Question 1

Which of the following integrals is best solved using the substitution $$x = a \sin 	heta$$?

Accepted Answer

$$\int \frac{dx}{\sqrt{a^2$ - x^2$}}$$

Question 2

Given the parametric equations $$x = t^2$ - 1$$, $$y = t^2$(2t + 3)$$, what is $$\frac{dy}{dx} in $$terms of $$t$$?

Accepted Answer

$$\frac{d}{dt}y / \frac{d}{dt}x = \frac{2t(2t + 3) + t^2$ \cdot 2}{2t}$$

Question 3

Which of the following is the correct formula for the arc length of a curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b \sqrt{1 + \left(\frac{dy}{dx}\$$right)^2$$} dx$$

Question 4

Suppose $$y = \sqrt{1 + x}$$, find the differential $$dy.$$

Accepted Answer

$$dy = \frac{1}{2\sqrt{1 + x}} dx$$

Question 5

Which of the following is the correct Maclaurin series for $$e^x$$?

Accepted Answer

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

Question 6

Evaluate $$\$$int_0$$^1 x(1 + x^2$)^{10} dx$ using the substitution u = 1$$ + $$x^2$.

Accepted Answer

$$\frac{1}{22}[(1 + x^2$)^{11}$$]_$$0^1$$ = \frac{1}{22}$$(2^{11} - 1$$)$

Question 7

Which of the following is the integrating factor for the linear differential equation $$\frac{dy}{dx} + 2y = x$$?

Accepted Answer

$$e^{2x}$$

Question 8

Given $$z = x^2$ y \sin(x + y^2$)$$, what is $$\frac{\partial z}{\partial x}$$?

Accepted Answer

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

Question 9

Which of the following is the general solution to the separable differential equation $$\frac{dy}{dx} = y^2$ with y(0) = 1$?

Accepted Answer

$$y = \frac{1}{1 - x}$$

Question 10

If $$f(x, y) is a $$function with continuous second partial derivatives, which of the following is always true?

Accepted Answer

$$\frac{\$$partial^2$$ f}{\partial x \partial y} = \frac{\$$partial^2$$ f}{\partial y \partial x}$$

Question 11

Which of the following is the correct formula for the surface area generated by rotating $$y = f(x)$$ about the $$x-$$axis from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\$$right)^2$$} dx$$

Question 12

Which of the following is the correct Taylor series for $$\ln x$$ about $$x = 1$$?

Accepted Answer

$$\ln x = (x-1) - \frac{(x-1)^2$}{2} + \frac{(x-1)^3$}{3} - \frac{(x-1)^4$}{4} + \cdots$$

Question 13

Suppose $$z = f(x, y)$$, $$x = r \cos 	heta$$, $$y = r \sin 	heta. $$Which of the following expresses $$\frac{\partial z}{\partial r} in $$terms of $$x$$ and $$y$$?

Accepted Answer

$$\frac{\partial z}{\partial r} = \cos 	heta \frac{\partial z}{\partial x} + \sin 	heta \frac{\partial z}{\partial y}$$

Question 14

Which of the following is the correct test for exactness for the differential equation $$M(x, y)dx + N(x, y)dy = 0$$?

Accepted Answer

$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$

Calculus Study Guide: Techniques of Integration, Applications, Series, and Partial Differentiation

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