Q: Suppose the graph of $$f(x)$$ has a jump at $$x = 2$$ such that $$\lim_{x \to 2^-} f(x) = 3$$ and $$\lim_{x \to 2^+} f(x) = 5. $$What can you conclude about $$\lim_{x \to 2} f(x)$$?

$$\lim_{x \to 2} f(x)$$ does not exist

Question 1

Suppose the graph of $$f(x)$$ has a jump at $$x = 2$$ such that $$\lim_{x 	o 2^-} f(x) = 3$$ and $$\lim_{x 	o 2^+} f(x) = 5. $$What can you conclude about $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$\lim_{x 	o 2} f(x)$$ does not exist

Question 2

Evaluate the following limit using algebraic methods: $$\lim_{x 	o 2} \frac{2x^2$ - 3x - 2}{x^2$ + x - 6}.$$

Accepted Answer

$$1$$

Question 3

Given $$f(x) = \frac{x-4}{x^2$ + 2x}$$, which of the following is NOT a partition number for $$f(x)$$?

Accepted Answer

$$x = 2$$

Question 4

Solve the inequality $$\frac{x-4}{x^2$ + 2x} \u003c 0$$ and express your answer in interval notation.

Accepted Answer

$$(-\infty, -2) \cup (0, 4)$$

Question 5

Using the definition of the derivative, find $$f'(x)$$ for $$f(x) = 2x^2$ + 8.$$

Accepted Answer

$$4x$$

Question 6

Find the derivative of $$f(x) = 4\sqrt{x} + 2$$ using differentiation formulas.

Accepted Answer

$$2x^{-1/2}$$

Question 7

Given the price-demand equation $$x = 4000 - 40p$$, express the price $$p in $$terms of demand $$x.$$

Accepted Answer

$$p = 100 - 0.025x$$

Question 8

For the price-demand equation $$x = 4000 - 40p$$, what is the domain of the price function $$p(x)$$?

Accepted Answer

$$0 \leq x \leq 4000$$

Question 9

Given $$f(x) = \frac{2x^5$ - 4x^3$ + 2x}{x^3$}$$, find $$f'(x)$$ using differentiation formulas.

Accepted Answer

$$10x^2 - 12x$$ + $$2x^{-2}$$

Question 10

Suppose the revenue function for a product is $$R(x) = x \left(100 - 0.025x\right). $$Find the marginal revenue at $$x = 1600.$$

Accepted Answer

$$R'(1600) = 60$$

Question 11

Interpret the meaning of the marginal revenue $$R'(1600) = 60 in $$the context of the shoe company.

Accepted Answer

Selling one more pair when 1600 pairs are sold increases revenue by approximately $60.

Question 12

Find the derivative of $$y = \frac{5x^3$}{4} - \frac{2}{5}x^2$.

Accepted Answer

$$\frac{15x^2$}{4} - \frac{4x}{5}$$

Question 13

Evaluate $$\lim_{x 	o 0} \frac{2x^2$ - 3x - 2}{x^2$ + x - 6}.$$

Accepted Answer

$$\frac{-2}{-6} = \frac{1}{3}$$

Question 14

Which of the following best describes the relationship between one-sided limits and the two-sided limit at $$x = a$$?

Accepted Answer

The two-sided limit exists if and only if both one-sided limits exist and are equal.

Question 15

Find the derivative of $$f(x) = 3x - \frac{5}{x} + 2\sqrt{x}.$$

Accepted Answer

$$3 + 5x$$^{-2}$$ + x$$^{-1/2}$$

Question 16

If $$f(x) = 4x - x^2$, use the definition of the derivative to find f$$'(x)$$.

Accepted Answer

$$4 - 2x$$

Question 17

Given $$f(x) = x^2$ + 2x$$, which of the following is the correct sign chart for $$f(x)$$?

Accepted Answer

Negative for $$x  -2$$

Question 18

Suppose the revenue function is $$R(x) = x(100 - 0.025x). $$For what value of $$x is $$the marginal revenue zero?

Accepted Answer

$$x = 2000$$

Question 19

If $$f(x) = \sqrt{x}$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{1}{2}x$$^{-1/2}$$

Business Calculus Exam 1 Study Guide: Limits, Derivatives, and Marginal Analysis

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