Question 1

Find the derivative of the function $$f(x) = 3x^2$ \cos(2x).$$

Accepted Answer

$$f'(x) = 6x \cos(2x) - 6x^2$ \sin(2x)$$

Question 2

Given $$g(x) = x \ln(4x)$$, find $$g'(x).$$

Accepted Answer

$$g'(x) = \ln(4x) + 1$$

Question 3

Suppose $$f(x) = e^x - x \sin^2$(x). $$Find $$f'(x).$$

Accepted Answer

$$f'(x) = e^x - \sin^2$(x) - 2x \sin(x) \cos(x)$$

Question 4

If $$g(x) = x \arccos(x)$$, find $$g'(x).$$

Accepted Answer

$$g'(x) = \arccos(x) - \frac{x}{\sqrt{1 - x^2$}}$$

Question 5

Let $$f(x) = 2x \csc(x^2$ + x). $$Find $$f'(x).$$

Accepted Answer

$$f'(x) = 2 \csc(x^2$ + x) - 2x \cot(x^2$ + x) \csc(x^2$ + x) (2x + 1)$$

Question 6

If $$f is a $$one-to-one function with $$f(3) = 5$$ and $$f'(3) = 10$$, find $$\$$left(f^{-1}$$\right)'(5).$$

Accepted Answer

$$\frac{1}{10}$$

Question 7

The tangent line to a one-to-one function $$f is$$ $$y = 10x - 25$$ when $$x = 3. $$Find $$f(3)$$ and $$f'(3).$$

Accepted Answer

$$f(3) = 5$$, $$f'(3) = 10$$

Question 8

The volume of a sphere is $$V = \frac{4}{3}\pi r^3$. If the rate of change of the volume is 9$ cubic inches per second when the radius is 3$ inches, what is $$\frac{dr}{dt}$$?

Accepted Answer

$$\frac{1}{12\pi}$$\ inches/sec

Question 9

A 13-foot ladder is leaning against a house. The top is sliding down at $$3 ft/sec $$when the top is $$12 ft $$above the ground. How fast is the base moving away from the wall at that instant?

Accepted Answer

$$\frac{36}{5} ft/sec$$

Question 10

Suppose a particle moves according to $$s(t) = 5t^2$ - 10t + 6. $$What is the acceleration function $$a(t)$$?

Accepted Answer

$$a(t) = 10$$

Question 11

Given the cost function $$C(x) = \frac{1}{4}x^2$ - 20x + 150$$, what is the marginal cost when producing the first 10 units?

Accepted Answer

$$C'(10) = -15$$

Question 12

Find the derivative of $$f(x) = x e^x$$ using logarithmic differentiation.

Accepted Answer

$$f'(x) = e^x (x + 1)$$

Question 13

If $$y is a $$function of $$x$$ and $$e^{xy}$$ + $$x^2 y$$ + $$y^2$$ = $$x^2$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$\frac{dy}{dx} = \frac{2x - y e$$^{xy}$$ - 2x y}{x e$$^{xy}$$ + x^2$ + 2y}$$

Question 14

Find the equation of the tangent line to the curve $$x^2 y - x y + 2 x$$ $$y^2 = 2$ at the point (2$$, 1)$$.

Accepted Answer

$$y - 1 = -\frac{2}{5}(x - 2)$$

Question 15

Find the absolute maximum and minimum values of $$f(x) = 3x^2$ - x on$$ $$[-3, 3].$$

Accepted Answer

Absolute max at $$x = 3$$, $$f(3) = 24$$; absolute min at $$x = -3$$, $$f(-3) = 30$$

Question 16

Find the location and value of the absolute extrema of $$f(x) = x^2$ + \arccos(x) on$$ $$[-1, 1].$$

Accepted Answer

Absolute max at $$x = -1$$, $$f(-1) = 1 + \pi$$; absolute min at $$x = 1$$, $$f(1) = 1$$

Step-by-Step Calculus Study Guide for MAT 131 Exam 2

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