Question 1

A ball is thrown vertically upward from the ground with an initial velocity of $$20 m/s. $$Its height at time $$t$$ seconds is given by $$s(t) = 20t - 4.9t^2$. What is the velocity of the ball after 2$ seconds?

Accepted Answer

$$-9.8 m/s$$

Question 2

Find the derivative of the function $$f(x) = x^2$ \sin(x).$$

Accepted Answer

$$2x \sin(x) + x^2$ \cos(x)$$

Question 3

A bacteria population grows according to $$P(t) = 500e$$^{0.03t}$$, where t$ is in hours. What is the rate of change of the population at t = 10$ hours?

Accepted Answer

$$500e^{0.3}$$ 	imes 0.03$$

Question 4

Evaluate the limit: $$\lim_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 5

Find $$\frac{dy}{dx} if$$ $$x^2$$ + $$y^2 = 25$.

Accepted Answer

$$-\frac{x}{y}$$

Question 6

Differentiate $$f(x) = \ln(x^2$ + 1).$$

Accepted Answer

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

Question 7

If $$y = 	an$$^{-1}$$(3x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{3}{1 + 9x^2$}$$

Question 8

Let $$f(x) = e$$^{x^2}$$. What is f$$'(x)$$?

Accepted Answer

$$2x e$$^{x^2}$$

Question 9

A ladder $$10 ft $$long is leaning against a wall. The bottom is sliding away from the wall at $$2 ft/s. $$How fast is the top of the ladder sliding down the wall when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-1.5 ft/s$$

Question 10

Find the derivative of $$f(x) = \ln(\sin(x)).$$

Accepted Answer

$$\cot(x)$$

Question 11

If $$y = x^x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$x^x (\ln(x) + 1)$$

Question 12

Find the derivative of $$f(x) = \frac{e^x}{x}.$$

Accepted Answer

$$\frac{e^x(x - 1)}{x^2$}$$

Question 13

Suppose $$f$$ and $$g$$ are differentiable functions. Which of the following is the correct expression for the derivative of $$f(g(x))$$?

Accepted Answer

$$f'(g(x)) \cdot g'(x)$$

Question 14

Find the derivative of $$f(x) = \log_3$(x).$$

Accepted Answer

$$\frac{1}{x \ln(3)}$$

Question 15

If $$y = \sin(x^2$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2x \cos(x^2$)$$

Question 16

A tank is being filled with water so that the volume $$V (in $$liters) at time $$t (in $$minutes) is $$V = 100 \sqrt{t}. $$How fast is the water being added when $$t = 9$$ minutes?

Accepted Answer

$$\frac{50}{3}$$ liters/min

Question 17

Find the derivative of $$f(x) = 5^x.$$

Accepted Answer

$$5^x \ln(5)$$

Question 18

If $$y = \cos$$^{-1}$$(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$-\frac{1}{\sqrt{1 - x^2$}}$$

Question 19

Suppose $$y is $$defined implicitly by $$x^3$$ + $$y^3 = 6xy$. Find $$\frac{dy}{dx}$$.

Accepted Answer

$$\frac{2x - y^2$}{y^2$ - 2x}$$

Question 20

A medication is administered so that the amount in the bloodstream at time $$t (in $$hours) is $$A(t) = 200e$$^{-0.2t}$$. What is the rate at which the medication is leaving the bloodstream at t = 5$ hours?

Accepted Answer

$$-40e^{-1}$$

MA 161 Exam 2 Study Guide: Differentiation and Applications

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