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

$$f'(x) = \lim\limits_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Q: Which of the following is NOT a property of limits?

$$\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a} f(x+h)$$ for any $$h$$

Q: If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

$$\lim\limits_{x \to a} f(x)$$ exists and equals $$f(a)$$

Q: Which of the following limits evaluates to $$1$$?

$$\lim\limits_{x \to 0} \frac{\sin x}{x}$$

Q: Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$, & x \u003c 0 \\ 2x, & x \geq 0 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

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

Question 1

Which of the following sets is closed under addition, subtraction, and multiplication, but not necessarily under division?

Accepted Answer

$$\mathbb{Z} ($$the integers)

Question 2

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

Accepted Answer

$$f'(x) = \lim\limits_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$

Question 3

Suppose $$f(x) = x^2$. What is the slope of the tangent to the graph at x = 3$?

Accepted Answer

$$6$$

Question 4

Which of the following statements about the function $$f(x) = \sqrt{x} is $$true?

Accepted Answer

Its domain is $$x \geq 0$$ and its range is $$y \geq 0.$$

Question 5

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

Accepted Answer

$$26$$

Question 6

Which of the following is NOT a property of limits?

Accepted Answer

$$\lim\limits_{x 	o a} f(x) = \lim\limits_{x 	o a} f(x+h)$$ for any $$h$$

Question 7

If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

Accepted Answer

$$\lim\limits_{x 	o a} f(x)$$ exists and equals $$f(a)$$

Question 8

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

Accepted Answer

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

Question 9

A function $$f is $$defined implicitly by $$x^2$$ + $$y^2 = 1$. Which of the following is the correct formula for $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = -\frac{x}{y}$$

Question 10

Which of the following best describes the Intermediate Value Theorem?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$N is $$any number between $$f(a)$$ and $$f(b)$$, then there exists $$c in$$ $$[a, b]$$ such that $$f(c) = N.$$

Question 11

Suppose $$f(x) = 2x + 1$$ and $$g(x) = x^2$. What is (f$$ \circ g)(x)$$?

Accepted Answer

$$2x^2 + 1$

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 population of bacteria doubles every hour. If the initial population is $$P_0$$, which function models the population after $$t$$ hours?

Accepted Answer

$$P(t) = 2$$^{t}$$P_0$

Question 14

Which of the following limits evaluates to $$1$$?

Accepted Answer

$$\lim\limits_{x 	o 0} \frac{\sin x}{x}$$

Question 15

If $$f(x) is $$differentiable at $$x = a$$, which of the following must be true?

Accepted Answer

$$f(x) is $$continuous at $$x = a$$

Question 16

Which of the following is the correct formula for the derivative of $$f(x) = x^n$$?

Accepted Answer

$$f'(x) = nx$$^{n-1}$$

Question 17

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$, & x \u003c 0 \ 2x, & x \geq 0 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

Accepted Answer

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

Question 18

A ball is thrown upward with position $$s(t) = 10t - 4.9t^2$. What is the velocity when t = 2$ seconds?

Accepted Answer

$$-9.6 m/s$$

Question 19

Which of the following is an example of a function that is continuous everywhere but not differentiable at $$x = 0$$?

Accepted Answer

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

Calculus I: Foundations, Limits, and Derivatives

Guida di studio - Domande di pratica

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