Question 1

A tank has the form of an inverted cone, with height 24 feet and radius 5 feet. Water enters the tank at a constant rate of $$2\pi$$ cubic feet per minute. At what rate is the depth of water in the tank increasing at the instant when the tank holds $$288\pi$$ cubic feet of water?

Accepted Answer

$$\frac{1}{3} ft/$$min

Question 2

Let $$f(x) = \sin^2$(x). $$What is $$f''(x)$$?

Accepted Answer

$$2\cos(2x)$$

Question 3

Suppose $$f(x) = \begin{cases} x^2$-1, & x \leq 1 \ 2x-1, & x \u003e 1 \end{cases}. $$For what value(s) of $$a is$$ $$f(x)$$ differentiable at $$x=1$$?

Accepted Answer

$$a=0$$

Question 4

A 25-foot ladder rests against a vertical wall. The bottom of the ladder slides away from the wall at a rate of 2 ft/sec. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 7 feet from the wall?

Accepted Answer

$$-\frac{7}{24} ft/sec$$

Question 5

Let $$f(x) = \ln(x^2$+1). At $$what value(s) of $$x$$ does the graph of $$f$$ have a horizontal tangent line?

Accepted Answer

$$x=0$$

Question 6

Given $$f(x) = 	an$$^{-1}$$(x)$$, what is $$\frac{d}{dx}f(x)$$?

Accepted Answer

$$\frac{1}{1+x^2$}$$

Question 7

A particle moves along the x-axis according to the position function $$x(t) = 2t^3$ - 6t^2$. For t$$\u003e0$$, at what time is the particle's velocity zero?

Accepted Answer

$$t=2$$

Question 8

Let $$f(x) = \sin(\sqrt{x}). $$What is $$f'(x)$$?

Accepted Answer

$$\frac{1}{2\sqrt{x}}\cos(\sqrt{x})$$

Question 9

Suppose $$y^2$$ - $$3x^2 = 4$ is a rotated ellipse. Using implicit differentiation, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{3x}{y}$$

Question 10

Let $$f(x) = \frac{x}{x+1}. $$How many tangent lines to the curve pass through the point $$(1,2)$$?

Accepted Answer

2

Question 11

Given $$f(x) = \sin(x)$$, use the symmetric difference quotient to estimate $$f'(0). $$Which of the following is closest to the value?

Accepted Answer

1

Question 12

A rock is dropped and allowed to fall so that its vertical position is $$h(t) = 70 - gt^2$ meters in t$ seconds, where g$ is a constant. Suppose the rock falls 70 meters to the bottom of a crevasse in a$ seconds. What is the value of g$?

Accepted Answer

$$g = \frac{70}{a^2$}$$

Question 13

Let $$f(x) = \frac{x^2$-1}{x-1}. $$What is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

2

Question 14

For which value(s) of $$x$$ does the graph of $$f'(x)$$, the derivative of $$f$$, have a horizontal tangent, given that $$f'(x) is $$shown in the graph and $$-10 \u003c x \u003c 10$$?

Accepted Answer

Where $$f''(x) = 0$$

Question 15

Let $$f(x) = 2x^4$ - x^3$. For what value(s) of x$ is the graph of f$ concave down?

Accepted Answer

$$x  \frac{3}{8}$$

Question 16

Given $$f(x) = \sin(x)$$, what is the equation of the tangent line to $$f at$$ $$x = \frac{\pi}{4}$$?

Accepted Answer

$$y = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(x - \frac{\pi}{4})$$

Question 17

If $$f(x) is $$continuous on $$[a,b]$$ and differentiable on $$(a,b)$$, which theorem guarantees that there exists $$c \in (a,b)$$ such that $$f'(c) = \frac{f(b)-f(a)}{b-a}$$?

Accepted Answer

Mean Value Theorem

Question 18

Let $$f(x) = \ln(x). $$For which value(s) of $$x$$ does $$f$$ have a vertical tangent?

Accepted Answer

$$x=0$$

Question 19

Suppose $$f(x) is a $$function such that $$f'(x) \u003e 0$$ for $$x  1. $$What can you conclude about $$f at$$ $$x=1$$?

Accepted Answer

$$f$$ has a local maximum at $$x=1$$

BC Calculus Review: Differentiation, Applications, and Graphical Analysis

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