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

$$\lim_{x \to 0} \frac{\sin x}{x}$$

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

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

Q: If $$\vec{a} = (1, 2, 3)$$ and $$\vec{b} = (4, 0, -1)$$, what is the dot product $$\vec{a} \cdot \vec{b}$$?

$$1 \times 4 + 2 \times 0 + 3 \times (-1) = 1$$

Q: Which of the following statements about limits is true?

If $$\lim_{x \to a} f(x)$$ exists, then $$f(x)$$ approaches a unique value as $$x$$ approaches $$a.$$

Question 1

Given the function $$f(x) = x^3$ - 6x^2$ + 9x + 2$$, find the local minimum value of $$f(x). $$What is the $$x-$$coordinate where it occurs?

Accepted Answer

$$x = 2$$

Question 2

Which of the following is the correct Taylor series expansion for $$e^x$$ centered at $$x=0$$?

Accepted Answer

$$e^x = 1 + x + \frac{x^2$}{2!} + \frac{x^3$}{3!} + \cdots$$

Question 3

A chemical reaction rate is modeled by the function $$R(t) = 5e$$^{-0.2t}$$. What is the instantaneous rate of change of R$ at t = 0$?

Accepted Answer

$$-1$$

Question 4

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

Accepted Answer

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

Question 5

Given the vector $$\vec{v} = (2, -1, 3)$$, what is the magnitude $$|\vec{v}|$$?

Accepted Answer

$$\sqrt{14}$$

Question 6

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 7

A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$. After how many hours will the population double?

Accepted Answer

$$t = \frac{\ln 2}{0.05}$$

Question 8

Which of the following is the correct definition of the derivative of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 9

If $$A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$$, what is $$\det(A)$$?

Accepted Answer

$$-2$$

Question 10

A projectile is launched with height $$h(t) = -4.9t^2$ + 20t + 1.5. At $$what time does it reach its maximum height?

Accepted Answer

$$t = 2.04$$

Question 11

Which of the following is the correct antiderivative of $$f(x) = 3x^2$?

Accepted Answer

$$x^3 + C$

Question 12

If $$\lim_{x 	o 2} f(x) = 5$$ and $$\lim_{x 	o 2} g(x) = -3$$, what is $$\lim_{x 	o 2} [2f(x) - 3g(x)]$$?

Accepted Answer

$$19$$

Question 13

Which of the following is the correct formula for the sum of a geometric series with first term $$a$$ and ratio $$r$$ ($$|r| \u003c 1$$)?

Accepted Answer

$$S = \frac{a}{1 - r}$$

Question 14

Given $$f(x) = x^2$ and g(x) = 2x + 1$, what is (f$$ \circ g)(x)$$?

Accepted Answer

$$(2x + 1)^2$

Question 15

If $$\vec{a} = (1, 2, 3)$$ and $$\vec{b} = (4, 0, -1)$$, what is the dot product $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$1 	imes 4 + 2 	imes 0 + 3 	imes (-1) = 1$$

Question 16

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

Accepted Answer

$$\cos(x)$$

Question 17

A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.1t}$$. What is the concentration after 10 hours?

Accepted Answer

$$50e^{-1}$$

Question 18

Which of the following statements about limits is true?

Accepted Answer

If $$\lim_{x 	o a} f(x)$$ exists, then $$f(x)$$ approaches a unique value as $$x$$ approaches $$a.$$

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