# 022 Exam 1 Sample A

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This is a sample, and is meant to represent the material usually covered in Math 22 up to the first exam. An actual test may or may not be similar. Click on the blue problem numbers to go to a solution.

## Definition of the Derivative

Problem 1. Use the definition of derivative to find the derivative of ${\displaystyle f(x)={\sqrt {x-5}}}$.

## Implicit Differentiation

Problem 2. Use implicit differentiation to find ${\displaystyle dy/dx}$ at the point ${\displaystyle (1,0)}$ on the curve defined by ${\displaystyle x^{3}-y^{3}-y=x}$.

## Continuity and Limits

Problem 3. Given a function ${\displaystyle g(x)={\frac {x+5}{x^{2}-25}}}$ ,

(a) Find the intervals where ${\displaystyle g(x)}$ is continuous.
(b). Find ${\displaystyle \lim _{x\rightarrow 5}g(x)}$.

## Increasing and Decreasing

Problem 4. Determine the intervals where the function  ${\displaystyle h(x)=2x^{4}-x^{2}}$ is increasing or decreasing.

## Marginal Revenue and Profit

Problem 5. Find the marginal revenue and marginal profit at ${\displaystyle x=4}$, given the demand function

${\displaystyle p={\frac {200}{\sqrt {x}}}}$

and the cost function

${\displaystyle C=100+15x+3x^{2}.}$

Should the firm produce one more item under these conditions? Justify your answer.

## Related Rates (Word Problem)

Problem 6. A 15-foot ladder is leaning against a house. The base of the ladder is pulled away from the house at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when the base of the ladder is 9 feet from the house.

## Slope of Tangent Line

Problem 7. Find the slope of the tangent line to the graph of ${\displaystyle f(x)=x^{3}-3x^{2}-5x+7}$ at the point ${\displaystyle (3,-8)}$.

## Quotient and Chain Rule

Problem 8. Find the derivative of the function ${\displaystyle f(x)={\frac {(3x-1)^{2}}{x^{3}-7}}}$. You do not need to simplify your answer.

## Marginal Cost

Problem 9. Find the marginal cost to produce one more item if the fixed cost is \$400, the variable cost formula is ${\displaystyle x^{2}+30x}$, and the current production quantity is 9 units.