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: A function $$f is $$defined as $$f(x) = \begin{cases} x^2$ & x \leq 1 \\ 2x-1 & x \u003e 1 \end{cases}. Is$$ $$f$$ continuous at $$x=1$$?

Yes, because $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1).$$

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

$$f is $$continuous at $$a if$$ $$\lim_{x \to a} f(x) = f(a).$$

Question 1

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

Accepted Answer

The domain is all real numbers $$x$$ such that $$x \geq 2.$$

Question 2

Suppose $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$. What is the domain of the composite function h(x) = f(g(x$$))$$?

Accepted Answer

All real numbers $$x$$ such that $$x \geq 0.$$

Question 3

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 4

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

Accepted Answer

$$24 m/s$$

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

$$4$$

Question 7

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 8

If $$f(x) = x^3$, what is f$$'(x)$$?

Accepted Answer

$$3x^2$$

Question 9

Which of the following statements about the limit $$\lim_{x 	o 0} \frac{\sin(x)}{x} is $$true?

Accepted Answer

The limit is $$1.$$

Question 10

A function $$f is $$defined as $$f(x) = \begin{cases} x^2$ & x \leq 1 \ 2x-1 & x \u003e 1 \end{cases}. Is$$ $$f$$ continuous at $$x=1$$?

Accepted Answer

Yes, because $$\lim_{x 	o 1^-} f(x) = \lim_{x 	o 1^+} f(x) = f(1).$$

Question 11

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 12

Let $$f(x) = |x|. $$Which statement about its differentiability is correct?

Accepted Answer

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

Question 13

Suppose $$f(x) = \frac{1}{x}$$ and $$g(x) = x^2$. What is $$\frac{d}{dx}[f(g(x))]$$?

Accepted Answer

$$-2x^{-3}$$

Question 14

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

Accepted Answer

$$f is $$continuous at $$a if$$ $$\lim_{x 	o a} f(x) = f(a).$$

Question 15

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

Accepted Answer

$$-1$$

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

If $$f(x) = x^n$$ for $$n a $$positive integer, what is $$f'(x)$$?

Accepted Answer

$$nx^{n-1}$$

Question 18

A patient’s temperature $$T(t) is $$modeled by $$T(t) = 37 + 0.5\sin(\pi t/12)$$, where $$t is $$hours after midnight. What is the rate of change of temperature at $$t=6$$?

Accepted Answer

$$0.5 \cdot \frac{\pi}{12} \cos(\pi/2) = 0$$

Calculus I: Numbers, Functions, Limits, and Derivatives

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