Q: The derivative of a function $$f(x) at$$ $$x = a is $$defined as:

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

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: $$ \lim_{x 	o 2} (3x^2$ - 4x + 1) $$

Accepted Answer

$$5$$

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 a function $$f(x) at$$ $$x = a is $$defined as:

Accepted Answer

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

Question 5

Find the derivative of $$f(x) = 5x^3$ - 2x + 7.$$

Accepted Answer

$$15x^2 - 2$

Question 6

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

Accepted Answer

$$26$$

Question 7

Which of the following is the correct application of the chain rule for $$y = \sin(3x^2$)$$?

Accepted Answer

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

Question 8

A company wants to minimize the cost $$C(x) = x^2$ - 8x + 20. $$What value of $$x$$ minimizes the cost?

Accepted Answer

$$x = 4$$

Question 9

Which of the following is the correct expression for the definite integral representing the area under $$f(x) = x^2$ from x = 1$ to x = 3$?

Accepted Answer

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

Question 10

Evaluate $$\$$int_0$$^2 (3x + 1) dx.$$

Accepted Answer

$$10$$

Question 11

A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

$$-5e^{-1}$$

Question 12

Which of the following statements about the Fundamental Theorem of Calculus is correct?

Accepted Answer

It connects differentiation and integration, stating that differentiation and integration are inverse processes.

Question 13

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

Accepted Answer

$$25$$ liters

Question 14

Which of the following is the correct linear approximation of $$f(x) = \sqrt{x} at$$ $$x = 4$$?

Accepted Answer

$$f(x) \approx 2 + \frac{1}{4}(x-4)$$

Question 15

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

Accepted Answer

Mean Value Theorem

Question 16

A ball is thrown upward with velocity $$v(t) = 20 - 9.8t. $$How long does it take for the ball to reach its maximum height?

Accepted Answer

$$t = 2.04$$ seconds

Question 17

Which of the following best describes the process of optimization in calculus?

Accepted Answer

Finding the maximum or minimum values of a function, often subject to constraints.

Question 18

Suppose $$f(x) is $$differentiable and $$f'(x) \u003e 0$$ for all $$x in an $$interval. What can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$increasing on that interval.

Question 19

A biologist models the population of a bacteria colony by $$P(t) = 100e$$^{0.3t}$$. What is the instantaneous rate of growth at t = 0$?

Accepted Answer

$$30$$

Question 20

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

Accepted Answer

$$2x^3 + C$

Calculus I Syllabus and Course Overview – North Carolina A&T State University

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