Q: Which of the following is the correct definition of the derivative of a function $$f at a $$point $$a$$?

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

Q: Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x \leq 0 \\ x & x \u003e 0 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

Yes, because $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0).$$

Question 1

Which of the following statements about the domain and range of the function $$f(x) = \sqrt{x-2} is $$correct?

Accepted Answer

The domain is $$x \geq 2$$ and the range is $$y \geq 0.$$

Question 2

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

Accepted Answer

$$v(3) = 26$$

Question 3

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

Accepted Answer

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

Question 4

Given $$f(x) = x^3$ - 2x + 1$$, what is $$f'(x)$$?

Accepted Answer

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

Question 5

Which of the following is the correct definition of the derivative of a function $$f at a $$point $$a$$?

Accepted Answer

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

Question 6

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x \leq 0 \ x & x \u003e 0 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

Accepted Answer

Yes, because $$\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^+} f(x) = f(0).$$

Question 7

Evaluate $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

2

Question 8

Which of the following is an example of an implicitly defined function?

Accepted Answer

$$x^2$$ + $$y^2 = 1$

Question 9

A chemical reaction's concentration $$C(t)$$ changes according to $$C(t) = 5e$$^{-0.2t}$$. What is the rate of change of concentration at t = 0$?

Accepted Answer

$$C'(0) = -1$$

Question 10

Which of the following statements about the vertical line test is true?

Accepted Answer

If a vertical line intersects a graph more than once, the graph does not represent a function.

Question 11

Find the derivative of $$f(x) = \frac{1}{x}$$ using the power rule.

Accepted Answer

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

Question 12

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

Question 13

A patient’s temperature $$T(t) is $$modeled by $$T(t) = 37 + 0.5\sin(\pi t)$$, where $$t is in $$hours. What is the rate of change of temperature at $$t = 2$$ hours?

Accepted Answer

$$T'(2) = -0.5\pi$$

Question 14

Which of the following limits does NOT exist?

Accepted Answer

$$\lim_{x 	o 0} \frac{1}{x}$$

Question 15

If $$f(x) = |x|$$, which statement about its differentiability is correct?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 0.$$

Question 16

Which of the following is the correct application of the quotient rule for differentiation?

Accepted Answer

$$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$

Question 17

A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t = 0$?

Accepted Answer

$$P'(0) = 5$$

Question 18

Which of the following is the correct statement of the Sandwich (Squeeze) Theorem?

Accepted Answer

If $$g(x) \leq f(x) \leq h(x)$$ for all $$x$$ near $$a$$, and $$\lim_{x 	o a} g(x) = \lim_{x 	o a} h(x) = L$$, then $$\lim_{x 	o a} f(x) = L.$$

Question 19

Which of the following is true about the continuity of polynomial functions?

Accepted Answer

Polynomial functions are continuous everywhere on $$\mathbb{R}.$$

Calculus I: Functions, Limits, and Derivatives – Structured Study Notes

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