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

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

Question 1

Which of the following sets contains all real numbers $$x$$ such that $$x^2$$ \u003c 4$$?

Accepted Answer

$$(-2, 2)$$

Question 2

Given the function $$f(x) = x^2$, what is the average rate of change of f$ from x = 1$ to x = 3$?

Accepted Answer

$$5$$

Question 3

Which of the following is the correct definition of the derivative of a function $$f at a $$point $$a$$?

Accepted Answer

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

Question 4

If $$f(x) = 3x + 2$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$3$$

Question 5

Which of the following functions is NOT continuous at $$x = 0$$?

Accepted Answer

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

Question 6

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

Accepted Answer

$$6$$

Question 7

Which of the following statements about the limit $$\lim_{x \to 0} \frac{\sin(x)}{x} is $$true?

Accepted Answer

The limit exists and equals $$1$$

Question 8

If $$f(x) = x^3$, what is f$$''(x)$$, the second derivative?

Accepted Answer

$$6x$$

Question 9

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

$$(fg)' = f'g + fg'$$

Question 10

Suppose $$f(x) is $$continuous at $$x = a$$ and $$g(x) is $$continuous at $$x = a. $$Which of the following is true about $$f(x) + g(x)$$?

Accepted Answer

$$f(x) + g(x) is $$continuous at $$x = a$$

Question 11

What is the domain of the function $$f(x) = \sqrt{x-2}$$?

Accepted Answer

$$[2, \infty)$$

Question 12

If $$f(x) = x^2$ and g(x) = x + 1$, what is the derivative of f(g(x$$))$$?

Accepted Answer

$$2(x+1)$$

Question 13

Which of the following is the correct definition of a continuous function at $$x = a$$?

Accepted Answer

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

Question 14

A function $$f is $$given by $$f(x) = |x|. $$Which statement is true about its differentiability?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 0$$

Question 15

If $$f(x) = x^2$ and g(x$$) = $$x^3$$, what is the derivative of $$f(x)g(x)$$?

Accepted Answer

$$2x^3$$ + $$3x^4$$

Question 16

Which of the following is the correct application of the quotient rule for differentiation?

Accepted Answer

$$\frac{f'g - fg'}{g^2$}$$

Question 17

If $$f(x) = x^n$$, what is the derivative $$f'(x)$$ according to the power rule?

Accepted Answer

$$nx^{n-1}$$

Question 18

Which of the following is a real-world example of a function that is not continuous?

Accepted Answer

A step function representing a train schedule with sudden jumps in arrival times

Question 19

Suppose $$f(x) = x^2$ and g(x) = 2x$. What is the value of $$\lim_{x \to 1} (f(x) + g(x))$$?

Accepted Answer

$$5$$

Question 20

Which of the following is true about the function $$f(x) = \frac{1}{x}$$?

Accepted Answer

$$f(x) is $$not defined at $$x = 0$$

Calculus I: Numbers, Functions, Limits, and Derivatives – Structured Study Notes

Study Guide - Practice Questions