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

Chain Rule: $$\lim_{x \to a} f(g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$$

Q: A function $$f(x) is $$continuous at $$x = a if$$:

$$\lim_{x \to a} f(x)$$ exists and $$\lim_{x \to a} f(x) = f(a)$$

Question 1

Which of the following correctly represents the formula for the $$nth $$term of an arithmetic sequence?

Accepted Answer

$$a_n = a_{n-1} + d$$

Question 2

A biologist is tracking the population of a bacteria culture that doubles every hour. If the initial population is $$P_0$$, which formula best models the population after $$n$$ hours?

Accepted Answer

$$P_n = P_0$ \cdot 2^n$$

Question 3

Which of the following is the correct formula for the sum of the first $$n$$ terms of a finite geometric sequence with initial term $$g_1$$ and common ratio $$r 
eq 1$$?

Accepted Answer

$$S_n = \dfrac{g_1$(1 - r^n)}{1 - r}$$

Question 4

A recursively defined sequence is given by $$f_{n+2} = f_{n+1} + f_n$$ with $$f_1 = 1$ and f_2$ = 1. $$What is $$f_5$$?

Accepted Answer

5

Question 5

Which of the following is NOT a property of limits?

Accepted Answer

Chain Rule: $$\lim_{x 	o a} f(g(x)) = \lim_{x 	o a} f(x) \cdot \lim_{x 	o a} g(x)$$

Question 6

A function $$f(x) is $$continuous at $$x = a if$$:

Accepted Answer

$$\lim_{x 	o a} f(x)$$ exists and $$\lim_{x 	o a} f(x) = f(a)$$

Question 7

Which of the following is the correct derivative of $$f(x) = e$$^{2x}$$?

Accepted Answer

$$f'(x) = 2e$$^{2x}$$

Question 8

A pharmaceutical company models the concentration of a drug in the bloodstream by $$C(t) = 5e$$^{-0.2t}$$, where t$ is time in hours. What is the instantaneous rate of change of concentration at t = 3$ hours?

Accepted Answer

Question 9

Which of the following is the correct application of the chain rule for $$f(x) = \sin(3x^2$)$$?

Accepted Answer

$$f'(x) = \cos(3x^2$) \cdot 6x$$

Question 10

Given $$f(x) = x^2$ + 2xy + y^2$, which of the following is the correct expansion for (x$$ + $$y)^2$$?

Accepted Answer

$$x^2 + 2xy$$ + $$y^2$$

Question 11

If $$a_k = 2k$$, what is $$\sum_{k=1}^{4} a_k$$?

Accepted Answer

16

Question 12

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

$$f'(x) = \dfrac{2x}{x^2$ + 1}$$

Question 13

A car's position is given by $$s(t) = 3t^2$ + 2t + 1. $$What is the instantaneous velocity at $$t = 2$$ seconds?

Accepted Answer

14

Question 14

Which of the following is the correct derivative of $$f(x) = 	an(x)$$?

Accepted Answer

$$f'(x) = \sec^2$(x)$$

Question 15

If $$f(x) = x^3$ and g(x$$) = $$x^2$$, what is the derivative of $$f(x) \cdot g(x)$$?

Accepted Answer

Question 16

Which of the following is the correct expansion for $$(x + y + z)^2$?

Accepted Answer

$$x^2$$ + $$y^2$$ + $$z^2 + 2xy + 2xz + 2yz$

Question 17

Which of the following is the correct derivative of $$f(x) = \arcsin(x)$$?

Accepted Answer

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

Question 18

A company’s revenue over time is modeled by $$R(t) = 1000 \ln(t+1). $$What is the marginal revenue at $$t = 4$$?

Accepted Answer

$$R'(4) = 1000 \cdot \dfrac{1}{5}$$

Study Notes: Sequences, Limits, and Derivatives (Calculus I)

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