Q: A function $$f(x)$$ has a jump discontinuity at $$x = 2. $$Which of the following statements is true?

$$\lim_{x \to 2^-} f(x) \neq \lim_{x \to 2^+} f(x)$$

Question 1

A pharmaceutical company models the concentration of a drug in the bloodstream by $$f(x) = \frac{1}{x^2$ - 4}. $$Which of the following correctly describes the domain of $$f(x)$$?

Accepted Answer

$$(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$$

Question 2

A biologist is studying the growth rate of bacteria, modeled by $$f(x) = \sqrt{2x^2$ + 1}. $$What is the domain of $$f(x)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 3

Evaluate the limit: $$\lim_{h 	o 0} \frac{\sqrt{9 + h} - 3}{h}$$, which models the rate of change of a population at a specific time.

Accepted Answer

$$\frac{1}{6}$$

Question 4

A chemical reaction's concentration is modeled by $$f(x) = \frac{7x^3$ + 2}{-14x^3$ + 5x}. $$What is $$\lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$-\frac{1}{2}$$

Question 5

A medical device records a signal modeled by $$f(x) = |x| \cos\left(\frac{1}{x}ight). $$What is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$0$$

Question 6

Consider the piecewise function $$f(x) = \begin{cases} x^2$ - 1 & x  1 \end{cases}. $$Which type of discontinuity occurs at $$x = 1$$?

Accepted Answer

Jump discontinuity

Question 7

A climate scientist models temperature with $$f(x) = x^3$ - 5. $$Use the Intermediate Value Theorem to determine if $$f(x) = 10$$ has a solution on $$[2, 3].$$

Accepted Answer

Yes, because $$f(x) is $$continuous and $$f(2) = 3$$, $$f(3) = 22 ($$sign change between 3 and 22)

Question 8

If $$f(x) = 3x^4$ - \cos(x) + 5x$$, what is $$f''(x)$$, the second derivative, which could represent acceleration in a physics experiment?

Accepted Answer

$$12x^2$$ + \cos(x)$$

Question 9

Differentiate $$y = \frac{\cos x}{x^3$ + 1}$$, which models the intensity of light in an experiment.

Accepted Answer

$$\frac{-\sin x (x^3$ + 1) - 3x^2$ \cos x}{(x^3$ + 1)^2$}$$

Question 10

Compute $$\frac{d}{dx} \left( \sqrt{x^2$ + 1} \right)$$, which could represent the rate of change of a distance in a navigation system.

Accepted Answer

$$\frac{x}{\sqrt{x^2$ + 1}}$$

Question 11

A population model is $$f(x) = 3x^2$. Using the limit definition, what is f$$'(x)$$?

Accepted Answer

$$6x$$

Question 12

For $$f(x) = \frac{4}{x}$$, what is the equation of the tangent line at $$x = 4$$?

Accepted Answer

$$y - 1 = -\frac{1}{4}(x - 4)$$

Question 13

Find the $$x-$$values where the tangent is horizontal for $$f(x) = x^4$ - 8x^2$, which could represent the height of a roller coaster track.

Accepted Answer

$$x = 0, x = \pm 2$$

Question 14

Suppose $$s(t) = \sin(t^4$)$$ models the position of a particle. What is the formula for its velocity $$v(t)$$?

Accepted Answer

$$v(t) = 4t^3$ \cos(t^4$)$$

Question 15

A car's position is given by $$s(t) = \frac{2t - 1}{t + 1}$$ for $$t 
eq -1. $$What is the average velocity on $$[0, 2]$$?

Accepted Answer

$$\frac{1}{3}$$

Question 16

A function $$f(x) is $$continuous everywhere except at $$x = 5$$, where $$\lim_{x 	o 5} f(x) = 2$$ but $$f(5) = 7. $$What type of discontinuity is this?

Accepted Answer

Removable discontinuity

Question 17

Differentiate $$y = (x^2$ + 1) \sin(x^3$)$$, which could model the amplitude of a vibrating string.

Accepted Answer

$$2x \sin(x^3$) + (x^2$ + 1) \cos(x^3$) \cdot 3x^2$

Question 18

A function $$f(x)$$ has a jump discontinuity at $$x = 2. $$Which of the following statements is true?

Accepted Answer

$$\lim_{x 	o 2^-} f(x) 
eq \lim_{x 	o 2^+} f(x)$$

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