Q: Which of the following statements about continuity is correct?

A function is continuous at $$x = a if$$ $$ \lim_{x \to a} f(x) = f(a) .$$

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 approaches a certain point.

Question 2

Evaluate the following limit: $$ \lim_{x 	o 2} (3x^2$ - 4) $$

Accepted Answer

$$ 8 $$

Question 3

Which of the following statements about continuity is correct?

Accepted Answer

A function is continuous at $$x = a if$$ $$ \lim_{x 	o a} f(x) = f(a) .$$

Question 4

Find the derivative of $$ f(x) = x^3$ - 5x + 2 .$$

Accepted Answer

$$ f'(x) = 3x^2$ - 5 $$

Question 5

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

Accepted Answer

$$ 26  m/s$$

Question 6

Which rule is used to differentiate the product of two functions?

Accepted Answer

Product Rule: $$ (fg)' = f'g + fg' $$

Question 7

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

Accepted Answer

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

Question 8

A company’s profit function is $$ P(x) = -2x^2$ + 40x - 100 . At $$what value of $$ x  is $$the profit maximized?

Accepted Answer

$$ x = 10 $$

Question 9

Which of the following is the correct antiderivative of $$ f(x) = 6x^2$ $$?

Accepted Answer

$$ 2x^3$ + C $$

Question 10

Evaluate the definite integral $$ \$$int_0$$^2 (3x^2$) dx .$$

Accepted Answer

$$ 8 $$

Question 11

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It relates differentiation and integration, stating that integration can be reversed by differentiation.

Question 12

A tank is being filled with water at a rate given by $$ r(t) = 5t $$ liters per minute. How much water is added between $$ t = 0 $$ and $$ t = 4 $$ minutes?

Accepted Answer

$$ 40 $$ liters

Question 13

Which of the following is an example of a function that is not continuous at $$ x = 2 $$?

Accepted Answer

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

Question 14

Find the local minimum of $$ f(x) = x^2$ - 4x + 5 .$$

Accepted Answer

$$ (2, 1) $$

Question 15

Which substitution would you use to evaluate $$ \int 2x \cos(x^2$) dx $$?

Accepted Answer

Let $$ u = x^2$ $$

Question 16

A medication is absorbed into the bloodstream at a rate modeled by $$ r(t) = 8e$$^{-0.5t}$$  mg/$$hour. How much is absorbed in the first 2 hours?

Accepted Answer

$$ \$$int_0$$^2 8e$$^{-0.5t}$$ dt $$

Question 17

Which of the following is the correct application of the Chain Rule?

Accepted Answer

If $$ y = (3x + 2)^5$ $$, then $$ \frac{dy}{dx} = 5(3x + 2)^4$ \cdot 3 $$

Question 18

Which of the following integrals represents the area under the curve $$ y = x^2$ $$ from $$ x = 1  to$$ $$ x = 3 $$?

Accepted Answer

$$ \$$int_1$$^3 x^2$ dx $$

Question 19

If $$ f'(x) = 4x - 7 $$, what is the general form of $$ f(x) $$?

Accepted Answer

$$ f(x) = 2x^2$ - 7x + C $$

Calculus I (MATH 211) Syllabus and Course Structure Overview

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