Q: Which of the following best describes the concept of a limit as $$x$$ approaches $$a$$ for a function $$f(x)$$?

$$\displaystyle \lim_{x \to a} f(x) is $$the value that $$f(x)$$ approaches as $$x$$ gets arbitrarily close to $$a$$, regardless of $$f(a).$$

Q: A biologist models the population of a bacteria culture by $$P(t) = 100e$$^{0.2t}$$, where t$ is in hours. What is the instantaneous rate of growth of the population at t = 5$ hours?

$$P'(5) = 0.2 \times 100e$$^{0.2 \times 5}$$

Question 1

Which of the following best describes the concept of a limit as $$x$$ approaches $$a$$ for a function $$f(x)$$?

Accepted Answer

$$\displaystyle \lim_{x 	o a} f(x) is $$the value that $$f(x)$$ approaches as $$x$$ gets arbitrarily close to $$a$$, regardless of $$f(a).$$

Question 2

A biologist models the population of a bacteria culture by $$P(t) = 100e$$^{0.2t}$$, where t$ is in hours. What is the instantaneous rate of growth of the population at t = 5$ hours?

Accepted Answer

$$P'(5) = 0.2 	imes 100e$$^{0.2 	imes 5}$$

Question 3

Which of the following statements about continuity is correct?

Accepted Answer

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

Question 4

Suppose $$f(x) = x^3$ - 2x + 1. $$What is the equation of the tangent line to $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$y = 10x - 17$$

Question 5

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

Accepted Answer

$$a(3) = 8$$

Question 6

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

Accepted Answer

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

Question 7

A chemical reaction's temperature in degrees Celsius is modeled by $$T(t) = 50 - 10e$$^{-0.5t}$$, where t$ is in minutes. What is the rate of change of temperature at t = 0$?

Accepted Answer

$$T'(0) = 5$$

Question 8

Which of the following is NOT a valid interpretation of the derivative $$f'(a)$$?

Accepted Answer

The average rate of change of $$f(x)$$ from $$x = a to$$ $$x = b.$$

Question 9

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 10

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

Accepted Answer

$$f(x) = |x|$$

Question 11

A medication's concentration in the bloodstream is modeled by $$C(t) = 20e$$^{-0.3t}$$ mg/L. How fast is the concentration changing at t = 2$ hours?

Accepted Answer

$$C'(2) = -6e$$^{-0.6}$$

Question 12

Which of the following is the correct derivative of $$f(x) = x^2$ \ln(x)$$?

Accepted Answer

$$f'(x) = 2x\ln(x) + x$$

Question 13

A company’s profit (in dollars) is given by $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units sold. At what number of units is the profit maximized?

Accepted Answer

$$x = 10$$

Question 14

Which of the following is the correct application of the Product Rule for $$y = x^2$ e^x$$?

Accepted Answer

$$y' = 2x e^x + x^2$ e^x$$

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 tank is being filled with water so that the volume at time $$t is$$ $$V(t) = 3t^2$ + 2t$$ liters. What is the rate at which water is being added at $$t = 4$$ minutes?

Accepted Answer

$$V'(4) = 26$$ liters/min

Question 17

Which of the following best describes the process of using derivatives to solve optimization problems in real life?

Accepted Answer

Find the derivative, set it to zero to locate critical points, and test these points to find maximum or minimum values relevant to the scenario.

Question 18

Which of the following is the correct derivative of $$f(x) = \frac{1}{x^2$}$$?

Accepted Answer

$$f'(x) = -2x$$^{-3}$$

Math 210 (Calculus I) Syllabus and Course Overview

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