AP Calculus AB Unit 6 Packet Antiderivatives. Antiderivatives

Transcription

1 Antiderivatives Name In mathematics, we use the inverse operation to undo a process. Let s imagine undoing following everyday processes. Process Locking your car Going to sleep Taking out your calculator Inverse Giving Mrs. Sapp a gift You have learned many inverse operations throughout your algebra courses. Write down as many as you can think of: Function Inverse As you can probably imagine, antidifferentiation refers to the process of undoing the derivative. The act of taking antiderivatives is called. We use the symbol to represent the integral. NOTATION: C = + C The antiderivative of with respect to equals + C Note: It is important to attach the +C to every indefinite integral. Notice that when we take the integral of, we are looking for a function whose derivative is. There are an infinite number of functions with a derivative of. Write down some of them:

2 Common functions and their antiderivatives: Function n Antiderivative cos sin sec e Find the antiderivatives of the following functions: ( + ) ( 4) (4 sin + ) ( ) ( 1 4 ) (e ) (5 7 1 ) ( 1 π)

3 Sometimes integrals can look quite difficult. If that is the case and you are unsure where to start, try one of these hints: HINT 1: Multiply Out ( 5) ( + ) ( + 5) HINT : Split into individual fractions ( 1)(4 + 5) 4 HINT #: Use a trig identity cos sin cos cot

4 Find the original function f(). 1. f () = 1, f() =. f () = +, f() = 1. f () =, f (4) = 1, f( 1) = 4. f () =, f ( 5) = 0, f() = 1 5. Given that the graph of f() passes through the point (, 1) and the slope of its tangent line is 4, find the original function f(). 6. Given that the graph of f() passes through the point (1, 6) and the slope of its tangent line is + 1, find f(6). APPLICATIONS: Suppose we know the acceleration of an airplane and wish to find its velocity. Or perhaps we know that rate at which a pool is leaking water and we need to know the amount of water lost. In these cases, the derivative of a function is known and we need to find the original function (called the ). EXAMPLE 1: Roy is traveling to school at 60 mph (88 ft/sec). Suppose she spots a deer in the road and needs to stop quickly. If applying the brakes produces a constant deceleration of ft/sec², how far will the car travel before coming to a stop?

5 EXAMPLE : Suppose Blake wants to show off his football skills. He leads his friends up to the top of the MHS School building (154 ft) and throws the football with an initial velocity of 64 ft/sec. When will the ball reach the maimum height? When will it hit the ground? Are his friends impressed??? PRACTICE: Find the general antiderivative of the given function e e 5. sec 1 7. Find three different antiderivatives for the function f() =. 8. The graph of the function g passes through the point (1, ) and g () =. Find the original function g.

6 9. Suppose a baseball is thrown vertically upward from a position 144 feet above the ground with an initial velocity of 96 ft/sec. Find: a) the baseball s distance above the ground after t seconds. b) the length of time the baseball rises. c) when and with what velocity the baseball hits the ground.

7 REVIEW: U-Substitution Antiderivative Rule: k = Eample: = n = = e = ( e 4) = 1 = + 1 = b = (5 e ) = cos = ( cos + ) = sin = ( 5 sin + ) = sec = ( sec ) =

8 U-substitution is an integration method designed to transform an epression in order to the integration. Eamples: = t. t 1 dt. cos( ) = 4. (1 + sin ) cos = 5. ln 6. 1 = 8. sin 7. 1

9 Practice: Integrate using u-substitution e e. cos sin cos sin cos 4

10 For each integral, decide which of the following is needed before integrating: U-Substitution, Algebra, Trig identity, Nothing, or Can t be done by the techniques learned in Calculus AB. Then evaluate each integral (ecept for the 5 th type) ( 1)

11 cos sin cos cos sin 1. tan sec. tan cos. sec tan 4. tan 1 5. e 6. e e