Q: Which of the following is the correct definition of the derivative at a point $$x = a$$?

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

Q: Which of the following is the correct application of L'Hôpital's Rule for $$\lim_{x \to 0} \frac{\sin(x)}{x}$$?

$$\lim_{x \to 0} \frac{\cos(x)}{1}$$

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 value.

Question 2

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

Accepted Answer

$$8$$

Question 3

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

Accepted Answer

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

Question 4

The derivative of $$f(x) = x^3$ is:

Accepted Answer

$$3x^2$$

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

$$26$$

Question 6

Which rule is used to differentiate $$f(x) = (2x^2$ + 3x)(x^3$ - 1)$$?

Accepted Answer

Product Rule

Question 7

Find the derivative of $$f(x) = \sin(x) \cdot e^x.$$

Accepted Answer

$$\cos(x) \cdot e^x + \sin(x) \cdot e^x$$

Question 8

Which of the following is the correct application of the Chain Rule for $$f(x) = \sin(3x^2$)$$?

Accepted Answer

$$f'(x) = \cos(3x^2$) \cdot 6x$$

Question 9

A tank is being filled with water such that the volume $$V (in $$liters) at time $$t (in $$minutes) is $$V(t) = 5t^2$. What is the rate at which water is entering the tank at t = 4$ minutes?

Accepted Answer

$$40$$ liters/min$

Question 10

Which of the following statements is true about the Mean Value Theorem?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, then there exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 11

Evaluate $$\int 2x e$$^{x^2}$$ dx.$$

Accepted Answer

$$e^{x^2} + C$

Question 12

Which of the following is the correct definition of the derivative at a point $$x = a$$?

Accepted Answer

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

Question 13

A company wants to minimize the cost $$C(x) = x^2$ - 10x + 25$$ for producing $$x$$ units. What is the optimal number of units to minimize cost?

Accepted Answer

$$x = 5$$

Question 14

Which of the following is an antiderivative of $$f(x) = 3x^2$?

Accepted Answer

$$x^3 + C$

Question 15

If $$f(x) = \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 16

A medication is administered such that the amount in the bloodstream at time $$t is$$ $$A(t) = 100e$$^{-0.2t}$$. What is the rate of change of the medication at t = 5$ hours?

Accepted Answer

$$-20e^{-1}$$

Question 17

Which of the following is the correct application of L'Hôpital's Rule for $$\lim_{x 	o 0} \frac{\sin(x)}{x}$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{\cos(x)}{1}$$

Question 18

Find the area under the curve $$y = x$$ from $$x = 0 to$$ $$x = 2.$$

Accepted Answer

$$2$$

Question 19

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

Accepted Answer

It connects differentiation and integration, stating that if $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_a^b f(x) dx = F(b) - F(a).$$

Calculus I (MATH 2600) Syllabus and Course Structure – Study Guide

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