Question 1

Given the function $$z(u, x, y) = 9$$uxy^2$$ - \frac{x^6$y^4$$}{u} + $$2u^2y$, what is $$\frac{\partial z}{\partial u}$?

Accepted Answer

$$9xy^2$$ - \frac{$$x^6y^4$}{u^2$} + 4uy$

Question 2

Given $$y = -\frac{1}{12}x^4$ + 2x^3$ + x^2$ - 5x + 30$$, what is the third derivative $$\frac{d^3$y}{dx^3$}$$?

Accepted Answer

$$-2x + 12$$

Question 3

Given $$u = \frac{1}{3-v}$$, what is the second derivative $$u''(v)$$?

Accepted Answer

$$\frac{2}{(3-v)^3$}$$

Question 4

Given $$g(x) = x^3$ + 5x^2$ - 7$$, what is $$g''(-4)$$?

Accepted Answer

$$-14$$

Question 5

Given $$v = \sqrt{y^3$}$$, what is $$\frac{d^2$v}{dy^2$}$$ evaluated at $$y = 1$$?

Accepted Answer

$$-\frac{2}{9}$$

Question 6

A car travels according to $$s(t) = t^3$ + 2t - 1. $$What is the average speed between $$t = 2$$ and $$t = 4$$?

Accepted Answer

$$30$$

Question 7

For the function $$s(t) = 112t - 16t^2$, what is the velocity at t = 4$ seconds?

Accepted Answer

$$-16 ft/sec$$

Question 8

Given $$f(800) = 150$$ and $$f'(800) = 3$$, use the derivative to approximate $$f(802).$$

Accepted Answer

$$156$$

Question 9

For the function $$s(t) = 112t - 16t^2$, at what time(s) is the object on the ground?

Accepted Answer

$$t = 0 sec $$and $$t = 7 sec$$

Question 10

For $$s(t) = 112t - 16t^2$, what is the maximum height reached by the object and at what time?

Accepted Answer

At $$t = 3.5 sec$$, $$s(3.5) = 196 ft$$

Question 11

If $$f'(x)  0$$ for $$x \u003e -10$$, what can be said about $$f(x) at$$ $$x = -10$$?

Accepted Answer

$$f(x)$$ has a relative minimum at $$x = -10$$

Question 12

If $$f''(x) \u003e 0$$ for $$x  8$$, what does this indicate about $$f(x) at$$ $$x = 8$$?

Accepted Answer

$$f(x)$$ has an inflection point at $$x = 8$$

Question 13

A function $$f(x) is $$increasing and concave up on the interval $$-3 \u003c x \u003c 3. $$Which of the following statements is true?

Accepted Answer

$$f'(x) \u003e 0$$ and $$f''(x) \u003e 0$$ for $$-3 \u003c x \u003c 3$$

Question 14

Given the function $$y = \sqrt{x + 5}$$, what is the second derivative $$y''$$?

Accepted Answer

$$-\frac{1}{4(x+5$$)^{3/2}$$}$$

Question 15

For the function $$s(t) = 112t - 16t^2$, what is the acceleration at any time t$?

Accepted Answer

$$-32 ft/sec$^2$$

Question 16

If $$f'(x) = 0 at$$ $$x = 2$$ and $$f''(x) \u003c 0 at$$ $$x = 2$$, what does this indicate about $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$f(x)$$ has a relative maximum at $$x = 2$$

Question 17

A function $$f(x)$$ has a point $$(0, -1)$$, is increasing and concave up on $$-3 \u003c x \u003c 3. $$Which of the following could be a possible sketch?

Accepted Answer

A curve passing through $$(0, -1)$$, rising and curving upwards between $$x = -3$$ and $$x = 3$$

Question 18

If $$f'(x) = 0 at$$ $$x = 2$$ and $$f''(x) = 0 at$$ $$x = 2$$, what can be said about $$f(x) at$$ $$x = 2$$?

Accepted Answer

It could be a point of inflection, but more information is needed

Business Calculus Study Guide: Derivatives, Function Behavior, and Applications

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