) to \(t = 6\,\text{h}\). Interpret the sign of your result in context.

Use the graph of \(f\) to find the average rate of change on each interval. Read the needed points directly from the coordinate plane. a) \([0, 2]\) b) \([-2, 0]\)

Use the graph of \(g\) to find the average rate of change on each interval. a) \([0, 2]\) b) \([1, 3]\)

The function \(h\) is defined by \(h(x) = 3x^2 - 6x + 2\). Find the average rate of change of \(h\) on each interval. Use the graph to illustrate your results. a) \([1, 3]\) b) \([-1, 1]\)

The function \(k\) is defined by \(k(x) = -x^2 + 4x\). Find the average rate of change of \(k\) on each interval. What do you notice when you compare the results? a) \([0, 2]\) b) \([2, 4]\)

The graph of \(f\) includes tangent lines at \(x = -1\) and \(x = 2\). a) Use the tangent lines to find \(f'(-1)\) and \(f'(2)\). b) State whether \(f'(0)\), \(f'(3)\), and \(f'(-2)\) are positive, negative, or zero. c) Estimate \(f'(1)\) from the graph.

The graph of \(f\) and two tangent lines are shown. a) Use the drawn tangent lines to find \(f'(1)\) and \(f'(-2)\). b) Estimate the tangent-line slopes at \(A(2, 0)\) and \(B(0, -2)\).

A tank is being filled with water. The graph shows the water height \(h\), in inches, as a function of time \(t\), in minutes. Find the average rate of change on each interval. a) \([0, 5]\) b) \([5, 15]\) c) \([0, 15]\)

Use the drawn tangent lines to find the derivative of \(f\) at \(x_1 = 0\), \(x_2 = 2\), and \(x_3 = 4\).

The graph of \(g\) includes tangent lines at points \(A\), \(B\), and \(C\). Find \(g'(-2)\), \(g'(0)\), and \(g'(2)\).

Use the graph of \(h\) to find the average rate of change on each interval. a) \([0, 2]\) b) \([2, 4]\) c) \([0, 4]\)

The figure shows the graph of a periodic function \(k\). Find the average rate of change of \(k\) on each interval. a) \([0, 2]\) b) \([0, 4]\) c) \([2, 6]\)

Use the tangent line in the graph to find the instantaneous rate of change of \(g\) at point \(P\).

Find the slope of the graph of \(h\) at \(P(4, 2)\) using the tangent line.

Use the given axis scale and tangent line to find the slope of the function at point \(P\).

The graph of \(g\) has marked points \(A\), \(B\), and \(C\). At which point is the derivative value smallest? Justify your answer by comparing tangent slopes.

A hot-air balloon rises and later descends. The function \(h\) gives its height above the ground, in meters, as a function of time \(t\), in minutes. Find the balloon’s average rate of change in height on each interval. a) \([0, 1]\) b) \([1, 4]\)

The graph of \(V\) shows the volume of water in a storage tank, in liters, as a function of time \(t\), in minutes. Find the average rate of change of the volume on \([2, 8]\). Include the correct units and interpret the result in context.

The graph of a function \(f\) is shown. a) Find the average rate of change of \(f\) on \([0, 1]\) and on \([-2, 0]\). b) Give an interval on which the average rate of change is \(0\).

The graph of a function \(g\) is shown. a) Find the average rate of change on \([-1, 1]\) and on \([1, 2]\). b) Give an interval on which the average rate of change is \(0\).

Consider the function \(g(x) = \frac{6}{x}\) for \(x > 0\). Find the average rate of change on \([1, 3]\) and on \([2, 5]\).

A cup of hot tea cools over time. The graph shows the temperature \(T\), in degrees Fahrenheit, as a function of time \(t\), in minutes. Find the average rate of change of the temperature on each interval. a) \([0, 20]\) b) \([20, 40]\)

A sunflower’s growth was tracked for \(20\) weeks. The graph shows its height \(h\), in inches, as a function of time \(t\), in weeks. Find the average growth rate during each period. a) The first \(10\) weeks, \([0, 10]\) b) The next \(10\) weeks, \([10, 20]\)

A tangent line is drawn to the graph of \(f\) at \(x = 0\). Find \(f'(0)\).

Use the graph to determine the instantaneous rate of change of \(f\) at \(x_0=2\).

The graph shows a square-root function \(f\). Find \(f^{\prime}(x_0)\) at the marked point.

Use the graph and its tangent line to find \(f'(-2)\).

The function is \(f(x)=x^2-4x+5\). Use the drawn tangent to determine \(f^{\prime}(1)\) graphically.

A small ball rolls down a model track. The track’s vertical profile is approximated by \(h(x) = -0.2x^2 + 5\), where \(x\) is the horizontal distance from the start and \(h(x)\) is the height above the floor, both in feet. Find the average slope of the track from \(x = 1\) to \(x = 4\). Interpret the value and its sign in context.

A remote-controlled car starts from rest, reaches a maximum speed, and then slows to a stop. The graph shows velocity \(v\) as a function of time \(t\). a) Use the drawn tangent lines to find the instantaneous rate of change of velocity at \(t = 0\) and \(t = 20\) seconds. b) When is the instantaneous rate zero? What is true about the velocity at that time? c) What is the physics term for the instantaneous rate of change of velocity?

The graph of a function \(g\) is shown. Find the average rate of change of \(g\) on each interval. a) \([1, 3]\) b) \([-2, 0]\)

The distance-time graph shows an eight-hour hike. The variable \(t\) is time in hours, and \(s\) is distance traveled in miles. 1. Find the hiker’s average speed for the entire hike. 2. During which interval, \([0, 4]\) or \([4, 8]\), was the hiker faster on average? Justify your answer using the graph. 3. Explain what the slope of the secant line through \(P(0, 0)\) and \(Q(8, 14)\) means in context.

The graph shows a function \(f\) and tangent line \(t\) at point \(P(2, 4)\). Find \(f'(2)\) by reading the slope of the tangent line from the graph.

A water tank is being drained. The volume of water remaining, in liters, is modeled by \(V(t) = 0.5t^2 - 20t + 200\) for \(0 \le t \le 20\), where \(t\) is the number of minutes since the drain was opened. a) Find the average rate of change of the volume during the first 10 minutes, \([0, 10]\), and during the second half of the draining process, \([10, 20]\). b) Compare the two values and explain what they mean for the rate at which water leaves the tank.

Let \(f(x) = \frac{6}{x - 2}\). State the domain of \(f\). Then find the average rate of change of \(f\) over \(I_1 = [-4, -1]\), \(I_2 = [-1, 1]\), and \(I_3 = [-4, 1]\).

Let \(f(x) = \sqrt{x + 1}\). State the domain of \(f\). Then find the average rate of change of \(f\) over \(I_1 = [0, 3]\), \(I_2 = [3, 8]\), and \(I_3 = [0, 8]\).

An initial deposit of \(\$1000\) is placed in a savings account that earns \(4\%\) interest compounded annually. a) Write a function \(K(t)\) that gives the account balance, in dollars, after \(t\) years. b) Find the average rate of change of the balance over each interval: (1) \(I_1 = [0, 10]\) (2) \(I_2 = [20, 30]\) c) Interpret the values in context. Explain why they differ even though the interest rate stays the same.

After a medication is taken, the concentration of its active ingredient in the blood decreases. The concentration, in milligrams per liter, is modeled by \(C(t) = 200 \cdot 0.8^t\), where \(t\) is the number of hours since the medication was taken. a) Find the average rate of change of the concentration over \(I_1 = [0, 2]\) and \(I_2 = [4, 6]\). b) Interpret the sign of each result in context. c) Compare the magnitudes of the two rates and explain the difference in terms of the decay process.

A weather balloon rises through the atmosphere. From \(t_1 = 12\,\text{min}\) to \(t_2 = 28\,\text{min}\) after launch, its average upward velocity is \(3.5\,\frac{\text{m}}{\text{s}}\). At time \(t_2\), the balloon is \(7450\,\text{m}\) above the ground. a) Find the height of the balloon at time \(t_1\). b) Interpret the average upward velocity geometrically on a height-versus-time graph. Explain why the height at \(t = 20\,\text{min}\) cannot be determined exactly without knowing a function for the balloon’s height.

In a chemistry experiment, a liquid cools over time. The temperature is described by a function \(T\), where \(t\) is the number of minutes since the experiment began. From \(t = 5\,\text{min}\) to \(t = 15\,\text{min}\), the average rate of change of the temperature is \(-1.8\,\frac{\text{K}}{\text{min}}\). After \(15\) minutes, the temperature is \(42.5\,^\circ\text{C}\). a) Find the temperature at \(t = 5\,\text{min}\). b) Write the difference quotient for \([5, 15]\) and explain how its meaning differs from the instantaneous rate of change \(T'(10)\).

Let \(f(x) = 2x^2 - x\). Find the average rate of change of \(f\) over \([2, 2 + h]\) in terms of \(h\), where \(h > 0\). Simplify your expression completely.

Let \(g(x) = \frac{3}{x}\), where \(x \ne 0\). Find the slope of the secant line through \(P(1, g(1))\) and \(Q(1 + a, g(1 + a))\), where \(a > 0\). Write your answer as a simplified expression in terms of \(a\).

The height of a ball thrown straight upward is modeled by \(h(t) = -5t^2 + 20t + 2\), where \(t\) is time in seconds and \(h(t)\) is height in meters. 1. Find the average rate of change of the height over \([0.5, 1.5]\). 2. Identify the physical quantity represented by this value and state its units. 3. Find a time \(b > 0\) such that the average rate of change of the height over \([0, b]\) is exactly \(10\,\frac{\text{m}}{\text{s}}\).

During its first 10 weeks of growth, the height of a wheat plant is modeled by \(h(t) = -0.1t^2 + 2t + 5\), where \(t\) is measured in weeks and \(h(t)\) in centimeters. 1. Find the plant’s average growth rate during the first 4 weeks and from week 4 to week 8. 2. Explain the geometric meaning of the average growth rate over \([0, 4]\) on the graph of \(h\). 3. Use the calculated rates to determine whether the plant’s growth is speeding up or slowing down in this model.

A delivery van travels from a distribution center to a customer. The table shows the van’s distance from the distribution center over time. <table> <tr> <td>Time \(t\) (in \(\text{min}\))</td> <td>\(0\)</td> <td>\(10\)</td> <td>\(25\)</td> <td>\(30\)</td> <td>\(50\)</td> <td>\(65\)</td> <td>\(80\)</td> </tr> <tr> <td>Distance \(s\) (in \(\text{mi}\))</td> <td>\(0\)</td> <td>\(9\)</td> <td>\(9\)</td> <td>\(15\)</td> <td>\(33\)</td> <td>\(33\)</td> <td>\(48\)</td> </tr> </table> a) Find the van’s average speed for the entire trip in miles per minute. b) Find the average speed during the first \(30\) minutes and during the remaining time after minute \(30\). c) Compare the average speed during the first \(10\) minutes with the average speed during the final \(15\) minutes.

During a \(20\)-second motor test, a test object moves in a fixed direction, and its velocity \(v\) is increased in stages. The table gives its velocity at selected times \(t\). <table> <tr> <td>Time \(t\) (in \(\text{s}\))</td> <td>\(0\)</td> <td>\(4\)</td> <td>\(10\)</td> <td>\(11\)</td> <td>\(15\)</td> <td>\(18\)</td> <td>\(20\)</td> </tr> <tr> <td>Velocity \(v\) (in \(\frac{\text{m}}{\text{s}}\))</td> <td>\(0\)</td> <td>\(12\)</td> <td>\(12\)</td> <td>\(18\)</td> <td>\(18\)</td> <td>\(25\)</td> <td>\(30\)</td> </tr> </table> a) Find the average acceleration over the entire test. b) Find the average acceleration during the first half, from \(0\) to \(10\) seconds, and during the second half, from \(10\) to \(20\) seconds. c) During which interval, \([0, 4]\) or \([18, 20]\), is the average acceleration greater? Support your answer with calculations.

Let \(f(x) = 0.5x^2 + 2\). The fixed point \(P(2, 4)\) lies on the graph of \(f\), and a second point \(Q(x, f(x))\) moves along the graph. a) Write the difference quotient for the slope of the secant line through \(P\) and \(Q\). b) Simplify the expression from part a as much as possible. c) Find the secant slope when \(x = 2.2\).

Let \(g(x) = x^2 - 5x\). a) Write the average rate of change of \(g\) over \([x_0, x_0 + h]\) in terms of \(x_0\) and \(h\), where \(h > 0\). b) Simplify the expression completely. c) Evaluate the average rate of change when \(x_0 = 3\) and \(h = 0.1\).

Let \(f(x) = 2x^2 - 3\). The point \(P(-1, -1)\) lies on the graph of \(f\). Find and simplify the slope of the secant line through \(P\) and any other point \(Q(x, f(x))\) on the graph.

Let \(g(x) = x^3 + 2\). Find a simplified expression for the slope of the secant line through the fixed point \(A(2, 10)\) and the variable point \(B(x, g(x))\) on the graph of \(g\).

A chemist records the temperature \(T\) of a hot liquid at several times \(t\). <table> <tr><td>Time \(t\) (in \(\text{min}\))</td><td>\(0\)</td><td>\(5\)</td><td>\(10\)</td><td>\(20\)</td><td>\(30\)</td><td>\(50\)</td></tr> <tr><td>Temperature \(T\) (in \(^\circ\text{C}\))</td><td>\(85.0\)</td><td>\(72.5\)</td><td>\(62.0\)</td><td>\(46.0\)</td><td>\(35.0\)</td><td>\(23.0\)</td></tr> </table> a) Find the average rate of change of the temperature during the first 10 minutes. b) Find the average rate of change from \(t = 20\) to \(t = 50\). c) During which interval, \([0, 5]\), \([5, 10]\), \([10, 20]\), \([20, 30]\), or \([30, 50]\), does the liquid cool fastest on average? Support your answer by calculating each rate.

The table shows the size of a wildlife population in a protected area over several years. <table> <tr><td>Year</td><td>\(2015\)</td><td>\(2017\)</td><td>\(2018\)</td><td>\(2020\)</td><td>\(2023\)</td></tr> <tr><td>Number of animals</td><td>\(1200\)</td><td>\(1550\)</td><td>\(1820\)</td><td>\(1740\)</td><td>\(2100\)</td></tr> </table> a) Find the average annual rate of change of the population from 2015 to 2023. b) Find the average annual rates of change from 2017 to 2018 and from 2018 to 2020. c) Interpret the sign of the rate from 2018 to 2020 in context. d) During which interval between consecutive measurements was the average annual growth greatest?

Find the slope of the tangent line to \(g(x) = |x|+2x\) at \(x = a\) in each case. a) \(a > 0\) b) \(a < 0\)

The total cost of removing pollutants at a wastewater treatment plant depends on the amount removed. The function \(C(x)\) gives the total cost in dollars of removing \(x\) kilograms of pollutants. Interpret each expression in context and give its units. a) \(C(x)-C(x_0)\) b) \(\frac{C(x)-C(x_0)}{x-x_0}\) c) \(C'(x_0)\)

Suppose \(h'(x)=-0.5\) for every real number \(x\). 1. What type of function is \(h\)? Describe the graph of \(h\). 2. Find an equation for \(h\) if \(h(4)=2\). 3. Another function is defined by \(g(x)=h(x)+10\). Compare \(g'\) and \(h'\), and explain the result geometrically.

A sled moves down a snowy hill. Its distance traveled, in meters, is modeled by \(s(t) = \frac{1}{2}at^2\), where \(t\) is time in seconds and \(a\) is the constant acceleration down the hill. a) Derive a simplified expression for the average velocity \(v_{\text{avg}}\) over \([t_1, t_2]\), where \(t_1 < t_2\). b) Find the average velocity when \(a = 2.4\,\frac{\text{m}}{\text{s}^2}\), \(t_1 = 5\,\text{s}\), and \(t_2 = 15\,\text{s}\).

Let \(f(x) = 0.2x^2 + 3\). a) Find the slope of the secant line through \(P(2, f(2))\) and \(Q(7, f(7))\). b) Find \(x_2 > 0\) such that the average rate of change of \(f\) over \([0, x_2]\) is exactly \(1.6\).

Under favorable conditions, an algae patch on a lake grows exponentially. At the beginning of the observation, it covers \(2.0\,\text{m}^2\). The covered area increases by \(25\%\) each week. a) Write a function in the form \(A(t) = ab^t\) that gives the area \(A\), in square meters, after \(t\) weeks. b) Find the area after \(4\) weeks and after \(8\) weeks. c) Find the average rate of change of the area during the first \(4\) weeks and from week \(4\) to week \(8\).

A new electric bicycle costs \(\$5000\). Because of age and use, its market value decreases by \(15\%\) each year. The value is modeled by an exponential function \(V(t) = ab^t\), where \(t\) is the number of years since purchase and \(V\) is measured in dollars. a) Write the function \(V(t)\). b) Find the average rate of change of the value during the first two years. c) Find the average rate of change from year \(5\) to year \(7\). Compare it with the result from part b and explain the difference.

A newly installed solar energy system has a value of \(\$15{,}000\). Because of aging and technological advances, its value decreases by \(4\%\) each year. a) Write a function \(W(t) = W_0 \cdot a^t\) that gives the value \(W\), in dollars, after \(t\) years. b) Find the average annual rate of change of the value during the first five years and from year \(10\) to year \(15\). c) Compare the two results and explain why they differ even though the annual percent loss is constant.

The two panels show the graphs of \(f\) and \(g\). Difference quotient I: \(\frac{2.5 - (-1.5)}{3 - 1}\) Difference quotient II: \(\frac{0 - 1}{2 - 0}\) 1. Match each difference quotient to the appropriate graph. Justify each match by identifying the corresponding points on the graph. 2. Calculate the average rate of change for each function over its corresponding interval.

The graph of \(f\) includes tangent line \(t_1\) at \(x = -2\) and tangent line \(t_2\) at \(x = 0\). a) Use the tangent lines to find \(f'(-2)\) and \(f'(0)\). b) Estimate \(f'(2)\) and \(f'(1)\) from the shape of the graph.

Use the graph of \(f\) to complete both parts. **Part 1: Average rate of change** Find the average rate of change of \(f\) over each interval: a) \([-1, 3]\) b) \([0, 4]\) c) \([3, 5]\) **Part 2: Statements about the derivative** Determine whether each statement about \(f'\) is true or false. Briefly justify your answer. (1) \(f'(2) < 0\) (2) \(f'(1) = 0\) (3) On \([1, 3]\), the graph of \(f'\) lies above the x-axis.

Panels 1, 2, and 3 show the graphs of three functions \(f\), \(g\), and \(h\). Each graph matches exactly one difference-quotient card: * **Card A:** \(\frac{1 - 1}{4 - 0} = 0\) * **Card B:** \(\frac{1 - (-1)}{3 - (-1)} = 0.5\) * **Card C:** \(\frac{-1 - 0}{2 - 0} = -0.5\) Match each graph to its card. Justify each match by stating the two graph points used in the difference quotient.

The graph shows the elevation profile of a mountain-bike trail. Horizontal distance from the trailhead is shown on the x-axis, and elevation above sea level is shown on the y-axis. The trail starts at \(1300\,\text{m}\) and ends at \(820\,\text{m}\). a) Find the average grade, as a percent, over the entire trail. b) Find the average grade over the first half of the horizontal distance, from \(0\) to \(800\) meters, and over the second half, from \(800\) to \(1600\) meters. Compare the results. c) A rider claims, “Because the average grade over the first half is \(-45\%\), no part of that section can be steeper than \(-50\%\).” Explain why this claim is not generally valid.

For the first \(10\) days after sprouting, a seedling's height is modeled by \(H(t) = -0.015625t^2+0.375t+0.5\), where \(t\) is measured in days and \(H(t)\) is measured in inches. a) Use the tangent line shown at day \(4\) to estimate the instantaneous growth rate. Explain how you found the estimate. b) Find the average growth rate during the first \(4\) days, and compare it with the instantaneous growth rate on day \(4\). c) Explain how the graph shows that the growth rate decreases throughout the \(10\)-day period.

The time-height graph shows a 60-minute hot-air-balloon flight. a) Describe the flight during \([0, 20]\), \([20, 40]\), and \([40, 60]\), with time measured in minutes. b) Find and interpret the average rate of change of the height, in meters per minute, over each interval. c) An observer claims, “Because the average upward velocity over \([0, 20]\) is \(15\,\frac{\text{m}}{\text{min}}\), the balloon must have been at \(250\,\text{m}\) after exactly \(10\) minutes.” Evaluate both the numerical conclusion and the reasoning.

The graph shows a function \(f\) and its tangent line \(t\) at \(x = 2\). a) Use the graph to find \(f(0)\) and \(f(6)\). b) Find the average rate of change of \(f\) over \([0, 6]\). Explain its geometric meaning. c) Find the instantaneous rate of change \(f'(2)\). Explain its geometric meaning.

The graph shows the water level \(h\), in feet, in a harbor during a \(12\)-hour tide cycle. Time \(t\) is measured in hours after midnight, and points \(A\), \(B\), \(C\), and \(D\) mark selected times. a) Interpret the instantaneous rate of change of the water level and give its units. b) Estimate the instantaneous rates of change at \(A(3, 10)\) and \(B(6, 16)\). Tangent lines are drawn at \(A\) and \(D\). c) A student claims, “At \(t = 10\), point \(D\), the water level is falling especially fast at about \(4\) feet per hour.” Evaluate the claim by estimating the instantaneous rate at \(D(10, 7)\) from the tangent line.

The graph shows the water level \(h(t)\), in meters, in a large tank over time \(t\), in hours. The dashed line is tangent to the graph at the inflection point \(W(2, 6)\). a) Find the average rate of change of the water level on \([0, 4]\) and on \([4, 6]\). b) Find the instantaneous rate of change at \(t = 4\) from the graph, and interpret it in context. c) Use the drawn tangent line to find the maximum instantaneous rate of change during the process. When does it occur?

The graph of \(f\) has marked points \(P_1\), \(P_2\), and \(P_3\) at \(x_1 = -1\), \(x_2 = 0\), and \(x_3 = 2\). Match each x-value with its instantaneous rate of change. Value A: \(-1\) Value B: \(0.5\) Value C: \(3\) Give your answer in the form “\(x_1\): Value ..., \(x_2\): Value ..., \(x_3\): Value ...”.

The graph shows the elevation profile of a 12K trail run. Horizontal distance is measured in kilometers, and elevation is measured in meters. a) Find the average grade, as a percent, for each section: 1. From the start to point \(B\) at kilometer \(4\) 2. From point \(B\) to point \(C\) at kilometer \(8\) 3. From point \(C\) to the finish at kilometer \(12\) b) Find the average grade over the entire course. Briefly assess whether this value accurately represents the effort required by the run.

The water level in a small mountain reservoir is modeled by \(h(t) = -0.1t^3 + 0.9t^2 - 1.8t + 3\) for \(0 \le t \le 8\), where \(t\) is measured in hours and \(h(t)\) in meters. The graph of \(h\) is shown. a) Find the average rate of change of the water level over each interval: (1) \([1, 5]\) (2) \([0, 6]\) (3) \([5, 8]\) b) Use the graph to decide whether each statement about the instantaneous rate of change is true or false. Briefly justify each answer. (1) At \(t = 3\), the water level is rising. (2) The water level is falling throughout \([0, 3]\). (3) The instantaneous rate of change is positive throughout \([2, 4]\).

The distance-time graph shows a cyclist’s first \(10\) minutes of a training ride. Time \(t\) is measured in minutes, and distance \(s\) is measured in miles. a) Find the cyclist’s average speed for the entire ride, from \(t = 0\) to \(t = 10\), in miles per hour. b) Find the average speed from \(t = 2\) to \(t = 8\). Use the graph values \(s(2) \approx 0.38\,\text{mi}\) and \(s(8) \approx 2.12\,\text{mi}\). c) The cyclist’s instantaneous speed is greatest at the inflection point \(t = 5\). Use the tangent slope there to determine whether the cyclist ever travels faster than \(20\,\text{mph}\).

Match each graph to the card showing its difference quotient. Verify each match by reading two graph points and calculating the average rate of change. **Cards:** * Card A: \(\frac{0.8 - (-0.2)}{3 - (-2)} = 0.2\) * Card B: \(\frac{0.5 - 2}{4 - 1} = -0.5\) * Card C: \(\frac{2 - 0}{2 - 0} = 1\)

The figure shows four graphs labeled 1 through 4. Match each graph to the card that describes its average rate of change over the indicated interval. Justify each match by reading the endpoint coordinates. Card A: \(\frac{2-(-1)}{2-0}=1.5\) Card B: \(\frac{1-3}{2-0}=-1\) Card C: \(\frac{0-0}{3-1}=0\) Card D: \(\frac{2-0}{3-(-1)}=0.5\)

The graph of \(f\) includes tangent lines at \(P(2, 2)\) and \(Q(-2, -2)\). a) Use the tangent lines to find \(f'(2)\) and \(f'(-2)\). b) Estimate the slope of the graph at \(C(1, 4)\) and \(D(4, 1)\).

The graph shows a function \(f\) and two transformed functions \(g\) and \(h\). Function \(g\) is a vertical translation of \(f\), and \(h\) is a horizontal translation of \(f\). 1. Use the dashed tangent to find the slope of \(f\) at \(x=3\). 2. Explain why \(g\) has the same slope as \(f\) at \(x=3\). 3. At what x-value does \(h\) have the same slope that \(f\) has at \(x=3\)? Explain.

The graphs of \(f\) and \(g\) are shown. Find the average rate of change of each function on each interval. a) \([-2, 0]\) b) \([0, 4]\)

The graph models the motion of a remote-controlled car. It shows distance traveled \(s\), in meters, as a function of time \(t\), in seconds. a) Find the average speed on \([0, 10]\). b) Find the average speed on \([10, 20]\). c) Explain the difference between the two values using the shape of the graph.

A water tank is filled over \(10\) minutes. The graph shows the volume of water in liters. a) Find the average inflow rate over the full \(10\)-minute period, in liters per minute. b) During which shown interval—\([0, 2]\), \([2, 5]\), \([5, 8]\), or \([8, 10]\)—was the average inflow rate greatest? Calculate that rate. c) What would have to happen for the graph to have a negative slope over an interval?

Match each difference-quotient card to Graph 1, 2, or 3. Justify each match by identifying the corresponding points on the graph. **Card A:** \(\frac{5-0}{5-0} = 1\) **Card B:** \(\frac{2-2}{2-(-2)} = 0\) **Card C:** \(\frac{1-4}{4-1} = -1\)

Two function graphs and two difference-quotient cards are shown. Match each card to a graph, and explain how the card’s average rate of change appears on that graph. **Card A:** \(\frac{6-1.5}{4-2} = 2.25\) **Card B:** \(\frac{1-2}{3-0} = -\frac{1}{3}\)

The graph shows the water depth \(h\) in a rain barrel during a \(12\)-hour period that includes a heavy rainstorm. a) Describe how the water depth changes over time. b) Find the average rate of change on \([0, 4]\), \([4, 8]\), and \([8, 12]\). Include appropriate units. c) Interpret each value and its sign in context. What might have happened during each time period?

A runner records a training route with a fitness app. The distance-time graph shows distance \(s\), in miles, as a function of time \(t\), in minutes. a) Find the runner’s average speed on \([0, 20]\), \([20, 30]\), and \([30, 40]\), in miles per hour. b) During which interval was the runner fastest on average? Explain how you can tell directly from the graph without calculating. c) A friend says, “You ran exactly \(4\) miles in \(40\) minutes, so you will run \(8\) miles in exactly \(80\) minutes.” Evaluate this claim using the graph.

The graph shows a periodic function \(h\) with marked points \(P\), \(Q\), \(R\), and \(S\). a) At which point is the instantaneous rate of change smallest, meaning the graph is decreasing most steeply? b) Order \(P\), \(Q\), \(R\), and \(S\) from greatest slope to smallest slope. c) What is the slope at \(R\)?

The temperature of a cup of coffee is modeled by a function \(T\), where \(t\) is the number of minutes after the coffee is poured and \(T(t)\) is measured in degrees Fahrenheit. a) Use the drawn tangent line to estimate the instantaneous cooling rate after \(10\) minutes, and explain your method. b) Explain how the graph shows that the coffee cools more slowly as time passes.

A water tank is being filled. The function \(V(t)\) gives the water volume in gallons after \(t\) minutes. a) Use the drawn tangent line to estimate the instantaneous inflow rate at \(t = 4\) minutes. b) Use the shape of the graph to decide whether the inflow rate increases or decreases over time.

The graph shows the temperature \(T\), in degrees Fahrenheit, during a sunny morning. The variable \(t\) is the number of hours since measurements began at 6:00 a.m. 1. Use the graph to find \(T(0)\) and \(T(6)\). 2. Calculate \(\frac{T(6)-T(0)}{6-0}\). 3. Explain the meaning of this value in context and include the correct units. 4. Use the tangent line shown at \(t = 2\) to estimate the instantaneous rate of change of the temperature.

The graph of \(g\) is shown. 1. Find the average rate of change of \(g\) on \([0, 5]\). 2. Find the instantaneous rate of change at \(x = 3\). 3. State the geometric meaning of the values from parts 1 and 2. 4. At what x-value in \([0, 5]\) is the instantaneous rate of change greatest? Briefly justify your answer.

The graph shows the height \(h\), in meters, of a weather balloon as a function of time \(t\), in minutes. Estimate the instantaneous rate of change at each marked time by comparing the curve's local steepness. Match each time with one possible rate. Times: \(t_1=0\) \(t_2=2\) \(t_3=4\) \(t_4=5\) Possible rates: A. \(-1.8\,\frac{\text{m}}{\text{min}}\) B. \(-1.0\,\frac{\text{m}}{\text{min}}\) C. \(-0.2\,\frac{\text{m}}{\text{min}}\) D. \(0.5\,\frac{\text{m}}{\text{min}}\) E. \(1.5\,\frac{\text{m}}{\text{min}}\)

The graph of \(f(x) = 0.1x^3+1\) is shown. We are investigating the instantaneous rate of change at \(P(1, 1.1)\). a) Find the slopes of the secant lines through \(P\) and \(Q(a,f(a))\) for the values of \(a\) in the table. Complete the last row. <table> <tr><th>\(a\)</th><td>\(2\)</td><td>\(1.5\)</td><td>\(1.1\)</td></tr> <tr><th>\(f(a)\)</th><td>\(1.8\)</td><td>\(1.3375\)</td><td>\(1.1331\)</td></tr> <tr><th>\(m_s\)</th><td>...</td><td>...</td><td>...</td></tr> </table> b) Predict the tangent-line slope at \(P\). Justify your prediction using the secant slopes.

The graph shows the elevation profile of a bike ride. Elevation \(h\), in feet, is plotted against horizontal distance \(s\), in miles. A tangent line is drawn at point \(P\). a) Interpret the local rate of change at \(P\) in context. b) Use the tangent line to find the grade at \(P\), first in feet per mile and then as a percentage. c) Estimate the local rate of change at the start, \(s = 0\), and at the highest point of the ride. Justify your estimates.

A water tank is being filled. The graph shows the volume \(V\), in gallons, as a function of time \(t\), in minutes. a) Find the average inflow rate during the first \(20\) minutes. b) Use the drawn tangent line to estimate the instantaneous inflow rate at \(t = 10\) minutes. c) Compare the inflow rates at \(t = 5\) and \(t = 25\). At which time is water entering the tank faster? Justify your answer from the graph.

The graph shows the temperature \(T(t)\), in degrees Celsius, of a chemical during a \(10\)-minute experiment. a) Find the maximum temperature and when it occurs. b) Use the drawn tangent lines to estimate the instantaneous temperature rates at \(t = 4\) and \(t = 6\). c) When is the substance cooling most rapidly? d) What does the instantaneous rate at \(t = 2\) mean for the temperature?

The graph shows the water depth \(h\) in a stormwater retention basin during a heavy rainstorm. a) Use the drawn tangent line to find the instantaneous rate of change of the depth at \(t = 5\) hours. b) When is the instantaneous rate greatest? How is this point characterized on the graph? c) What does an instantaneous rate of \(0\,\frac{\text{m}}{\text{h}}\) mean in context?

The graph shows the elevation profile of a hiking trail. Horizontal distance is measured in miles, and elevation is measured in feet. a) Find the average grade, as a percent, for each section: - mile \(0\) to mile \(2\) - mile \(2\) to mile \(3.5\) - mile \(3.5\) to mile \(5.5\) b) A trail guide warns, “Although some average grades are moderate, the section from mile \(3.5\) to mile \(5.5\) contains short, steep stretches.” Explain this statement using the graph.

The graph shows the temperature during a sunny fall day from midnight to midnight. a) Find the average rate of change of the temperature, in degrees Fahrenheit per hour, from 4:00 a.m. to noon and from noon to 8:00 p.m. b) During what time period between midnight and midnight does the temperature appear to change fastest? Justify your answer using the graph.

The graph shows the distance \(s\), in feet, traveled by a vehicle during the first \(10\) seconds. 1. Use the drawn tangent line to estimate the instantaneous speed at \(t = 5\) seconds. 2. Convert the result to miles per hour. 3. Describe how the speed changes during the first \(10\) seconds. Justify your answer from the graph's curvature.

Let \(f(x) = x^2 - 4x\). a) Find the average rate of change of \(f\) over \(I_1 = [1, 5]\) and \(I_2 = [0, 6]\). b) Show that, for any \(a < b\) and \(k > 0\), the average rate of change of \(f\) over \([a, b]\) equals the average rate of change over \([a - k, b + k]\). c) Interpret this result geometrically in terms of secant lines to the parabola.

After a medication is taken, the amount of its active ingredient in a patient’s blood decreases continuously. The amount decreases by \(15\%\) every \(4\) hours. At \(t = 0\), the blood contains \(200\,\text{mg}\) of the active ingredient. a) Find a function in the form \(f(t) = ca^t\) that gives the amount \(f(t)\), in milligrams, after \(t\) hours. Determine \(c\) and \(a\). b) Interpret \(f(t) - f(t_0)\) and \(\frac{f(t) - f(t_0)}{t - t_0}\) in context. c) Find the average rate of change of the amount over \([0, 4]\) and \([12, 16]\).