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AP Calculus AB Lesson 29: The First Fundamental Theorem of Calculus Lesson 29: The First Fundamental Theorem of Calculus In the last lesson we learned that if the derivative of F is f, then we call F an antiderivative of f . For example, since the derivative of F ( x ) x 2 is f ( x ) 2 x , we say that x 2 is an antiderivative of 2x . Notice, however, that 2x actually has many antiderivatives. x 2 , x 2 1, x 2 2 , etc. all have a derivative of 2x . In fact, if C is any d 2 x C 2 x 0 2 x , so any function of the form x 2 C is an constant, we have dx antiderivative of 2x . The function f x 2 x has a family of antiderivatives. We x discovered some members of that family by evaluating the definite integral 2t dt and 0 x 2t dt . By starting at different points, we came up with equations that were of the form 2 x 2 C , where C represented different y-intercepts, but all the equations had the same slope. Up until now we have evaluated the definite integral by approximating the area under the curve using left, right, or midpoint Riemann Sums or the Trapezoidal Rule. The smaller the change in x, or in other words, the more rectangles or trapezoids used to approximate the area for more and more intervals, better the approximation to the area under the curve. 3 Example Compute 2 x dx using left- and right-hand Riemann sums, with n = 100 1 and n = 500 intervals. Theorem 5.1: The Fundamental Theorem of Calculus or First Fundamental Theorem of Calculus b If f is continuous on the interval [a, b] and f t F' t then f t dt F b F a a b F b F a Total change in F t between t = a and t = b = F' t dt . a In words, the definite integral of a rate of change gives the total change. We can use this idea about the Fundamental Theorem of Calculus to estimate or to calculate the exact values. AP Calculus AB Lesson 29: The First Fundamental Theorem of Calculus Example 1: Your velocity is vt ln t 2 1 ft/sec for 0 t 3 . Estimate the distance traveled during this time. Example 2: The population of Mexico can be modeled by the function t P f ( t ) 67.381.026 where P is in millions of people and t is in years since 1980. Use this function to predict the population change of Mexico between the years 2000 and 2020. Example 3: Let F t represent a bacteria population which is 5 million at time t=0. After t hours, the population is growing at an instantaneous rate of 2t million bacteria per hour. Estimate the total increase in the bacteria population during the first hour, and the population at t = 1. Example 4: Using the First Fundamental Theorem of Calculus, we can compute the integral exactly. Since f x 2 x . We know that if F x x 2 , then F' x 2 x. 3 So, 2 x dx 1 1 Example 5: Calculate the exact value of t 4 dt . 0 Example 6: Calculate example 3 exactly. p. 270 (10 – 13) p. 274 (32, 33)