022 Exam 1 Sample A

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  boxed problem numbers  to go to a solution.

Definition of the Derivative

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

Implicit Differentiation

 Problem 2.  Use implicit differentiation to find $dy/dx$ at the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,0)} on the curve defined by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{3}-y^{3}-y=x} .

Continuity and Limits

 Problem 3.  Given a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)=\frac{x+5}{x^{2}-25}}  ,

(a) Find the intervals where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)} is continuous.

(b). Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow5}g(x)} .

Increasing and Decreasing

 Problem 4.  Determine the intervals where the function  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=4} , given the demand function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=\frac{200}{\sqrt{x}}}

and the cost function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^{3}-3x^{2}-5x+7} at the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3,-8)} .

Quotient and Chain Rule

 Problem 8.  Find the derivative of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{2}+30x} , and the current production quantity is 9 units.