CALCULUS AP AB Q401.Chapter 5B Lesson 1: Fundamental Theorem (Part 1) Fundamental Theorem of Calculus (Part I)

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1 CALCULUS AP AB Q401.Chapter 5B Lesson 1: Fundamental Theorem (Part 1) Fundamental Theorem of Calculus (Part I)

2 CALCULUS AP AB Q401.Chapter 5B Lesson 1: Fundamental Theorem (Part 1) APPLICATION (1, 4) 2 (2, 1) -2 2 (4, -1) -2 The graph of the function f, consisting of three line segments, is given above. Let ( ) ( ). A. Compute g(4) and g(-2) B. Find the instantaneous rate of change of g, with respect to x, at x = 1. C. Find the absolute minimum value of g on the closed interval [-2, 4]. Justify your answer. D. The second derivative of g is not defined at x = 1 and x = 2. How many of these values are x- coordinates of points of inflection of the graph of g? Justify your answer. g x = x 1 f t dt

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6 Approximating a Definite Integral with a Riemann Sum RECTANGLE APPROXIMATION b n k = 1 ( ) [ ] f ( x) dx f ck xk on a, b where f ( c k ) is the value of f at x = c on the kth interval. a TRAPEZOIDAL APPROXIMATION b x f ( x) dx ( y0 + 2y1+ 2y yn 1+ yn) where y = f( x) and x is constant. 2 a

7 1. Consider the area under the curve (bounded by the x-axis) of 2 f ( x) = x from = 1 x to x = 5. Use 4 equal rectangles whose heights are the left endpoint of each rectangle to approximate the area. (LRAM) Use 4 equal rectangles whose heights are the right endpoint of each rectangle to approximate the area. (RRAM) Use 4 equal rectangles whose heights are the midpoint of each rectangle to approximate the area. (MRAM)

8 2. Use LRAM, RRAM, and MRAM with n = 3 rectangles to estimate the area under the curve of f (x) over the interval [2,3.5] when the function values are as given in the table. x y

9 3. t (hours) R(t) (gallons per hours) The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time t. The table above measured every 3 hours for a 24-hour period. 24 A. Using correct units, explain the meaning of the integral R( t) dt in terms of water flow B. Use a trapezoidal approximation with 4 subdivisions of equal length to approximate R ( t) dt. 0 C. Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate the average of R(t). 1 2 D. The rate of water flow R(t) can be approximated by Q( t) ( t t ) =. Use Q(t) to 79 approximate the average rate of water flow during the 24-hour time period. Indicate the units of measure.

10 Time (days) Oil Consumption Rate Q (t) (liters/hour) 4. A diesel generator runs continuously, consuming oil at a gradually increasing rate Q (t), in liters per hour at time t days, until it must be temporarily shut down to have the filters replaced. Select values of Q for 0 t 7 are shown in the table above. Hint: Convert time to hours. A. Use a right-endpoint Riemann sum with seven subintervals to approximate correct units, explain the meaning of your answer in terms of oil Q() t dt. Using B. Use part A to estimate the average consumption rate over the week from which the data was collected. 5. Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at the rate of y( t) = t + 1 gallons per minute, for 0 t 30 minutes. At time t = 0, the tank contains 30 gallons of water. 30 A. Estimate y ( t) dt using a midpoint Riemann sum with 6 subdivisions of equal length. Using 0 correct units, explain the meaning of your answer in terms of water flow. B. Use part A to estimate the average leak rate.

11 6. An automobile computer gives a digital readout of fuel consumption in gallons per hour. During a trip, a passenger recorded the fuel consumption every 5 minutes for a full hour of travel. Hint: Convert time to hours. time (minutes) gal/h A. Use a midpoint Rieman sum with six subintervals of equal length and values from the table to approximate the total fuel consumption during the hour. B. Use a trapezoidal sum with six subintervals of equal length and values from the table to approximate the total fuel consumption during the hour. C. Use part B to estimate the fuel efficiency (average miles per gallon) over the hour. The automobile traveled 60 miles in the hour # FormB # #2