Q: A function $$f(x) is $$said to be continuous at $$x = a if $$which of the following conditions are met?

$$f(a) is $$defined, $$\lim_{x \to a} f(x)$$ exists, and $$\lim_{x \to a} f(x) = f(a)$$

Q: Refer to the image of the inscribed polygons in a circle. As the number of sides $$n of $$the inscribed polygon increases, what happens to the area of the polygon?

It becomes equal to the area of the circle as $$n \to \infty.$$

Question 1

Which of the following best describes the concept of a limit in calculus?

Accepted Answer

The value a function approaches as the input gets close to a particular point.

Question 2

Given the function $$g(x) = \frac{x^2$ - 2}{x + 2}$$, what is $$\lim_{x 	o -2} g(x)$$?

Accepted Answer

$$-4$$

Question 3

A function $$f(x) is $$said to be continuous at $$x = a if $$which of the following conditions are met?

Accepted Answer

$$f(a) is $$defined, $$\lim_{x 	o a} f(x)$$ exists, and $$\lim_{x 	o a} f(x) = f(a)$$

Question 4

Refer to the image of the inscribed polygons in a circle. As the number of sides $$n of $$the inscribed polygon increases, what happens to the area of the polygon?

Accepted Answer

It becomes equal to the area of the circle as $$n 	o \infty.$$

Question 5

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

Accepted Answer

$$2$$

Question 6

Which of the following is an indeterminate form?

Accepted Answer

$$0 	imes \infty$$

Question 7

If $$f(x) = 3x + 4$$, what is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$10$$

Question 8

Given $$f(x) = x^2$ + 5x - 3$$, for which value of $$x is $$the function momentarily at rest?

Accepted Answer

$$x = -\frac{5}{2}$$

Question 9

Find the derivative of $$y = 4x^3$ using the power rule.

Accepted Answer

$$12x^2$$

Question 10

A company’s cost function is $$C(x) = 5x + 10. $$What does the derivative $$\frac{dC}{dx}$$ represent in this context?

Accepted Answer

The rate at which cost increases per additional item produced.

Question 11

Given $$y = 2x^2$ - 4x + 5$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$4x - 4$$

Question 12

If the tangent to the curve $$y = 5x^2$ - 2x + 5 is $$parallel to the line $$y = 8x + 5$$, at what $$x-$$coordinate does this occur?

Accepted Answer

$$x = 1$$

Question 13

What is the equation of the tangent to $$y = 2x^2$ + 5x at $$the point $$(2, 18)$$?

Accepted Answer

$$y = 13x - 8$$

Question 14

For the function $$f(x) = x^3$ - x^2$ - 8x - 5$$, at which $$x-$$values is the function momentarily at rest?

Accepted Answer

$$x = 2$$ and $$x = -\frac{4}{3}$$

Question 15

If $$y = \frac{1}{x^2$}$$, what is the derivative $$\frac{dy}{dx}$$?

Accepted Answer

$$-2x^{-3}$$

Question 16

The average rate of change of $$f(x) = x^2$ from x = 1$ to x = 2$ is:

Accepted Answer

$$3$$

Question 17

Which of the following statements about the limit $$\lim_{x 	o 1} \frac{x+1}{x-1} is $$true?

Accepted Answer

The limit does not exist because the denominator approaches zero.

Question 18

If $$f(x) = 2x^3$ - 3x^2$ - 36x + 7$$, for which intervals is $$f(x)$$ increasing?

Accepted Answer

$$(-\infty, -2)$$ and $$(3, \infty)$$

Question 19

Given $$y = 3x^2$ - 2x + 4$$, what is the equation of the tangent at $$(2, 12)$$?

Accepted Answer

$$y = 10x - 8$$

Limits and Differentiation: Foundations of Calculus for Physics with Algebra

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