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 $$f'(3)$$?

Accepted Answer

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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