Question 1

Evaluate the derivative of the function $$f(x) = \sin$$^{-1}$$(2x^4$$)$. Which of the following is correct?

Accepted Answer

$$f'(x) = \dfrac{8x^3$}{\sqrt{1 - 4x^8$}}$$

Question 2

Suppose $$\$$int_3$$^8 f(x)\,dx = 13$$ and $$\$$int_3$$^8 g(x)\,dx = 7. $$What is $$\$$int_3$$^8 (f(x) - g(x))\,dx$$?

Accepted Answer

$$6$$

Question 3

Find the derivative of $$A = 730(1.759)^t$$ with respect to $$t.$$

Accepted Answer

$$A' = 730(1.759)^t \ln(1.759)$$

Question 4

Given $$f(x) = (x-6)^2$ + 2$$ for $$x \geq 6$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x) = 6$$ + \sqrt{x-2}$$

Question 5

Evaluate $$\dfrac{d}{dx}\left(\ln(\ln(11x))\right).$$

Accepted Answer

$$\dfrac{1}{x\ln(11x)}$$

Question 6

Use substitution to evaluate $$\int 4x^3$ \sqrt{1 - 8x^4$}\,dx.$$

Accepted Answer

$$-\dfrac{1}{4}(1 - 8x^4$)^{1/2} + C$

Question 7

Find the derivative of $$f(x) = (8x-9)x$$^{3/2}$$.

Accepted Answer

$$\dfrac{(40x - 27)\sqrt{x}}{2}$$

Question 8

Suppose $$f(x)$$ has a slope of $$\dfrac{1}{5} at$$ $$(1,4). $$What is $$(f^{-1}$$)'(4)$$?

Accepted Answer

$$5$$

Question 9

Find the area bounded by $$f(x) = \sin x$$ and the $$x-$$axis on $$\left[-\dfrac{2\pi}{3}, \dfrac{3\pi}{4}\right].$$

Accepted Answer

$$\dfrac{5 + \sqrt{2}}{2}$$

Question 10

Evaluate $$\int x\sqrt[4]{6x+1}\,dx.$$

Accepted Answer

$$\dfrac{(6x+1$$)^{9/4}$$}{81} - \dfrac{(6x+1$$)^{5/4}$$}{45} + C$$

Question 11

A rectangular pen is built with one side against a barn. If 400 m of fencing are used for the other three sides, what dimensions maximize the area?

Accepted Answer

Sides perpendicular to the barn: $$100 m$$, side parallel to the barn: $$200 m$$

Question 12

Evaluate $$\int (\sec^2$ x - 2)\,dx.$$

Accepted Answer

$$	an(x) - 2x + C$$

Question 13

Evaluate $$\int 28t^6$ e$$^{-t^7}$$\,dt.$$

Accepted Answer

$$-4e^{-t^7} + C$

Question 14

Find the linear approximation $$L(x) to$$ $$f(x) = -2x^2$ - x at$$ $$a = 3.$$

Accepted Answer

$$L(x) = 18 - 13x$$

Question 15

Suppose $$f'(x) = x^2$(x-5)(x+2). At $$which $$x-$$value does $$f$$ have a local maximum?

Accepted Answer

$$x = -2$$

Question 16

Evaluate $$\int \dfrac{dx}{8x+1}.$$

Accepted Answer

$$\dfrac{1}{8}\ln|8x+1| + C$$

Question 17

A square-based box-shaped shipping crate is to have volume $$150 ft$^3. $$The base material costs $$\dfrac{8}{5} as $$much as the sides, and the top costs $$\dfrac{4}{5} as $$much as the sides. What are the dimensions that minimize cost?

Accepted Answer

Base side $$= 5 ft$$, height $$= 6 ft$$

Question 18

Evaluate $$\lim_{x 	o 0} x^2$ \csc x.$$

Accepted Answer

$$0$$

Question 19

Use the substitution $$u = x^5$ + 3x to $$evaluate $$\int (5x^4$ + 3)\sqrt{x^5$ + 3x}\,dx.$$

Accepted Answer

$$\dfrac{2}{3}(x^5$ + 3x$$)^{3/2}$$ + C$$

Question 20

Evaluate $$\int x^3$ \cos(x^4$)\,dx.$$

Accepted Answer

$$\dfrac{1}{4}\sin(x^4$) + C$$

Step-by-Step Calculus Study Guidance: Derivatives, Integrals, Limits, and Optimization

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