$$\lim_{x \to a} f(x)$$ exists and equals $$f(a)$$

Question 1

Given the function $$f(x)$$ graphed in the figure, which of the following statements is true about $$\lim_{x 	o 7} f(x)$$?

Accepted Answer

$$\lim_{x 	o 7} f(x) = 2$$

Question 2

Evaluate $$\lim_{x 	o 4} \frac{x}{x-4}.$$

Accepted Answer

$$\infty$$

Question 3

Find the derivative of $$y = 3x^2$ + \sec^2$ x.$$

Accepted Answer

$$6x + 2\sec^2$ x 	an x$$

Question 4

Using the definition of the derivative, what is $$\frac{d}{dx} \sqrt{x}$$?

Accepted Answer

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

Question 5

The position of an object is given by $$s(t) = 10 + 12t - t^3$. What is the acceleration when the velocity is zero?

Accepted Answer

$$-6$$

Question 6

Find $$\frac{dy}{dx}$$ for the curve $$(y-x)^2 = 1 + x$.

Accepted Answer

$$\frac{1 + 2y - 2x}{2y - 2x}$$

Question 7

What is the equation of the tangent line to $$(y-x)^2 = 1 + x$ at the point (0$$,1)$$?

Accepted Answer

$$y = \frac{3}{2}x + 1$$

Question 8

Evaluate $$\lim_{x 	o \infty} \frac{x+3}{x^2$+6}.$$

Accepted Answer

$$0$$

Question 9

Find the derivative of $$y = x^2$ \sin(3x).$$

Accepted Answer

$$2x \sin(3x) + 3x^2$ \cos(3x)$$

Question 10

If $$y = \frac{e^x}{\sqrt{1 + x^2$}}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{e^x(1 + x^2$) - e^x x^2$}{(1 + x^2$)^{3/2}$$}$

Question 11

Evaluate $$\lim_{x 	o 0} \frac{\sin 3x}{\sin 2x}.$$

Accepted Answer

$$\frac{3}{2}$$

Question 12

Which of the following best describes the process used to find $$\frac{dy}{dx}$$ for $$(y-x)^2 = 1 + x$?

Accepted Answer

Implicit differentiation and the product rule

Question 13

If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ exists and equals $$f(a)$$

Question 14

Given $$y = e$$^{\sin x + 2x + 1}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$e^{\sin x + 2x + 1}$$(\cos x + 2)$$

Question 15

A car's position is given by $$s(t) = 10 + 12t - t^3$. What is the velocity function v(t$$)$$?

Accepted Answer

$$v(t) = 12 - 3t^2$

Calculus Midterm Study Guidance

Study Guide - Practice Questions