Q: Given $$y = 2 - x^2$, x = 2$, and $$\Delta x = dx = 0.01$$, what is the value of dy$?

$$dy = -4 \times 0.01 = -0.04$$

Q: The radius of a spherical balloon is measured as 6 inches, with a possible error of 0.02 inch. Use differentials to approximate the maximum error in the surface area $$S = 4\pi r^2$.

$$\Delta S \approx 48\pi \times 0.02$$

Q: Given $$y = sec^2$ x / (x^2$ + 1)$$, what is the differential $$dy$$?

$$dy = \frac{2 \sec x \tan x (x^2$ + 1) - 2x \sec^2$ x}{(x^2$ + 1)^2$} dx$$

Question 1

Which of the following statements best describes the absolute maximum of a function $$f(x) on a $$closed interval $$[a, b]$$?

Accepted Answer

It is the largest value of $$f(x)$$ for all $$x in$$ $$[a, b].$$

Question 2

Given the function $$f(x) = (x-5)^2$ on the interval $$[0, 7]$$, what are the absolute maximum and minimum values?

Accepted Answer

Absolute maximum: $$(0, 25)$$; Absolute minimum: $$(5, 0)$$

Question 3

A hotel rents 180 rooms when the daily price is $$45. For each 3 $$increase in price, four fewer rooms are rented. What price maximizes daily revenue?

Accepted Answer

$60

Question 4

A rancher has 100 feet of fencing to enclose two adjacent rectangular corrals as shown. What should $$x be to $$maximize the total area?

Accepted Answer

$$x = \frac{100}{6}$$

Question 5

Given $$C(t) = \frac{100t}{t^2$ + 64}$$, where $$t is $$time in hours, at what time does the drug reach its maximum concentration?

Accepted Answer

$$t = 8$$

Question 6

Which of the following is the correct first step in implicit differentiation for the equation $$x^2 + 2xy$$ + $$4y^2 = 12$?

Accepted Answer

Differentiate both sides with respect to $$x to $$get $$2x + 2y \frac{dy}{dx} + 2x \frac{dy}{dx} + 8y \frac{dy}{dx} = 0$$

Question 7

Find $$\frac{dy}{dx}$$ for $$x^2 + 2xy$$ + $$4y^2 = 12$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = \frac{-x - y}{x + 4y}$$

Question 8

What is the equation of the tangent line to $$f(x) at$$ $$x = a$$?

Accepted Answer

$$y = f(a) + f'(a)(x - a)$$

Question 9

Given $$y = 2 - x^2$, x = 2$, and $$\Delta x = dx = 0.01$$, what is the value of dy$?

Accepted Answer

$$dy = -4 	imes 0.01 = -0.04$$

Question 10

The radius of a spherical balloon is measured as 6 inches, with a possible error of 0.02 inch. Use differentials to approximate the maximum error in the surface area $$S = 4\pi r^2$.

Accepted Answer

$$\Delta S \approx 48\pi 	imes 0.02$$

Question 11

Which of the following best describes the difference between $$dy$$ and $$\Delta y$$ for a differentiable function $$y = f(x)$$?

Accepted Answer

$$dy is $$the change in the tangent line approximation, while $$\Delta y is $$the actual change in the function.

Question 12

If $$f(x) is $$continuous on $$[a, b]$$, which of the following must be checked to find the absolute extrema?

Accepted Answer

Critical points in $$(a, b)$$ and endpoints $$a$$ and $$b$$

Question 13

Given $$y = sec^2$ x / (x^2$ + 1)$$, what is the differential $$dy$$?

Accepted Answer

$$dy = \frac{2 \sec x 	an x (x^2$ + 1) - 2x \sec^2$ x}{(x^2$ + 1)^2$} dx$$

Question 14

Which of the following best describes the linearization $$L(x) of a $$function $$f(x) at$$ $$x = a$$?

Accepted Answer

$$L(x) = f(a) + f'(a)(x - a)$$

Question 15

If $$f(x) is $$differentiable at $$x = a$$, which of the following is true about the tangent line at $$(a, f(a))$$?

Accepted Answer

The tangent line is the best linear approximation to $$f(x)$$ near $$x = a.$$

Question 16

For the function $$f(x) = (x-5)^2$ on $$[0, 7]$$, which x$-values should be checked to find the absolute extrema?

Accepted Answer

$$x = 0, 5, 7$$

Applications of the Derivative: Extrema, Optimization, Implicit Differentiation, and Differentials

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드