Question 1

Given the implicit equation $$x^2y^3$ = 3x + 2$$, find $$\frac{dy}{dx}$.

Accepted Answer

$$\frac{6x - 2y^3$}{3x^2$y^2$$}$

Question 2

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin 2x}{x}.$$

Accepted Answer

$$2$$

Question 3

Evaluate $$\lim\limits_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 4

Which of the following is the correct integral expression for the exact area under $$f(x) = x^3$ + x^2$ - x + 1$$ between $$x = -2$$ and $$x = 1$$?

Accepted Answer

$$\int_{-2}^{1} (x^3$ + x^2$ - x + 1) \, dx$$

Question 5

Evaluate $$\int x e$$^{x^2}$$ \, dx.$$

Accepted Answer

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

Question 6

Find the slope of the tangent line to $$f(x) = x^4$ + x^2$ at x = 1$.

Accepted Answer

$$6$$

Question 7

For the function $$y = x^4$ - 4x^3$ + 6x^2$ + 1$$, which of the following is a critical value?

Accepted Answer

$$x = 2$$

Question 8

A 6-foot-tall woman walks away from a street light that is 18 ft above the ground at a rate of 4 ft/sec. At what rate is the length of her shadow increasing when she is 12 ft from the light?

Accepted Answer

$$1.5 ft/sec$$

Question 9

Which of the following best describes how a Riemann sum can be modified to better approximate the area under a curve?

Accepted Answer

Increase the number of subintervals

Question 10

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

Accepted Answer

$$\frac{2}{3}x^3$ + 2x + C$$

Question 11

For the function $$y = x^4$ - 4x^3$ + 6x^2$ + 1$$, on which interval is the function concave up?

Accepted Answer

$$x \u003e 2$$

Question 12

A rectangle is inscribed under the curve $$y = 25 - x^2$ with its base on the x$-axis and its upper corners on the curve. What is the maximum possible area of such a rectangle?

Accepted Answer

$$\frac{125}{2}$$

Question 13

Estimate the slope of $$f(x) at$$ $$x = 0$$ from a graph if the tangent appears horizontal.

Accepted Answer

$$0$$

Question 14

If $$f'(x) \u003e 0$$ for all $$x in an $$interval, what can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$increasing

Question 15

Evaluate $$\int x^2$ \, dx.$$

Accepted Answer

$$\frac{1}{3}x^3$ + C$$

Question 16

Which of the following is the value of $$\lim\limits_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

$$1$$

Question 17

For the function $$y = x^4$ - 4x^3$ + 6x^2$ + 1$$, what is the $$x$$ value of any inflection point?

Accepted Answer

$$x = 2$$

Question 18

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It relates differentiation and integration as inverse processes.

Calculus IB: Practice Final Exam Study Notes

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