Applications of Derivatives

Written Assignment #6 MAT 201 Calculus 1 Chapter 4 Applications of Derivatives (4.1, 4.2, 4.3) Name _______________________________ Directions: Print the assignment or copy the problems onto paper. Show all the steps in your work. There are 4 questions for a total of 50 points: #1 and #2 are worth 10 points each, and #3 and #4 are worth 15 points each. Problems are graded for accuracy. Scan your completed assignment or take a picture of the pages, then email to adelvitto@ccac.edu Note: Written Assignments are the only way that I can view and evaluate your written work. Write neatly and show all your steps. No Work = No Credit Clearly mark your final answer. If more workspace is needed, add additional sheet(s) of paper with the problem numbers clearly marked. Attach the additional sheet(s) β€œclosest” to the problem number, NOT at the end. 1) Find the Absolute Maximum and Absolute Minimum values of 3 𝑓(π‘₯ ) = π‘₯ βˆ’ √π‘₯ on the interval [-1, 3] using the Closed Interval Method. (4.1) 2) Choose one problem and solve it: (4.2) A) Verify the three hypotheses of Rolle’s Theorem are satisfied on the interval [-1, 1], and find all numbers β€œc” described by the theorem for 𝑓(π‘₯) = π‘₯ 2 βˆ’1 π‘₯+2 . B) Verify the two hypotheses of the Mean Value Theorem are satisfied on the interval [3, 8], and find all numbers β€œc” described by the theorem for 𝑓(π‘₯) = √π‘₯ + 1. 7 3) Consider the function 𝑓 (π‘₯ ) = √π‘₯ (π‘₯ + 8). Show all the steps to complete each category below, or write β€œnone” if applicable (there must be an answer for all categories). Use exact values in your answers. Show all steps when calculating derivatives. Show how you arrived at the conclusions of the First and/or Second Derivative Tests. Finally, complete the Summary Page at the end of the problem, and sketch the graph. (4.3) x-intercept(s): y-intercept First Derivative: Critical Number(s): Intervals of Increasing: Intervals of Decreasing: 3) Continued …………………………………. 7 Consider the function 𝑓 (π‘₯ ) = √π‘₯ (π‘₯ + 8). Show all the steps to complete each category below, or write β€œnone” if applicable (there must be an answer for all categories). Use exact values in your answers. Show all steps when calculating derivatives. Show how you arrived at the conclusions of the First and/or Second Derivative Tests. Finally complete the Summary Page at the end, and sketch the graph. Local Minima: Local Maxima: Second Derivative: 3) Continued ……………………… 7 Consider the function 𝑓 (π‘₯ ) = √π‘₯ (π‘₯ + 8). Show all the steps to complete each category below, or write β€œnone” if applicable (there must be an answer for all categories). Use exact values in your answers. Show all steps when calculating derivatives. Show how you arrived at the conclusions of the First and/or Second Derivative Tests. Finally complete the Summary Page at the end, and sketch the graph. Interval(s) of Concave Up: Interval(s) of Concave Down: Inflections Points: Summary Page for the function 7 𝑓(π‘₯ ) = √π‘₯ (π‘₯ + 8): x-intercept(s) _____________________________________ y-intercept _____________________________________ First Derivative _____________________________________ Critical Number(s) _________________________________ Interval(s) of Increasing ________________________________ Interval(s) of Decreasing ________________________________ Local Min Point(s ) _____________________________________ Local Max Point(s) _____________________________________ Second Derivative _____________________________________ Interval(s) of Concave Up ________________________________ Interval(s) of Concave Down ________________________________ Inflection Point(s) _____________________________________ 7 Sketch the graph of 𝑓 (π‘₯ ) = √π‘₯ (π‘₯ + 8), labeling all intercepts, inflection points, local max points, and local min points. Make sure an observer of your graph can see and identify the intervals of concavity by accentuating all β€œtwists and turns” in the graph. π‘₯ 3 βˆ’1 4) Consider the function 𝑓 (π‘₯ ) = 3 . Show all the steps to complete each π‘₯ +1 category below, or write β€œnone” if applicable (there must be an answer for all categories). Use exact values in your answers. Show all steps when calculating derivatives. Show how you arrived at the conclusions of the First and/or Second Derivative Tests. Finally, complete the Summary Page at the end of the problem, and sketch the graph. (4.3) Domain of 𝑓: x-intercept(s): y-intercept Vertical Asymptote(s): Horizontal asymptote: First Derivative: Critical Numbers: 4) Continued ……………. Consider the function 𝑓(π‘₯ ) = π‘₯ 3 βˆ’1 . Show all the steps to complete each π‘₯ 3 +1 category below, or write β€œnone” if applicable (there must be an answer for all categories). Use exact values in your answers. Show all steps when calculating derivatives. Show how you arrived at the conclusions of the First and/or Second Derivative Tests. Finally, complete the Summary Page at the end of the problem, and sketch the graph. Intervals of Increasing: Intervals of Decreasing: Local Min Point(s): Local Max Point(s): 4) Continued ……………. Consider the function 𝑓(π‘₯ ) = π‘₯ 3 βˆ’1 . Show all the steps to complete each π‘₯ 3 +1 category below, or write β€œnone” if applicable (there must be an answer for all categories). Use exact values in your answers. Show all steps when calculating derivatives. Show how you arrived at the conclusions of the First and/or Second Derivative Tests. Finally, complete the Summary Page at the end of the problem, and sketch the graph. Second Derivative: Intervals of Concave Up: Intervals of Concave Down: Inflections Point(s): Summary Page for the function 𝑓(π‘₯ ) = π‘₯ 3 βˆ’1 π‘₯ 3 +1 x-intercept(s) _____________________________________ y-intercept _____________________________________ Vertical Asymptote(s) __________________________________ Horizontal Asymptote _________________________________ First Derivative _____________________________________ Critical Number(s) _________________________________ Interval(s) of Increasing ________________________________ Interval(s) of Decreasing ________________________________ Local Min Point(s ) _____________________________________ Local Max Point(s) _____________________________________ Second Derivative _____________________________________ Interval(s) of Concave Up ________________________________ Interval(s) of Concave Down ________________________________ Inflection Point(s) _____________________________________ Sketch the graph of 𝑓(π‘₯ ) = π‘₯ 3 βˆ’1 , labeling all intercepts, asymptotes, inflection π‘₯ 3 +1 points, local max points, and local min points. Make sure an observer of your graph can see and identify the intervals of concavity by accentuating all β€œtwists and turns” in the graph.
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