Q: What is the definition of the derivative of a function $f$ at a number $a$?

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

Q: Which of the following is the correct formula for the slope of the tangent line to $y = f(x)$ at $x = a$?

$m = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}$

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

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

Q: Which of the following is a correct application of the limit definition of the derivative to $f(x) = x^2$ at $x = a$?

$f'(a) = \lim_{h \to 0} \frac{(a+h)^2 - a^2}{h}$

Question 1

Which of the following statements about continuous functions is TRUE?

Accepted Answer

The sum, product, and quotient (where the denominator is nonzero) of continuous functions are always continuous at a point where both functions are continuous.

Question 2

Given $f(x) = \sqrt{1-3x}$, what is the equation of the tangent line at the point $x = -1$?

Accepted Answer

$y = -\frac{3}{4}(x+1) + 2$

Question 3

Which of the following functions is NOT continuous everywhere in its domain?

Accepted Answer

$f(x) = \frac{1}{x-2}$

Question 4

What is the definition of the derivative of a function $f$ at a number $a$?

Accepted Answer

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

Question 5

Suppose $f(x) = x^2 + 6$. Using the limit definition, what is $f'(3)$?

Question 6

Given $f(x) = x^2 + 6$, use the limit definition $f'(a) = \lim_{x 	o a} \frac{f(x) - f(a)}{x-a}$ to find $f'(3)$. What is the answer?

Accepted Answer

$f'(3) = \frac{5}{9}$

Question 7

Which of the following is the correct formula for the slope of the tangent line to $y = f(x)$ at $x = a$?

Accepted Answer

$m = \lim_{x 	o a} \frac{f(x) - f(a)}{x-a}$

Question 8

A chemical reaction's rate is modeled by $f(x) = \sqrt{x-1}$. Using the limit definition, what is $f'(x)$?

Accepted Answer

$f'(x) = \frac{1}{2\sqrt{x-1}}$

Question 9

Which of the following is NOT a property of continuous functions at a point $a$?

Accepted Answer

If $f$ is continuous at $a$, then $f^2$ is not necessarily continuous at $a$.

Question 10

A medical researcher models the spread of a virus with $f(x) = x^3 - x$. Using the limit definition, what is $f'(x)$?

Accepted Answer

$f'(x) = 3x^2 - 1$

Question 11

Which of the following is the correct equation for the tangent line to $y = f(x)$ at $x = a$?

Accepted Answer

$y - f(a) = f'(a)(x - a)$

Question 12

Which of the following functions is continuous everywhere in its domain?

Accepted Answer

$f(x) = \sin(x)$

Question 13

A ball is thrown and its height is modeled by $f(x) = x^2 + 6$. What is the instantaneous rate of change of height at $x = 3$?

Accepted Answer

$f'(3) = \frac{5}{9}$

Question 14

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

Accepted Answer

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

Question 15

If $f(x) = \frac{3}{x}$, what is the slope of the tangent line at $x = 3$?

Accepted Answer

$-\frac{1}{3}$

Question 16

Which of the following is a correct application of the limit definition of the derivative to $f(x) = x^2$ at $x = a$?

Accepted Answer

$f'(a) = \lim_{h 	o 0} \frac{(a+h)^2 - a^2}{h}$

Question 17

A population grows according to $f(x) = \sqrt{x-1}$. What is the instantaneous rate of change at $x = 5$?

Accepted Answer

$\frac{1}{4}$

Question 18

Which of the following is NOT a type of function that is continuous everywhere in its domain?

Accepted Answer

Piecewise functions with a jump discontinuity

Question 19

If $f(x) = x^3 - x$, what is the equation of the tangent line at $x = 2$?

Accepted Answer

$y = 10(x-2) + 6$

Introduction to Tangents and Derivatives Using Limits

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