cos 5x dx e dt dx 20. CALCULUS AB WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND REVIEW Work the following on notebook paper. No calculator.

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1 WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND REVIEW Work the following on notebook paper. No calculator. Find the derivative. Do not leave negative eponents or comple fractions in our answers f f sin. cos 5 tan. f 6. sin 5 Evaluate the given integrals d. cos d 5 d sin 5 cos 5 d d. sin d 9. 5 sec tan sec. 4 8 d 4. Evaluate. d 5. sin t dt d d cos 8. t 5 dt 5 d d d 6. dt d 9. t 4 sec t dt d 4 d 7. t d tan e dt cos t dt d d. d

WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND FUNCTIONS DEFINED BY INTEGRALS. Evaluate. d sin t (a) d dt (c) dt d t d t (e) d cos t dt d 5 d t d (b) e dt d (d) ln t dt d d 7 (f) sin 4 t dt d tan.

2 WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND FUNCTIONS DEFINED BY INTEGRALS. Evaluate. d sin t (a) d dt (c) dt d t d t (e) d cos t dt d 5 d t d (b) e dt d (d) ln t dt d d 7 (f) sin 4 t dt d tan. The graph of a function f consists of a semicircle and two line segments as shown. Let g be the function given b g f tdt. (a) Find g, g, g,and g 5. (b) Find all values of on the open interval,5 at which g has [,5], a relative maimum. Justif our answers. (c) Find the absolute minimum value of g on the closed interval and the value of at which it occurs. Justif our answer. (d) Write an equation for the line tangent to the graph of g at =. (e) Find the -coordinate of each point of inflection of the graph of g on the open interval,5. Justif. (f) Find the range of g.. Let g f tdt, where f is the function whose graph is shown. (a) Evaluate g g g g g,,,, and 7. (b) Write an equation for the line tangent to the graph of g at = 4. (c) On what interval(s) is g increasing? Decreasing? Justif our answer. (d) For what value of does the graph of g have a relative maimum? Justif our answer. (e) For what value of does the graph of g have its absolute maimum value? What is the maimum value? Justif our answer. (f) For what value of does the graph of g have its absolute minimum value? What is the minimum value? Justif our answer. 4. Let g f t dt, where f is the function whose graph is shown. (a) On what intervals is g decreasing? Justif. (b) For what value(s) of does g have a relative maimum? Justif. (c) On what intervals is g concave down? Justif. (d) At what values of does g have an inflection point? Justif. Evaluate the given integrals. 5. d d 8. cos 4 6 d 9. sin d 9 d. cos 5 sin d

3 WORKSHEET ON FUNCTIONS DEFINED BY INTEGRALS Work the following on notebook paper. F. Find the equation of the tangent line to the curve where 7 at the point F t dt on the curve where =.. Suppose that 5 4 (a) What is c f t dt. f? (b) Find the value of c.. If F t t dt, for what values of is F decreasing? Justif our answer Let H f t dt where f is the continuous function with domain [, ] shown on the right. (a) Find. H (b) On what interval(s) of is H increasing? Justif our answer. (c) On what interval(s) of is H concave up? Justif our answer. (d) Is H positive or negative? Eplain. (e) For what value of does H achieve its maimum value? Eplain. 5. The graph of a function f consists of a semicircle and two line segments as shown on the right. Let g f t dt. (a) Find g, g, g. (b) On what interval(s) of is g decreasing? Justif our answer. (c) Find all values of on the open interval, 4 at which g has a relative minimum. Justif our answer. (d) Find the absolute maimum value of g on the interval, 4 and the value of at which it occurs. Justif our answer. (e) On what interval(s) of is g concave up? Justif our answer. (f) For what value(s) of does the graph of g have an inflection point? Justif our answer. (g) Write an equation for the line tangent to the graph of g at. 6. The graph of the function f, consisting of three line segments, is shown on the right. Let g f t dt. (a) Find g, g 4, g. (b) Find g and g. (c) Find the instantaneous rate of change of g with respect to at =. (d) Find the absolute maimum value of g on the interval, 4. Justif. (e) The second derivative of g is not defined at = and at =. Which of these values are -coordinates of points of inflection of the graph of g? Justif our answer. TURN->>>

4 f d 9, find f 5 d. 7. Given 4, 5 8. Given f. Evaluate: f d. 4, 9. Find d d given.. A tank contains gallons of oil at time t = hours. Oil is being pumped into the tank at a rate, where Rt in the table below. is measured in gallons per hour and t is measured in hours. Select values of Rt Rt are given Rt t (hours) 5 9 (gallons per hour) (a) Estimate the number of gallons of oil in the tank at t = hours b using a trapezoidal approimation with four subintervals and values from the table. Show the computations that lead to our answer. (b) Estimate the number of gallons of oil in the tank at t = hours b using a right Riemann sum with four subintervals and values from the table. Show the computations that lead to our answer. (c) A model for the rate at which oil is being pumped into the tank is given b the function Gt, where is is measured in gallons per hour and t is measured in hours. ln t Use the model to find the number of gallons of oil in the tank at t = hours. G t

5 WORKSHEET ON FUNCTIONS DEFINED BY INTEGRALS Work the following on notebook paper.. The function g is defined on the interval [, 6] b g f t dt where f is the function graphed in the figure. (a) For what values of, < < 6, does g have a relative maimum? Justif our answer. (b) For what values of is the graph of g concave down? Justif our answer. (c) Write an equation for the tangent line to g at the point where =. (d) Sketch a graph of the function g. List the coordinates of all critical point and inflection points.. Suppose that is a continuous function, that. Find the average value f f f, and that f 7 of over the interval [, ].. The graph of a differentiable function f on the closed interval [ 4, 4] is shown. Let G f t dt for (a) Find G 4. (b) Find G 4. (c) On which interval or intervals is the graph of G decreasing? Justif our answer. (d) On which interval or intervals is the graph of G concave down? Justif our answer. (e) For what values of does G have an inflection point? Justif our answer. 4. The function F is defined for all b (a) F F t 8 dt. Find: (b) F (c) F (d) F 5. The function F is defined for all b F where f is the function graphed in the figure. The graph of f is made up of straight lines and a semicircle. (a) For what values of is F decreasing? Justif our answer. (b) For what values of does F have a local maimum? A local minimum? Justif our answer. F, F, and F. (c) Evaluate f t dt, (d) Write an equation of the line tangent to the graph of F at = 4. (e) For what values of does F have an inflection point? Justif our answer. TURN->>>

6 5 6. If 6 F t t dt, on what intervals is F decreasing? Use our calculator or problems f f f cos and 4.5. Find..5t 8. Water is pumped out of a holding tank at a rate of e liters/minute, where t is in minutes since the pump is started. If the holding tank contains liters of water when the pump is started, how much water does it hold one hour later? 9. In a certain cit, the temperature, in F, t hours after 6 AM was modeled b the function t Tt 55sin. Find the average temperature during the period from 6 AM to 6 PM. b Remember the formula for average value: f AVE f ( ) d b a a Evaluate. d. sin t dt d d tan. dt d t

7 WORKSHEET ON Work the following on notebook paper. Do not use our calculator. Find the derivative.. f ln 5. 4 ln 4. 4 ln 5 ln f t 5 dt 5. g ln. f ln at the point 6. Write the equation of the tangent line to the graph of where =. d 4 7. Find given ln. d Evaluate d 6. tan d d sin cos d. d 8. 4 e ln d. d 9. 4 d d. cos d. sin. cos d. sin 6 4 d 4. sin d. cos e e d ln 5. 4 d 9. sec d

8 REVIEW WORKSHEET ON st & nd FUND. TH., AVE. VALUE, FUNCTIONS DEFINED BY INTEGRALS, & Work the following on notebook paper. Evaluate. No calculator. d d 5. 7 t dt d. dt 5 d t. The graph of a function f consists of a quarter circle and three line segments as shown. Let g be the function given b g f tdt. (a) Find g 4, g,and g 7. (b) Find g 4, g4, and g 4. (c) Find all values of on the open interval 4,7 at which g has a relative maimum. Justif our answers. (d) Find the absolute minimum value of g on the closed interval [ 4, 7] and the value of at which it occurs. Justif our answer. (e) Find the -coordinate of each point of inflection of the graph of g on the open interval. Justif our answer. 4. Given f t 5t 6dt. (a) Write the equation of the tangent line to f at the point where =. (b) For what values of is the graph of f increasing? Decreasing? Justif our answer. (c) Find all values of at which the graph of f has a relative maimum or a relative minimum. Justif our answers. (d) On what intervals is the graph of f concave up? Concave down? Justif our answer. (e) Find the -coordinate of each point of inflection of the graph of f. Justif our answer. Find the derivative. ln 7. ln 5. ln 6. f ln 8. g t 4dt 5 d 9. Given 5ln, find in terms of and. d Evaluate.. 4 d. tan d 6. cos d e 4 ln. d 4. e sec 6 d 7. cos sin d 8 5. d 5. 4 d TURN->>> 4,7

9 Use our calculator on problems 8. Give decimal answers correct to three decimal places. f sin and f. Find f f e f f and. Find.. A cup of coffee at 9 C is put into a C room when t =. The coffee s temperature is dropping at a rate.t of r t 5e C per minute, with t in minutes. Find the coffee s temperature when t = 8 minutes.,. Given f. Evaluate f d. (No calculator.), 7 7 f d 4, find f d. (No calculator.). Given

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS d d d d t dt 6 cos t dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 t dt d d a f t dt d d 6 cos t dt Second Fundamental