- Check that the theorem's two hypotheses are already supplied. - What slope is determined by the two endpoint values? - The theorem matches an instantaneous slope to that endpoint slope.

1. The hypotheses of the Mean Value Theorem are given: continuity on the closed interval and differentiability on the open interval. 2. The secant slope is \(\frac{f(5)-f(1)}{5-1}=\frac{11-3}{4}=2\). 3. Therefore, there is at least one \(c\in(1,5)\) such that \(f'(c)=2\).

\(f'(c)=2\) for at least one \(c\in(1,5)\).

- Which interval must include both endpoints in the theorem's conditions? - Can a function be continuous at a point outside its domain? - Check the endpoint separately from the interior.

1. The Mean Value Theorem requires continuity on the entire closed interval \([0,3]\). 2. Because the function is not defined at \(0\), it cannot be continuous on \([0,3]\). 3. The theorem's conditions are not met.

No. The function is not continuous on the whole closed interval \([0,3]\) because it is undefined at \(0\).

- Compare the two endpoint heights before thinking about the tangent. - What would the secant line look like when its endpoints have equal heights? - Which special case of the Mean Value Theorem uses equal endpoint values?

1. The endpoint heights agree, so the secant slope is \(\frac{p(3)-p(-1)}{3-(-1)}=0\). 2. Rolle's theorem applies because the function is continuous on the closed interval, differentiable inside, and has equal endpoint values. 3. Therefore at least one interior tangent has slope \(0\).

At least one interior tangent has slope \(0\), by Rolle's theorem.

- Focus on the words “at least one” in the theorem's conclusion. - Does an existence statement rule out additional points? - Compare the student's claim with the two derivative values given.

1. The Mean Value Theorem guarantees that at least one interior tangent has slope \(3\). 2. It does not say that there is exactly one such tangent. 3. The two stated derivative values are therefore consistent with the theorem, so Sienna is incorrect.

No. The theorem guarantees at least one matching tangent, not exactly one.

- Check continuity on the full closed interval first. - Then check differentiability only at interior points. - A corner and a point where a function is undefined cause different theorem hypotheses to fail.

1. \(f(x)=\sqrt{x}\) is continuous on \([1,4]\) and differentiable on \((1,4)\), so the theorem applies. 2. \(g(x)=|x|\) is continuous on \([-1,1]\) but is not differentiable at \(x=0\), so the theorem is not guaranteed to apply. 3. \(h(x)=\frac{1}{x-2}\) is not continuous on \([1,3]\) because it is undefined at \(x=2\), so the theorem is not guaranteed to apply.

a) Yes. b) No; differentiability fails at \(x=0\). c) No; continuity fails at \(x=2\).

- Start with the rate of change determined by the two endpoint values. - Ask what the theorem says about that rate somewhere inside the interval. - Separate the information needed for the theorem from information needed to solve an equation explicitly.

1. The average rate of change is \(\frac{17-5}{8-2}=2\). 2. Differentiability on \((2,8)\) together with continuity on \([2,8]\) satisfies the hypotheses of the Mean Value Theorem. 3. Therefore, for some \(c\in(2,8)\), \(f'(c)=2\).

a) \(f'(c)=2\) for at least one \(c\in(2,8)\). b) The Mean Value Theorem uses the endpoint values and the continuity/differentiability hypotheses; an explicit formula is not required.

- Find the change in height per unit of elapsed time over the whole interval. - Interpret the derivative of height in this context. - State why the theorem's hypotheses are satisfied.

1. The average vertical velocity is \(\frac{300\,\text{ft}}{20\,\text{s}}=15\,\text{ft/s}\). 2. The height function is differentiable and therefore continuous on the time interval. 3. By the Mean Value Theorem, at some interior time the instantaneous vertical velocity equals the average vertical velocity, so it is \(15\,\text{ft/s}\).

At some time strictly inside the \(20\,\text{s}\) interval, the elevator's instantaneous vertical velocity is \(15\,\text{ft/s}\).

- Compare the theorem's hypotheses with the type of function shown. - The relevant line slope comes from the two endpoints of the interval. - Parallel lines have equal slopes.

1. The polynomial \(f\) is continuous on \([0,4]\) and differentiable on \((0,4)\), so the Mean Value Theorem applies. 2. \(f(0)=3\) and \(f(4)=11\), so the secant slope is \(\frac{11-3}{4}=2\). 3. Since \(f'(x)=2x-2\), solve \(2c-2=2\), giving \(c=2\).

a) The theorem applies. b) The tangent is parallel to the secant at \(x=2\).

- Separate conditions on the function from conditions on its derivative. - Recall exactly which interval is closed and which is open in the theorem statement. - Ask whether the theorem says anything about continuity of the derivative itself.

1. The statement is false. 2. The Mean Value Theorem requires \(f\) to be continuous on the closed interval and differentiable on the open interval. 3. Continuity of \(f'\) is not one of the hypotheses. A derivative may exist throughout the open interval without itself being continuous.

False. The theorem requires continuity of \(f\) on \([a,b]\) and differentiability of \(f\) on \((a,b)\); it does not require \(f'\) to be continuous.

- Use only the endpoint rows to find the interval's overall rate of change. - Interpret the derivative of position in context. - The theorem guarantees an interior time even though that time is not listed in the table.

1. The average velocity is \(\frac{133-85}{20-12}=\frac{48}{8}=6\,\text{m/s}\). 2. Since \(s\) is differentiable, it is continuous on the interval. 3. By the Mean Value Theorem, there is some \(c\in(12,20)\) with \(s'(c)=6\,\text{m/s}\).

At some time \(c\in(12,20)\), the runner's instantaneous velocity is \(6\,\text{m/s}\).

- Use the change in position and elapsed time to find the interval's average velocity. - Connect that average velocity to an instantaneous derivative value using the theorem. - Convert the guaranteed velocity to speed before comparing it with the speed limit.

1. The average velocity over the interval is \(\frac{120\,\text{mi}}{1.5\,\text{h}}=80\,\text{mph}\). 2. The position function is continuous on the closed time interval and differentiable in its interior, so the Mean Value Theorem gives an interior time \(c\) with \(s'(c)=80\,\text{mph}\). 3. Since this velocity is positive, the instantaneous speed at that time is \(|s'(c)|=80\,\text{mph}\), which exceeds the \(70\,\text{mph}\) speed limit.

The instantaneous speed was \(80\,\text{mph}\) at some time during the trip, so the car exceeded a \(70\,\text{mph}\) speed limit.

- Rolle's theorem has the usual continuity and differentiability hypotheses plus one endpoint condition. - Compare the two endpoint function values. - The conclusion concerns a horizontal tangent in the open interval.

1. The polynomial is continuous on \([1,5]\) and differentiable on \((1,5)\). 2. The endpoint values are \(f(1)=0\) and \(f(5)=0\), so the endpoints have equal function values. 3. Differentiate: \(f'(x)=2x-6\). Solving \(f'(c)=0\) gives \(c=3\), which lies in \((1,5)\).

a) Yes. \(f\) is continuous on \([1,5]\), differentiable on \((1,5)\), and \(f(1)=f(5)=0\). b) \(c=3\)

- Is any point where the function is undefined included in the interval? - Compare the change in function values with the change in inputs. - What equation expresses that a tangent and a secant are parallel?

1. The function is continuous on \([1,4]\) and differentiable on \((1,4)\) because this interval does not contain \(0\). 2. The secant slope is \(\frac{f(4)-f(1)}{4-1}=\frac{\frac14-1}{3}=-\frac14\). 3. Since \(f'(x)=-\frac{1}{x^2}\), the tangent condition is \(-\frac{1}{c^2}=-\frac14\). 4. This gives \(c^2=4\), and the only value in \((1,4)\) is \(c=2\).

a) \(f\) is continuous on \([1,4]\) and differentiable on \((1,4)\). b) \(c=2\)

- What slope must occur somewhere inside the interval? - Which proposed ranges contain that slope? - Can you build a linear function whose slope equals the required value?

1. The Mean Value Theorem requires \(f'(c)=\frac{5-1}{2-0}=2\) for some \(c\in(0,2)\). 2. Claim A rules out a derivative of \(2\), and claim C also rules it out. 3. Claim B allows it. For example, \(f(x)=1+2x\) meets the endpoint conditions and has \(f'(x)=2\) everywhere.

B. The required derivative value is \(2\). One possible function is \(f(x)=1+2x\).

- Check the theorem's conditions separately on the closed interval and the open interval. - Compare the slope across the whole interval with a tangent-line slope inside the interval. - Translate “parallel” into a condition involving two slopes.

1. The polynomial \(f\) is continuous on \([0,2]\) and differentiable on \((0,2)\), so the Mean Value Theorem applies. 2. \(f(0)=0\) and \(f(2)=2\), so the secant slope is \(\frac{2-0}{2-0}=1\). 3. Since \(f'(x)=3x^2-3\), solve \(3c^2-3=1\). Thus \(c^2=\frac{4}{3}\), and the value in \((0,2)\) is \(c=\frac{2\sqrt{3}}{3}\).

a) The theorem applies because \(f\) is continuous on \([0,2]\) and differentiable on \((0,2)\). b) Secant slope: \(1\). c) \(c=\frac{2\sqrt{3}}{3}\).

- Relate total change across the interval to one derivative value inside it. - Use the derivative bounds to bound that total change. - After finding a candidate range, check whether a constant-slope function can realize every value in it.

1. By the Mean Value Theorem, there is some \(c\in(1,4)\) such that \(f(4)-f(1)=f'(c)(4-1)\). 2. Since \(2\le f'(c)\le5\), \(6\le f(4)-7\le15\). 3. Therefore, \(13\le f(4)\le22\). 4. Conversely, for any \(L\in[13,22]\), the linear function \(f(x)=7+\frac{L-7}{3}(x-1)\) has derivative \(\frac{L-7}{3}\in[2,5]\) and satisfies \(f(4)=L\). Thus every value in the interval is possible.

\(f(4)\in[13,22]\).

- Compute the slope determined by the two endpoint values. - Solve for every interior point whose derivative equals that slope. - Distinguish what the theorem guarantees from what the algebra happens to show for this function.

1. The polynomial is continuous on \([0,3]\) and differentiable on \((0,3)\). 2. \(f(0)=0\) and \(f(3)=6\), so the average rate of change is \(2\). 3. Since \(f'(x)=3x^2-6x+2\), solve \(3c^2-6c+2=2\), giving \(3c(c-2)=0\). The value in \((0,3)\) is \(c=2\). 4. The Mean Value Theorem guarantees at least one such value; uniqueness is not part of its conclusion. In this particular function, solving the derivative equation shows that the only interior value is \(2\).

a) \(c=2\). b) No. The theorem guarantees existence, not uniqueness; uniqueness here follows from solving the derivative equation.

- Do not start with the endpoints only; choose two arbitrary points in the interval. - Apply the theorem on the smaller interval determined by those two points. - Use the derivative information to determine the corresponding secant slope.

1. Choose any two points \(u<v\) in \([a,b]\). 2. The Mean Value Theorem applied on \([u,v]\) gives a point \(c\in(u,v)\) such that \(\frac{f(v)-f(u)}{v-u}=f'(c)\). 3. Since \(f'(c)=0\), \(f(v)-f(u)=0\), so \(f(v)=f(u)\). 4. Because the two points were arbitrary, \(f\) has the same value everywhere on \([a,b]\), so it is constant.

For arbitrary \(u<v\), the Mean Value Theorem gives \(\frac{f(v)-f(u)}{v-u}=0\), hence \(f(v)=f(u)\). Therefore, \(f\) is constant on \([a,b]\).

- Determine the slope forced by the two endpoint values. - Ask what derivative value the theorem then guarantees. - Compare that required value with the proposed derivative bound.

1. The average rate of change from \(0\) to \(4\) is \(\frac{15-2}{4}=\frac{13}{4}=3.25\). 2. By the Mean Value Theorem, some \(c\in(0,4)\) must satisfy \(f'(c)=3.25\). 3. This contradicts \(f'(x)\le3\) everywhere on the open interval. Therefore, no such function exists.

The claim is impossible because the Mean Value Theorem would require \(f'(c)=3.25\) for some \(c\in(0,4)\), contradicting \(f'(x)\le3\).

- For a piecewise function, check the joining point separately. - Continuity and differentiability require two different comparisons at that point. - After finding the interval's average slope, solve on each derivative piece.

1. Each piece is continuous and differentiable on its own interval. At \(x=1\), the left and right function values are both \(1\), so \(f\) is continuous there. 2. The left derivative at \(1\) is \(2\), and the right derivative is also \(2\), so \(f\) is differentiable at \(1\). Therefore, the Mean Value Theorem applies on \([0,3]\). 3. \(f(0)=0\) and \(f(3)=5\), so the average rate of change is \(\frac{5}{3}\). 4. On \((0,1)\), \(f'(x)=2x\), so \(2c=\frac{5}{3}\) gives \(c=\frac{5}{6}\). On \((1,3)\), \(f'(x)=2\), so there is no additional solution.

a) The theorem applies; the pieces match in both value and derivative at \(x=1\). b) \(c=\frac{5}{6}\).

- Break the full interval at the middle point. - Compare the endpoint values on each smaller interval. - The two theorem applications produce points in disjoint open intervals.

1. Apply the Mean Value Theorem on \([a,b]\). Since \(f(a)=f(b)\), the secant slope is \(0\), so there is \(u\in(a,b)\) with \(f'(u)=0\). 2. Apply the Mean Value Theorem on \([b,c]\). Since \(f(b)=f(c)\), there is \(v\in(b,c)\) with \(f'(v)=0\). 3. The intervals \((a,b)\) and \((b,c)\) are disjoint, so \(u\ne v\). Thus there are at least two distinct zeros of \(f'\).

There is a point \(u\in(a,b)\) and a distinct point \(v\in(b,c)\) such that \(f'(u)=f'(v)=0\).

- Compare the two functions by considering their difference. - What is the derivative of that difference? - Use the theorem to determine what a function with zero derivative can do across an interval.

1. Define \(h(x)=f(x)-g(x)\). Then \(h\) is continuous on \([0,5]\), differentiable on \((0,5)\), and \(h'(x)=f'(x)-g'(x)=0\). 2. For any \(x\in(0,5]\), apply the Mean Value Theorem to \(h\) on \([0,x]\). There is some \(c\in(0,x)\) such that \(h(x)-h(0)=h'(c)x=0\), so \(h(x)=h(0)\). 3. Since \(h(0)=f(0)-g(0)=0\), \(h(x)=0\) for every \(x\in(0,5]\); it is also \(0\) at \(x=0\). Therefore, \(f(x)=g(x)\) throughout \([0,5]\).

\(f(x)=g(x)\) for every \(x\in[0,5]\).