Applications of the Derivative

In Exercises $21-28$, graphs of $f,f,\text{ or }{f}^{\text{'}\text{'}}$ are given. Whichever is shown, sketch graphs of the remaining two functions. Label the locations of any roots, extrema, and inflection points on each graph.

Use the definition of the derivative to find f' for each function f. 

$f\left(x\right)=x^3+2$

Use L’Hôpital’s rule to prove that power functions with positive powers always dominate logarithmic functions.

Read the section and make your own summary of the material. 

Review of definitions and theorems: State each theorem or definition that follows in precise mathematical language. Then give an illustrative graph or example, as appropriate.

(a) f has a local maximum at x = c .

(b) f has a local minimum at x = c .

(c) f is continuous on [a, b] .

(d) f is differentiable on (a, b) .

(e) The secant line from (a, f (a)) to (b, f (b)) .

(f) The right derivative f ' +(c) at a point x = c .

(g) The left derivative f ' −(c) at a point x = c .

(h) The Extreme Value Theorem .

(i) The Intermediate Value Theorem .

Examples: Construct examples of the thing(s) described in

the following. Try to find examples that are different than

any in the reading .

(a)  A function with a local minimum at x = 3 that is continuous but not differentiable at x = 3 .

(b)  A function with a local maximum at x = −2 that is not differentiable at x = −2 because of a removable discontinuity .

(c)  A function with a local minimum at x = 1 that is not differentiable at x = 1 because of a jump discontinuity .

If f has a local maximum at x = 1, then what can you say about f '(1)?  What if you also know that f is differentiable at x = 1 .

If f has a local minimum at x = 0 and a local maximum at x = 2, what can you say about f '(0) and f '(2)? Is there anything else you can say about f ' .

Suppose that $f$ is defined on (−∞,∞) and differentiable everywhere except at $x=-2$ and $x=4$, and that $f\text{'}\left(x\right)=0$ only at $x=0$ and $x=5$. List all the critical points of $f$ and sketch a possible graph of $f$ . 

Suppose that $f$ is defined for $x=0$ and differentiable everywhere except at $x=0$ and $x=1$, and that $f\text{'}\left(x\right)=0$ only at $x$= ±2. List all the critical points of f and sketch a possible graph of $f$.

If a continuous, differentiable function f has zeroes at x = −4, x = 1, and x = 2, what can you say about f ' on [−4, 2]?

If a continuous, differentiable function f is equal to 2 at x = 3 and at x = 5, what can you say about f ' on [3, 5]?

If a continuous, differentiable function f has values f (−2) = 3 and f (4) = 1, what can you say about f ' on [−2, 4]?

Restate Rolle’s Theorem so that its conclusion has to do with tangent lines.

Restate Theorem 3.3 so that its conclusion has to do with

tangent lines.

Restate the Mean Value Theorem so that its conclusion has to do with tangent lines.

Sketch the graph of a function that satisfies the given description. Label or annotate your graph so that it is clear that it satisfies each part of the description.

A function that satisfies the hypothesis, and therefore the conclusion, of Rolle’s Theorem on $[2,6]$. 

A function that satisfies the hypothesis, and therefore the

conclusion, of the Mean Value Theorem.

A function f that satisfies the hypothesis of the Mean

Value Theorem on [0, 4] and for which there are exactly

three values c ∈ (0, 4) that satisfy the conclusion of the

theorem .

A functionf that is defined on [−2, 2] with f (−2) = f (2) = 0 such that f is continuous everywhere, differentiable everywhere except at x = −1, and fails the conclusion of Rolle’s Theorem .

A function f defined on [1, 5] with f (1) = f (5) = 0 such that f is continuous everywhere except for x = 2, differentiable everywhere except at x = 2, and fails the conclusion of Rolle’s Theorem .

A function f defined on [−3,−1] with f (−3) = f (−1) = 0 such that f is continuous everywhere except at x = −1 and differentiable everywhere except at x = −1, and fails the conclusion of Rolle’s Theorem.

A function f defined on [0, 4] such that f is continuous everywhere,differentiable everywhere except at x = 2, and fails the conclusion of the Mean Value Theorem with a = 0 and b = 4.

A function f defined on [−3, 3] such that f is continuous everywhere except at x = 1, differentiable everywhere except at x = 1, and fails the conclusion of the Mean Value Theorem with a = −3 and b = 3.

A function f defined on [−2, 0] such that f is continuous everywhere except at x = −2, differentiable everywhere except at x = −2, and fails the conclusion of the Mean Value Theorem with a = −2 and b = 0.

For the graph of f in the given figure , approximate all the values x ∈ (0, 4) for which the derivative of f is zero or does not exist. Indicate whether f has a local maximum, minimum, or neither at each of these critical points .

For the graph of f in the given figure, approximate all the values x ∈ (0, 4) for which the derivative of f is zero or does not exist. Indicate whether f has a local maximum, minimum, or neither at each of these critical points.

For the graph of f in the given figure, approximate all the values x ∈ (0, 4) for which the derivative of f is zero or does not exist. Indicate whether f has a local maximum, minimum, or neither at each of these critical points.

For the graph of f in the given figure, approximate all the values x ∈ (0, 4) for which the derivative of f is zero or does not exist. Indicate whether f has a local maximum, minimum, or neither at each of these critical points. 

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these.

f (x) =  (x − 1.7) (x + 3)

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical point. $f\left(x\right)=x^3+x^2+1$

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

$f\left(x\right)=3x^4+8x^3-18x^2$

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

$f\left(x\right)={\left(2x-1\right)}^5$

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

$f\left(x\right)=3x-2e^x$

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

$f\left(x\right)=3^x-2^x$

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points .

$f\left(x\right)= \frac{In 2x}{x}$

Find the critical points of f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

$f\left(x\right)=2^{1-In x}$

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

$f\left(x\right)=\cos x$

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

$f\left(x\right)=sex x$

For each graph of f in Exercises 37–40, explain why f satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. Then approximate any values c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

[a, b] = [−3, 1]

For each graph of f in Exercises 37–40, explain why f satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. Then approximate any values c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

[a, b] = [−3, 3]

For each graph of f in Exercises 37–40, explain why f satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. Then approximate any values c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

[a, b] = [0, 4]

For each graph of f in Exercises 37–40, explain why f satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. Then approximate any values c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

[a, b] = [−1, 1]

Determine whether or not each function f in Exercises 41–48 satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

$f\left(x\right)=x^3-4x^2+3x, \left[a,b\right]=\left[0,3\right]$

Determine whether or not each function f in Exercises 41–48 satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

$f\left(x\right)=x^3-4x^2+3x, \left[a,b\right]=\left[1, 3\right]$

Determine whether or not each function f in Exercises 41–48 satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem. 

$f\left(x\right)=x^4-3.2x^2-3.04, \left[a, b\right]=\left[-2, 2\right]$

Determine whether or not each function f in Exercises 41–48 satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem. 

$f\left(x\right)=\frac{x^2-4x}{x^2-4x+3}, \left[a, b\right]=\left[0, 4\right]$

Determine whether or not each function f in Exercises 41–48 satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

$f\left(x\right)=\cos \left(x\right), \left[a, b\right]=\left[-\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2}\right]$

Determine whether or not each function f in Exercises 41–48 satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

$f\left(x\right)=\sin \left(2x\right), \left[a, b\right]=\left[0, 2\mathrm{\pi }\right]$

Determine whether or not each function f in Exercises 41–48 satisfies the hypotheses of Rolle’s Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of Rolle’s Theorem.

$f\left(x\right)=e^x\left(x^2-2x\right), \left[a, b\right]=\left[0, 2\right]$