Q: A pharmaceutical company models the decay of a drug in the bloodstream using the formula $$A = A_0$ e$$^{-kt}$$, where A_0$ is the initial amount, k$ is the decay constant, and t$ is time in hours. If A_0$ = 100 mg$$, $$k = 0.2$$, and $$t = 5$$ hours, what is the amount of drug remaining?

$$A = 100 e$$^{-0.2 \times 5}$$ = 100 e$$^{-1}$$ \approx 36.79 mg$$

Q: A quadratic equation models the height $$h (in $$meters) of a thrown ball as $$h(t) = -4.9t^2$ + 20t + 2. At $$what time does the ball hit the ground?

$$t = \frac{-20 \pm \sqrt{20^2$ - 4 \times (-4.9) \times 2}}{2 \times (-4.9)} \approx 4.19$$ seconds

Q: A population of bacteria doubles every 3 hours. If the initial population is 500, what will it be after 9 hours?

$$P = 500 \times 2$$^{9/3}$$ = 500 \times 2^3$ = 500 \times 8 = 4000$$

Q: Given the function $$f(x) = x^3$ - 4x$$, determine the end behavior as $$x \to -\infty$$ and $$x \to \infty.$$

As $$x \to -\infty$$, $$f(x) \to -\infty$$; as $$x \to \infty$$, $$f(x) \to \infty$$

Q: A geometric sequence has first term $$a_1 = 5$ and common ratio r = 3$. What is the fourth term a_4$?

$$a_4 = 5$$ \times $$3^{3} = 5$$ \times 27 = 135$$

Q: A matrix $$A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} is $$multiplied by scalar $$k = 2. $$What is $$2A$$?

$$2A = \begin{pmatrix} 4 & 6 \\ 2 & 8 \end{pmatrix}$$

Question 1

A pharmaceutical company models the decay of a drug in the bloodstream using the formula $$A = A_0$ e$$^{-kt}$$, where A_0$ is the initial amount, k$ is the decay constant, and t$ is time in hours. If A_0$ = 100 mg$$, $$k = 0.2$$, and $$t = 5$$ hours, what is the amount of drug remaining?

Accepted Answer

$$A = 100 e$$^{-0.2 	imes 5}$$ = 100 e$$^{-1}$$ \approx 36.79 mg$$

Question 2

A quadratic equation models the height $$h (in $$meters) of a thrown ball as $$h(t) = -4.9t^2$ + 20t + 2. At $$what time does the ball hit the ground?

Accepted Answer

$$t = \frac{-20 \pm \sqrt{20^2$ - 4 	imes (-4.9) 	imes 2}}{2 	imes (-4.9)} \approx 4.19$$ seconds

Question 3

Which of the following functions has a vertical asymptote at $$x = 2$$ and a horizontal asymptote at $$y = 0$$?

Accepted Answer

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

Question 4

A population of bacteria doubles every 3 hours. If the initial population is 500, what will it be after 9 hours?

Accepted Answer

$$P = 500 	imes 2$$^{9/3}$$ = 500 	imes 2^3$ = 500 	imes 8 = 4000$$

Question 5

Given the function $$f(x) = x^3$ - 4x$$, determine the end behavior as $$x 	o -\infty$$ and $$x 	o \infty.$$

Accepted Answer

As $$x 	o -\infty$$, $$f(x) 	o -\infty$$; as $$x 	o \infty$$, $$f(x) 	o \infty$$

Question 6

A piecewise function is defined as $$f(x) = \begin{cases} x^2$ & x \u003c 0 \ 2x + 1 & x \geq 0 \end{cases}. $$What is $$f(-3)$$?

Accepted Answer

$$f(-3) = (-3)^2$ = 9$$

Question 7

Solve the system of equations using the addition method: $$2x + 3y = 7$$ and $$4x - 3y = 1.$$

Accepted Answer

$$x = 2, y = 1$$

Question 8

A geometric sequence has first term $$a_1 = 5$ and common ratio r = 3$. What is the fourth term a_4$?

Accepted Answer

$$a_4 = 5$$ 	imes $$3^{3} = 5$$ 	imes 27 = 135$$

Question 9

Which of the following is the correct domain for the function $$f(x) = \frac{x-1}{x^2$-4}$$?

Accepted Answer

All real numbers except $$x = 2$$ and $$x = -2$$

Question 10

A company’s revenue is modeled by $$R(x) = 2x^2$ + 5x + 10. $$What is the $$y-$$intercept of the graph?

Accepted Answer

$$y-$$intercept is $$R(0) = 10$$

Question 11

Given $$f(x) = x^2$ - 6x + 9$$, what is the vertex of the parabola?

Accepted Answer

Vertex is $$(3, 0)$$

Question 12

A sequence is defined by $$a_n = 4 + 2(n-1). $$What is the sum of the first four terms?

Accepted Answer

Sum is $$4 + 6 + 8 + 10 = 28$$

Question 13

Which of the following functions is even?

Accepted Answer

$$f(x) = x^2$

Question 14

If $$\log_{2} x = 5$$, what is $$x$$?

Accepted Answer

$$x = 32$$

Question 15

A matrix $$A = \begin{pmatrix} 2 & 3 \ 1 & 4 \end{pmatrix} is $$multiplied by scalar $$k = 2. $$What is $$2A$$?

Accepted Answer

$$2A = \begin{pmatrix} 4 & 6 \ 2 & 8 \end{pmatrix}$$

Question 16

Find the sum $$\sum_{k=1}^{4} (2k + 1).$$

Accepted Answer

$$3 + 5 + 7 + 9 = 24$$

Question 17

A rational function $$f(x) = \frac{3x}{x^2$ - 1}$$ has what type of symmetry?

Accepted Answer

Odd symmetry (origin symmetry)

Question 18

If $$f(x) = x^2$ - 4$$, what is the multiplicity of the zero at $$x = 2$$?

Accepted Answer

Multiplicity is 1 (odd)

Question 19

Solve for $$x$$ using the square root property: $$x^2 = 49$.

Accepted Answer

$$x = \pm 7$$

College Algebra Final Exam Study Guide

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