$$\lim\limits_{x \to 2} f(x)$$ does not exist.

$$\lim\limits_{x \to a} f(x) = f(a)$$

Yes, because $$\lim\limits_{x \to 2} f(x) = f(2).$$

$$f'(a) = \lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Which of the following statements best describes the meaning of $$\lim\limits_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ approaches $$L.$$

Question 2

Suppose $$\lim\limits_{x 	o 2^-} f(x) = 3$$ and $$\lim\limits_{x 	o 2^+} f(x) = 5. $$Which of the following is true?

Accepted Answer

$$\lim\limits_{x 	o 2} f(x)$$ does not exist.

Question 3

A function $$f(x) is $$continuous at $$x = a if$$:

Accepted Answer

$$\lim\limits_{x 	o a} f(x) = f(a)$$

Question 4

Which of the following is NOT a possible reason for a function to be discontinuous at a point $$x = a$$?

Accepted Answer

The function is differentiable at $$x = a.$$

Question 5

Evaluate $$\lim\limits_{x 	o 1} \frac{x^2$ - 1}{x - 1}.$$

Accepted Answer

2

Question 6

If $$f(x) = \begin{cases} x^2$ & x \u003c 2 \ 4 & x \geq 2 \end{cases}$$, is $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

Yes, because $$\lim\limits_{x 	o 2} f(x) = f(2).$$

Question 7

Which of the following is the correct limit definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f'(a) = \lim\limits_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 8

The slope of the tangent line to the curve $$y = f(x) at$$ $$x = a is $$given by:

Accepted Answer

$$f'(a)$$

Question 9

Suppose $$f(x) = x^3$ - 3x + 2. $$What is the equation of the tangent line to $$f(x) at$$ $$x = 1$$?

Accepted Answer

Question 10

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

Accepted Answer

$$\frac{2}{x}$$

Question 11

Which of the following is the correct derivative of $$f(x) = e$$^{2x}$$?

Accepted Answer

$$2e^{2x}$$

Question 12

If $$y = x^2$ \sin(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 13

Which of the following is the correct application of the chain rule for $$\frac{d}{dx} \left[ \sin(x^2$) \right]$$?

Accepted Answer

$$2x\cos(x^2$)$$

Question 14

A car's position at time $$t is $$given by $$s(t) = 3t^2$ + 2t. $$What is the instantaneous velocity at $$t = 4$$?

Accepted Answer

26

Question 15

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 16

Which of the following is the correct derivative of $$f(x) = 	an(x)$$?

Accepted Answer

$$\sec^2$(x)$$

Question 17

If $$y = x^x$$, what is $$\frac{dy}{dx}$$ using logarithmic differentiation?

Accepted Answer

$$x^x (\ln(x) + 1)$$

Question 18

Suppose $$f(x) = \frac{1}{x}$$ and $$g(x) = \ln(x). $$What is $$\frac{d}{dx} [f(x)g(x)]$$?

Accepted Answer

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

Question 19

Which of the following is the correct application of the product rule for $$\frac{d}{dx} [u(x)v(x)]$$?

Accepted Answer

$$u'(x)v(x) + u(x)v'(x)$$

Calculus I Exam 1 Review: Limits, Continuity, and Derivatives

Study Guide - Practice Questions