Question 1

Simplify the following expression and write your answer without negative exponents:

$$\frac{2x^3$y^2$$}{$$x^2y$$^{-3}$$}$$

Accepted Answer

$$2x^1y^5$

Question 2

Factor the polynomial $$x^2 - 4x - 21$ into the product of first-degree factors with integer coefficients.

Accepted Answer

$$(x - 7)(x + 3)$$

Question 3

Given the function $$f(x)$$ shown in the graph (piecewise linear with a jump at $$x=3$$), what is $$\lim\limits_{x 	o 3^-} f(x)$$?

Accepted Answer

$$2$$

Question 4

Given the same function $$f(x)$$ from the graph, what is $$\lim\limits_{x 	o 3^+} f(x)$$?

Accepted Answer

$$4$$

Question 5

For the function $$f(x) in $$the graph, does $$\lim\limits_{x 	o 3} f(x)$$ exist?

Accepted Answer

No, because the left and right limits are not equal

Question 6

Let $$f(x) = \frac{x+3}{x-3}. $$What is $$\lim\limits_{x 	o 3^-} f(x)$$?

Accepted Answer

$$-\infty$$

Question 7

Let $$f(x) = \frac{x+3}{x-3}. $$What is $$\lim\limits_{x 	o 3^+} f(x)$$?

Accepted Answer

$$\infty$$

Question 8

Let $$f(x) = \frac{x+3}{x-3}. $$Does $$\lim\limits_{x 	o 3} f(x)$$ exist?

Accepted Answer

No, because the left and right limits are not equal

Question 9

Suppose a business's profit $$P (in $$thousands of dollars) is given by $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold (in hundreds). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $$18 at$$ $$x=2$$

Question 10

A company’s revenue function is $$R(x) = 50x - x^2$, where x$ is the number of items sold. What is the marginal revenue when x = 10$?

Accepted Answer

$$30$$

Question 11

If the cost function for producing $$x$$ units is $$C(x) = 5x + 200$$, what is the average cost per unit when $$x = 20$$?

Accepted Answer

$$10$$

Question 12

Given $$f(x) = x^2$ - 4x + 3$$, find all $$x$$ such that $$f(x) = 0.$$

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 13

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 9}{x - 3}$$ for $$x 
eq 3. $$What is $$\lim\limits_{x 	o 3} f(x)$$?

Accepted Answer

$$6$$

Question 14

If $$g(x) = 2x^2$ - 8x + 6$$, what is the minimum value of $$g(x)$$?

Accepted Answer

$$-2$$

Question 15

Suppose the demand function for a product is $$p = 100 - 2x$$, where $$p is $$the price and $$x is $$the quantity sold. What is the price when $$x = 30$$?

Accepted Answer

$$40$$

Question 16

If $$h(x) = x^3$ - 6x^2$ + 9x$$, find the critical points.

Accepted Answer

$$x = 0$$ and $$x = 3$$

Question 17

A company’s cost function is $$C(x) = x^2$ + 10x + 100. $$What is the marginal cost when $$x = 5$$?

Accepted Answer

$$15$$

Question 18

If $$f(x) = \frac{1}{x}$$, what is $$\lim\limits_{x 	o 0^+} f(x)$$?

Accepted Answer

$$\infty$$

Question 19

Suppose the supply function for a product is $$p = 20 + 3x. $$What is the quantity supplied when the price is $$50$$?

Accepted Answer

$$15$$

Business Calculus: Limits, Function Evaluation, and Polynomial Factoring

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