Problem #3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page Mark Sparks 2012

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1 Problem # Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 490 Mark Sparks 01

2 Finding Anti-derivatives of Polynomial-Type Functions If you had to explain to someone how to find the derivative of a polynomial-type function, what would you say? To find the anti-derivative, you would do the opposite of each one of those operations and in the reverse order. Therefore, to find the anti-derivative of a polynomial-type function The anti-derivative of a function, f(x), is denoted by the notation f ( x) dx. So when finding the antiderivative of a function, you are finding the function of which f(x) is the first derivative. This will enable us, if given f ' or f " to be able to find f. However, if f '( x) dx f ( x), what problem do you foresee? Find each of the following anti-derivatives. x x dx x x4 dx x ( x )(x ) dx dx x Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 491 Mark Sparks 01

3 d dx sine and cosine are. d dx We learned that sin x cos x and cos x sin x. Similarly, write what the anti-derivatives of cos xdx sin xdx Find each of the following anti-derivatives. sin x cos x dx t sin t dt 4 x cos x dx x sin x dx Use the given information about f ' and f " to find f(x). 1. f "( x) f '() 5 f () = 10. / f "( x) x f '(4) f (0) = 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 49 Mark Sparks 01

4 An evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by the differential equation dh 1.5t 5, dt where t is the time in years and h is the height in centimeters. The seedlings are 1 centimeters tall when planted, at t = 0. a. Find the value of the differential equation above when t =. Using correct units of measure, explain what this value represents in the context of this problem. b. Find an equation for h(t), the height of the shrubs at any year t. Then, determine how tall the shrubs are when they are sold. A particle moves along the x axis at a velocity of 1 v( t), for t > 0. At time t = 1, its position is 4. t a. What is the acceleration of the particle b. What is the position of the particle when t = 9? when t = 9? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 49 Mark Sparks 01

5 Riemann Sums A Graphical Approach to Approximating the Definite Integral Calculating Riemann sums is a way to estimate the area under a curve for a graphed function on a particular interval. In this activity, you will learn to calculate four types of Riemann sums: Left Hand, Right Hand, Midpoint, and Trapezoidal Sums. Approximation #1 Left Hand Riemann Sum with intervals of length units Let s consider for a moment the function f(x) = 1 x x 5. Graph this function on the axes provided below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length units. Place the upper left hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = 1 x x 5, x = 1, x = 7, and the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 494 Mark Sparks 01

6 Approximation # Left Hand Riemann Sum with intervals of length 1 unit Again, consider for a moment the function f(x) = 1 x x 5. Graph this function on the axes provided below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length1 unit. Place the upper left hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = 1 x x 5, x = 1, x = 7, and the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 495 Mark Sparks 01

7 Approximation # Left Hand Riemann Sum with intervals of length ½ unit Again, consider for a moment the function f(x) = 1 x x 5. Graph this function on the axes provided below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length ½ unit. Place the upper left hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = 1 x x 5, x = 1, x = 7, and the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 496 Mark Sparks 01

8 Approximation #4 Right Hand Riemann Sum with intervals of length units Let s consider for a moment the function f(x) = 1 x x 5. Graph this function on the axes provided below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length units. Place the upper right hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = 1 x x 5, x = 1, x = 7, and the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 497 Mark Sparks 01

9 Approximation #5 Right Hand Riemann Sum with intervals of length 1 unit Let s consider for a moment the function f(x) = 1 x x 5. Graph this function on the axes provided below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 1 unit. Place the upper right hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = 1 x x 5, x = 1, x = 7, and the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 498 Mark Sparks 01

10 Approximation #6 Right Hand Riemann Sum with intervals of length ½ unit Let s consider for a moment the function f(x) = 1 x x 5. Graph this function on the axes provided below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length ½ unit. Place the upper right hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = 1 x x 5, x = 1, x = 7, and the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 499 Mark Sparks 01

11 Approximation #7 Midpoint Riemann Sum with intervals of length units Let s consider for a moment the function f(x) = 1 x x 5. Graph this function on the axes provided below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length units. Place the midpoint of each rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = 1 x x 5, x = 1, x = 7, and the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 500 Mark Sparks 01

12 Approximation #8 Midpoint Riemann Sum with intervals of length 1 unit Let s consider for a moment the function f(x) = 1 x x 5. Graph this function on the axes provided below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 1 unit. Place the midpoint of each rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = 1 x x 5, x = 1, x = 7, and the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 501 Mark Sparks 01

13 Approximation #9 Trapezoidal Riemann Sum with intervals of length units Let s consider for a moment the function f(x) = 1 x x 5. Graph this function on the axes provided below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into right trapezoids of length units. Place the upper left and right hand vertices of the trapezoids on the curve each time. Then, calculate the area of each trapezoid and sum the areas to approximate the area of the region under the curve bounded by f(x) = 1 x x 5, x = 1, x = 7, and the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 50 Mark Sparks 01

14 Approximation #10 Trapezoidal Riemann Sum with intervals of length 1 unit Let s consider for a moment the function f(x) = 1 x x 5. Graph this function on the axes provided below. On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into right trapezoids of length 1 unit. Place the upper left and right hand vertices of the trapezoids on the curve each time. Then, calculate the area of each trapezoid and sum the areas to approximate the area of the region under the curve bounded by f(x) = 1 x x 5, x = 1, x = 7, and the x-axis. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 50 Mark Sparks 01

15 Type of Riemann Sum Left Hand Riemann Sum Intervals of units Numerical Approximation Do you think this is an over or under approximation of the area? At this point, place a star next to the approximation that you feel is the most accurate to the actual area. In our next lesson, we will learn how to find the EXACT area of a region between the graph of a function and the x axis. Left Hand Riemann Sum Intervals of 1 unit Left Hand Riemann Sum Intervals of ½ unit In the space below, we will come back to this to find the exact area once we complete the next lesson in order to see which approximation to the left is the most accurate. Right Hand Riemann Sum Intervals of units Right Hand Riemann Sum Intervals of 1 unit Right Hand Riemann Sum Intervals of ½ unit Midpoint Riemann Sum Intervals of units Midpoint Riemann Sum Intervals of 1 unit Trapezoidal Riemann Sum Intervals of units Trapezoidal Riemann Sum Intervals of 1 unit Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 504 Mark Sparks 01

16 Given the table of values below, approximate each definite integral by finding the indicated Riemann Sum. 5 a. Approximate f ( x) dx using a midpoint sum 0 and three subintervals. x f(x) b. Approximate f ( x) dx 0 and four subintervals. using a left hand sum x f(x) c. Approximate f ( x) dx 4 and four subintervals. using a right hand sum x f(x) d. Approximate f ( x) dx using a trapezoidal sum 0 and three subintervals x f(x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 505 Mark Sparks 01

17 Applying the Fundamental Theorem of Calculus Connecting the Graphical, Analytical, and Numerical Approaches The Fundamental Theorem of Calculus, Part I Consider the function f(x) = x + whose graph is pictured below. Calculate each of the following definite integrals according to the Fundamental Theorem of Calculus. Then, shade the area of the region that the integral represents. 1 x dx. Find 5 Find x dx. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 506 Mark Sparks 01

18 4 Find 1 x dx. Find 1 x dx. 5 Find x dx. 0 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 507 Mark Sparks 01

19 Based on the results of the five previous examples, what inferences can you make about the value of the definite integral and the amount of area bounded by the graph of the integrand and the x axis? Find each of the following definite integrals applying the fundamental theorem of calculus. Show your work. Then, use your graphing calculator to verify your results. x x dx 1 xdx 1 x 4 1 x dx x x 0 cos x dx Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 508 Mark Sparks 01

20 Pictured below is a table of values show the values of a function, f(x), and its first and second derivative for selected values of x. Use the information in the table to answer the questions that follow. 1. What is the value of 1 f '( x) dx. 5. What is the value of f '( x) f ''( x) dx?. What is the value of f ''( x) dx? What is the value of 1 f '( x) f ''( x ) dx? 5. What is the equation of the tangent line to the graph of f(x) at x =? 6. Use the equation of the tangent line in #5 to approximate the value of f(.1). Is this an over or under approximation of f(.1)? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 509 Mark Sparks 01

21 Properties of Definite Integrals Given the integral statements, write what you think each is equivalent to. Be prepared to explain your reasoning with the rest of the class. a a 1. f ( x) dx. Given that a < c < b, f ( x) dx = b b. If f ( x) dx K, then f ( x) dx = a a 4. Given that b < a, then f ( x) dx = b a 5. If k is a constant, then b k f ( x) dx = a b 6. f ( x) g( x) dx = a 7. Given that f(x) is an even function, a f ( x) dx = a a b 8. Given that f(x) is an odd function, f ( x) dx = a a Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 510 Mark Sparks 01

22 7 If f ( x) dx 6 and f ( x) dx 8, determine the value of each of the following integrals using the 0 properties of definite integrals. Explain how you arrived at your answer for each. 0 7 f ( x) dx 0 f ( x) dx f ( x) dx f ( x) dx 7 7 ( f ( x)) dx f ( x) dx, if f(x) is an even function f ( x) dx, if f(x) is an odd function Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 511 Mark Sparks 01

23 Pictured to the right is the graph of a function f(x). What is the value of 0 f ( x) dx? 4 What is the value of 0 f ( x) dx? What is the value of f ( x) dx? If F(0) = 5, what is the value of F(), where F is the anti-derivative of f(x)? If F( ) =, what is the value of F(), where F is the anti-derivative of f(x)? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 51 Mark Sparks 01