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Clemson MATH 2070 - Chapter 5 Mixed Practice

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MATH 2070 (Fall 2017) Chapter 5 Mixed Practice 1 1. A small college is looking at data on the sale of tickets to their home football games. The following data shows the rate of change in the sale of tickets for a given ticket price. Ticket Price (dollars) 10 15 20 25 30 35 40 ROC of Sales (tickets/dollar) 183 138 103 80 58 45 33 a. Find an exponential model for the data. Define the model completely. b. Draw 4 left rectangles that you would use to estimate the change in sales from $10 to $30. d. Using the unrounded function, use four midpoint rectangles to find the change in sales when the ticket price increases from $10 to $30. Sketch the four rectangles on the graph below. Interpret your answer. f. Interpret your answer to d or e. c. Shade the area that represents the exact change in sales when the ticket price increases from $10 to $30.MATH 2070 (Fall 2017) Chapter 5 Mixed Practice 2 2. The editors of a popular consumer buying guide tested a particular brand of air purifier for its effectiveness in removing smoke from the air in a cigar bar. '( )S x gives the rate of change of the amount of smoke in the air, measured in percent (of the original amount) per minute, where x is the number of minutes since the test began. The graph of '( )S x is shown below as well as the areas of the three regions trapped between the curve and the x-axis. Use the information from the graph to complete the questions that follow. If it is not possible to determine a value, write NA. a. Over what time interval(s) did the amount of smoke in the air decrease? b. Over what time interval(s) did the amount of smoke in the air increase? c. At what time was the amount of smoke in the air decreasing most rapidly? d. At what time was the amount of smoke in the air increasing most rapidly? e. When was the amount of smoke in the air at a relative minimum? f. When was the amount of smoke in the air at a relative maximum? g. Interpret the area marked A1 above. h. Give the value for each definite integral. 180( ) ______S x dx 2518( ) _______S x dx 3025( ) ______S x dx i. Use definite integral notation to write an equation that gives the overall change in the amount of smoke over the 30 minute test. Be sure to include both sides of the equation (notation = value). j. Use definite integral notation to write an equation that gives the total area trapped between the curve and the x-axis from x = 0 to x = 30. Be sure to include both sides of the equation (notation = value). k. Suppose the concentration of smoke in the air at the beginning of the test was 0.5 mg/m3 . What was the concentration of smoke at the end of the 30 minute test? A1 ≈ 76 A2 ≈ 4 A3 ≈ 21MATH 2070 (Fall 2017) Chapter 5 Mixed Practice 3 On what interval would F(x) be: Increasing? ________________ Decreasing? _______________ Concave up? _______________ Concave down? _____________ At what x-value would F(x) have a: Relative Extrema?__________ Inflection point? ___________ x-intercept? ______________ 3. 3. ______________________ ___________________ ___________________ ___________________________ ________________________ ________________________ ___________________________ ________________________ ________________________ ___________________________ ________________________ ________________________ 4. Given the graph of ( )f t below, sketch the graph of the accumulation function 2( ) ( )xF x f t dt and identify the properties of the accumulation function. Consider the graph of f(t) shown to the left. The graph of -1( )xf t dt is _______ (a, b or c) For the two graphs you didn’t choose, explain why they are not the accumulation function -1( )xf t dt on the lines provided. f(t) t a b c 3.2 6.5MATH 2070 (Fall 2017) Chapter 5 Mixed Practice 4 5. Complete the following chart. ( )f x '( )f x ( )f x dx ( ) 5f x ( )f x x x  3( ) 5 7x xf x e e e   ( ) 2xf x 3( )5f xx ( ) lnf x x 6. Explain why each of the following are incorrect and then correctly find the general antiderivative. a. 5 5dt x C  b.  2 322 22 3x xx x dx x C       c. 24( 2 ) 4x xe x dx x e x C    d. 1221ln | | C1xdx x dx xx      e. 5 55 131 3 3ln | |5 5x xxe ee dx x x C x x Cx             f. 4 2322 52 54 22x xx xdx Cxx  7. Rework / Review your Section 5.4 and 5.5 learning activities for more practice with general antiderivatives.MATH 2070 (Fall 2017) Chapter 5 Mixed Practice 5 8. Determine each of the following. a. 53( )mdh t dtdm    b.  325 7xdt t dtdx     c. 2753( )dh t dtdm    d.  7 9dx dxdx    e.  5 29xdx e dxdx   f. Use the algebraic method to determine  33x14 e .x dx g. Find ( )F x given 4( )f xx and (1) 5F . 9. Given a function ( )f x, explain how you would find the total trapped area between ( )f x and the x – axis on the interval x = a to x = b. 10. Given the functions ( )f x and ( )g x, explain how you would find the total trapped area between ( )f x and ( )g x on the interval x = a to x = b. (Section 5.7)MATH 2070 (Fall 2017) Chapter 5 Mixed Practice 6 11. Find the total area between the function and the horizontal axis between a and b and then find ( )baf x dx. a. The rate of change in the U.S. petroleum production between 2000 and 2008 can be modeled as 0.056 0.031 0 5'( )0.12 .79 5 8t tP tt t       million barrels per day per year where t is the number of years since 2000. USE GEOMETRIC FORMULAS. [a = 0 and b = 8] b. 3 2( ) 1.3 0.93 0.49; 1.5, 2f x x x a b       12. Write a sentence of interpretation for your answer to number 11a.MATH 2070 (Fall 2017) Chapter 5 Mixed Practice 7 13. The function 2( ) 6.27 48.8 3.17f x x


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