Question 1

Evaluate $$\lim_{x 	o 2^-} (f(x) - g(x))$$ using the graphs provided in the second image.

Accepted Answer

$$1$$

Question 2

Given $$f(x) = \ln(x^2$ + 1)$$, what is $$f''(x)$$?

Accepted Answer

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

Question 3

Find the derivative of $$y = x \sin x.$$

Accepted Answer

$$\sin x + x \cos x$$

Question 4

Evaluate $$\lim_{x 	o 0} \frac{\sin(3x)}{x}.$$

Accepted Answer

$$3$$

Question 5

If $$f(x) = x^2$ e^x$$, what is $$f'(x)$$?

Accepted Answer

$$e^x (2x + x^2$)$$

Question 6

Find $$\lim_{x 	o \infty} \frac{5x^2$+1}{2x^2$-3}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 7

Evaluate $$\$$int_0$$^1 (3x^2$ + 2x) dx.$$

Accepted Answer

$$\frac{5}{2}$$

Question 8

Evaluate $$\$$int_1$$^2 (2x^3$ + \frac{1}{x^3$}) dx.$$

Accepted Answer

$$\frac{63}{8}$$

Question 9

Evaluate $$\$$int_0$$^2 x \sqrt{2x^2$ + 1} dx.$$

Accepted Answer

$$\frac{52}{3}$$

Question 10

Find the area enclosed between $$y(x) = x^2$ + 3$$ and the x-axis on $$[0, 2].$$

Accepted Answer

$$\frac{26}{3}$$

Question 11

A circular ripple is expanding in a pond so that its radius increases at a rate of $$2 cm/s. $$How fast is the area of the circle increasing when the radius is $$5 cm$$?

Accepted Answer

$$10\pi cm$^2/s$$

Question 12

A $$10-ft $$ladder is leaning against a vertical wall. The bottom of the ladder is sliding away from the wall at a rate of $$3 ft/s. $$How fast is the top of the ladder sliding down the wall when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-1.8 ft/s$$

Question 13

Let $$f(x) = \frac{2x^2$}{x^2$ - 4}. $$What are the vertical asymptotes of $$f(x)$$?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 14

Given $$f(x) = x^3$ - 3x^2$ + 2$$, find the critical points.

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 15

Evaluate $$\$$int_0$$^2 x e$$^{x^2}$$ dx.$$

Accepted Answer

$$\frac{1}{2}(e$$^{4}$$ - 1)$$

Question 16

Let $$F(x) = \$$int_1$$^x \sqrt{t^2$ + 1} dt. $$Find $$F'(x).$$

Accepted Answer

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

Question 17

Given $$f(x) = 	an$$^{-1}$$(x - 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 18

Find the derivative of $$g(x) = \cos$$^{-1}$$(\sqrt{x}).$$

Accepted Answer

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

Question 19

A rectangle is inscribed under the curve $$y = 12 - x^2$ above the x-axis. What is the maximum area of the rectangle?

Accepted Answer

$$16$$

Guided Practice: Evaluating Limits Using Graphs

Study Guide - Practice Questions

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