Ex. Find the derivative. Do not leave negative exponents or complex fractions in your answers.

Transcription

1 CALCULUS AB THE SECOND FUNDAMENTAL THEOREM OF CALCULUS AND REVIEW E. Find the derivative. Do not leave negative eponents or comple fractions in your answers. 4 (a) y 4 e 5 f sin (b) sec (c) g 5 (d) y 4 4 (e) h sin 5 E. Evaluate the given integrals. 5 (a) sin d 4 (b) d (c) cos d

2 6 (d) 0 sin d 0 (e) sin 5cos 5 0 d The Second Fundamental Theorem involves taking the derivative of an integral. Try the following problems and see if you can discover what the theorem says. d t dt d d d 6 cost dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 d d t dt 6 d d a cost dt f t dt Second Fundamental Theorem of Calculus (Chain Rule Version): d g f tdt d a E. Use the Second Fundamental Theorem to evaluate: d (a) t dt d d (b) t d tan dt (c) d d t dt (d) d d sin t dt

3 CALCULUS FUNCTIONS DEFINED BY INTEGRALS E. The graph of a function f consists of a quarter circle and line segments. Let g be the function given by g f tdt. 0 (a) Find g 0, g, g, g 5. Graph of f (b) On what interval(s) is the graph of g increasing? Justify your answer. (c) Find all values of on the open interval, 5 at which g has a relative maimum. Justify your answer. (d) Find the absolute minimum value of g on, 5 and the value of at which it occurs. Justify your answer. (e) On what interval(s) is the graph of g concave up? Justify your answer. (f) Find the -coordinate of each point of inflection of the graph of g on, 5. Justify your answer.

4 Second Fundamental Th. & Functions Defined by Integrals, Day f t t dt. For what values of is the graph of f decreasing? E. Given _ E. Suppose that f tdt. c (a) What is f? (b) Find the value of c. _ 5 5 f d 4, find f d. E. Given _, 0 E. Given f. Evaluate f d., 0

5 E. The rate at which water is being pumped into a tank is given by the function selected values of Rt contained 50 gallons of water when t = 0. t Rt. A table of, for the time interval 0 0 minutes, is shown below. The tank t (min.) (gal/min) Rt (a) Use data from the table and four subintervals to find a left Riemann sum to approimate the amount of water in the tank when t = 0 minutes. Draw the rectangles, and show your computation. (b) Use data from the table and four subintervals to find a trapezoidal approimation to approimate the amount of water in the tank when t = 0 minutes. Draw the trapezoids, and show your computation. (c) A model for the rate at which water is being pumped into the tank is given by the function t W t 5e W t is measured in gallons per minute. 0.0, where t is measured in minutes and Use the model to find the amount of water in the tank when t = 0 minutes. Homework: Worksheet & AP Review Prob.

6 Second Fundamental Th. & Functions Defined by Integrals, Day f cos and f 7. Find f. E. _ sin f e and f 5.. Find f 4. E. _ E. A pizza with a temperature of 95 C is put into a room to cool when t = 0. The pizza s temperature is 0.t decreasing at a rate of r t 6e C per minute. Estimate the pizza s temperature when t = 5 minutes. _ E. A study suggests that between the hours of :00 PM and 4:00 PM on a normal weekday, the speed of the traffic on a certain freeway eit is modeled by the formula S t t t 60t 0 where the speed is measured in kilometers per hour and t is the number of hours past noon. Compute the average speed of the traffic between the hours of :00 PM and 4:00 PM. Homework: Worksheet & AP Review Problems - 5

7 5. The Natural Logarithmic Function and Differentiation Definition of the Natural Logarithmic Function: The natural logarithmic function is defined by ln t dt, where > 0. The base of natural logs is the number e. e was named for a Swiss mathematician, Leonhard Euler. n n n By definition: e lim lim n n n n y ln and y e are inverses. Properties of Natural Logs: 0, ) Domain = Range =, ) The function is continuous, increasing, and one-to-one. ) The graph is concave downward. E. Sketch the graph of the given function, and state its domain. f ln (b) f ln (c) f ln (a) Other properties: If a and b are positive numbers and n is rational, then: ) ln = 0 ) ln e = ) ln ab = ln a + ln b a 4) ln ln a ln b b n 5) ln a nln a E. Write as a sum, difference, or multiple of logs: ln E. Write as a single log: ln ln

8 E. Use the properties of logarithms and the facts that ln 0.69 and ln.0986 to approimate the value of: (a) ln (b) ln d d du ln u, u 0 u d E. f ln f yln E. y E. f ln f f E. ln f y ln ln E. y y ln E. y E. y ln y Homework: P. :, 4, 5, 6, 9, 7 odd,,, odd & AP Review 6

9 5. The Natural Logarithmic Function and Differentiation, Day d d du ln u, u 0 u d E. Write the equation of the tangent line to the graph of y ln at the point (, 0). E. Write the equation of the tangent line to the graph of y 5ln., at the point ln5 E. If f t 7 dt, find f. f _ dy E. Given ln y y 4, find in terms of and y. d _ E. Show whether or not yln 4 is a solution to the differential equation y y 0.

10 E. Locate the relative etrema of ln y, and justify your answer. Homework: P. : even, all, 8, 8, 84, 89, 9 & AP Review 7-9

11 5. The Natural Log Function and Integration d du In the previous section we learned that ln u. d u d The corresponding integration formula is: du ln u C u E. d e E. d E. d E. d E. d

12 E. 5 4 d e ln E. d e 4 ln E. d e 5 E. d 0 Homework: P. 40: 5 odd, 5 59 odd, and AP Review 0

13 5. The Natural Log Function and Integration, Day Yesterday we learned the formula du ln u C u Today we will use this formula to derive formulas for the antiderivatives of tan u, cot u, sec u, and csc u. _ tan d _ cot d _ sec d _ csc d tanu du cot u du secu du cscu du

14 tan u du ln cos u C cot u du ln sin u C secu du ln secu tan u C cscu du ln cscu cot u C E. tan d sec d E. E. 5 cot d E. Find the average value of f sec on [0, ]. 6 dp 6000 E. A population of bacteria is growing at a rate of dt 0.5t where t is the time in days. The initial population is 000. (a) Find a function that gives the population at any time t. (b) Use your answer to (a) to find the population when t = 8 days. Homework: P. 40: 0,, 6, 0, 40 even, even, 67, 70, 9, 99, 0 & AP Review

15 5. 5. The Natural Logarithmic Function and Differentiation and Integration E. 4 d _ E. 4 d _ E. 4 d _ E. cos d _ 6 E. cos sin 8 d Homework: Worksheet & AP Review 4-6