The Fundamental Theorem of Calculus Part 3
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1 The Fundamental Theorem of Calculus Part
2 FTC Part Worksheet 5: Basic Rules, Initial Value Problems, Rewriting Integrands A. It s time to find anti-derivatives algebraically. Instead of saying the anti-derivative of is Cwe write d C. In general, the notation f ( d F( C means that F '( f (, and that f ( d is an indefinite integral of the function f. Why the d? Why are there no limits? b f ( d instead of f ( d a Why the +C? B. List of basic anti-derivatives. I will start a list, you continue with a partner. Write as many as you can without using resources. n n d C n C. All the properties of integrals apply: we can add, subtract, and multiply by a constant.. e 5cos d. 7 5sec d
3 D. Sometimes we can use knowledge of algebra or trig identities to make a problem easier.. ( 7( d 4. 6 ln( e d d 6. 5 d 7. d 8. cos sin d E. We can use anti-derivatives to solve initial value problems: If dy d and when and y. Find a rule for y.
4 Homework FTC Part Worksheet 5. Find the Anti-Derivative. a. t t t dt b. sin d e 8 d d. 5 sec c. d e. cos d. Find the Anti-Derivative. You may need to use a trig identity, a rule of logs, or do some multiplication or division first. a. b. (cos sin d d c. e d ln d. ( ( 9 d e, d cos f. d. Solve the following initial value problems: a. Let f '( and f ( 4 find a rule for f ( dy b. Let sin( d when =, y= find the equation for y 4. Use the identity sec tan to find tan d
5 FTC Part Worksheet 6: Guessing Anti-Derivatives involving Constants, Definite Integrals A. Definite Integrals: We can use the Fundamental Theorem of Calculus Part to evaluate definite integrals. b f ( d F ( b F ( a is the total change in F from a to b. a Evaluate without using a calculator. Use a calculator to check your answers. 4. 7d. e d. 4 sin d 4. d 5. d 6. e d
6 B. We can think backwards to find anti-derivatives off by a constant (a family of functions. 7. e 7 d 8. cos5 d 9. sec 8 d. sec4 tan 4 d C. Guessing with constants.. 4 e d. 7sin 4 d. 5csc cot d Summary: How can you use FTC to evaluate definite integrals? How can you check your antiderivative?
7 FTC Part Worksheet 6 Homework dy. If sin(, and y (, write a formula for y d. Evaluate the Definite Integral (no calculator a. d b. cos d 9 c. 4 d d. ( d 4 e. cos d 5 f. 9 d 5 g. 4 d h. d 5. Find the Anti-Derivative: 7 a. d b. csc ( d c. 4 e 4 d d. d 4. Use the identity cos sin to find sin d. 5. How is the answer to a definite integral different from the answer to an indefinite integral?
8 FTC Part Worksheet 7: Area Between Two Curves. For each problem below, * sketch a graph of the region R between ( f and ( g * find the values where the curves intersect (store values & record to 4 decimal places *write an integral (or integrals to represent the area between the curves * Use your calculator to find the area of R correct to decimal places (round or truncate a. (, sin ( g f b. g f ( ( c. ( 4 ( g f d. ( sin( ( g f e. ( ( g f f. ( ( g e f
9 . Let f ( 6 and g( a. Sketch a graph of the region R in the first quadrant bounded above by f and below by g. b. Write an integral epression which stands for the area of the region R described above Hint: You will write two definite integrals c. Use your calculator to evaluate your integral epression.. No Calculators. Use the FTC to find the eact value of the area between the given functions. Assume all areas are positive. What do you notice? a. f ( 6 and g (. b. p ( and q (.
10 FTC Part Worksheet 8: Displacement, Velocity, Acceleration, and Total Distance Traveled. Calculator. Given the velocity function and a time interval, find the net distance traveled and the total distance traveled. a. v ( t sec t, b. v( t.5t 5 e [, 4 ]. No calculator. Given an equation for the acceleration, find an equation for the velocity of a given particle. Then find the net distance traveled. a. a( t t feet/sec b. a( t 6t 8 feet/sec v( 4feet/sec v ( 6 feet/sec t= to t=9 seconds t= to t=5 seconds. No calculator. The acceleration at time t of a particle moving along the -ais is a( t cost sin t. At time t= the velocity of the particle is v (. a. Find the velocity of the particle at any time t. b. Find an equation for the position (t if (. c. What is the speed of the particle at time t?
11 4. Calculator. At time t, a particle is at position. It moves along the -ais with t velocity v( t t cos a. Is the speed of the particle increasing at time t? Eplain why or why not in a sentence. b. Find all times t in the open interval t where the instantaneous acceleration equals the average acceleration. Show your calculus reasoning. c. Find all times t in the open interval t where the instantaneous velocity equals the average velocity. Show your calculus reasoning. d. Find all times t in the open interval t where the particle changes direction. Justify your answer with calculus. e. Find (4, the position of the particle at time t 4. Show your calculus reasoning. f. Find all times t in the open interval t when the particle returns to its origin. Justify your answer with calculus. g. During the time t, what is the greatest distance between the particle and the origin? Show the work that leads to your answer.
12 AP Area and Motion You should be able to answer all questions without a calculator.. Let R be the region in the first quadrant bounded by the -ais and the graphs of f ( 8 g( sin as shown in the figure. and (a Find the area of R. (b The vertical line kdivides R into two regions of equal area. Write, but do not solve, an equation involving one integral whose solution gives the value of k.. Let f ( 6 4 and g( 4cos 4. Let R be the region bounded by the graphs of f and g, as shown in the figure. Find the area of R.
13 . The velocity of a particle moving along the -ais is modeled by a differentiable function v, where the position is measured in meters, and time t is measured in seconds. Selected values of v(t are given in the table. The particle is at position = 7 meters when t = seconds. (a Estimate the acceleration of the particle at t = 6 seconds. Show the computations that lead to your answer. Indicate units of measure. (b Using correct units, eplain the meaning of v ( t dt in the contet of this problem. Use a trapezoidal sum with the three subintervals indicated by the data in the table to approimate 4 v ( t dt. 4 (c For t 4, must the particle change direction in any of the subintervals indicated by the data in the table? If so, identify the subintervals and eplain your reasoning. If not, eplain why not. (d Suppose that the acceleration of the particle is positive for < t < 8 seconds. Eplain why the position of the particle at t = 8 seconds must be greater than = meters. 4. A squirrel starts at building A at time t = and travels along a straight wire connected to building B. For t 8, the squirrel s velocity is modeled by the piecewise-linear function defined by the graph to the right. (a At what times in the interval < t < 8, if any, does the squirrel change direction? Give a reason for your answer. (b At what time in the interval t 8 is the squirrel farthest from building A? How far from building A is the squirrel at this time? (c Find the total distance the squirrel travels during the time interval t 8. (d Write epressions for the squirrel s acceleration a(t, velocity v(t, and distance (t, from building A that are valid for the time interval 7 < t <.
14 FTC Part Worksheet 9: Undoing the Chain Rule A. What is the chain rule? B. Practice. d d (sec( + =. d d (tan5 =.. d d (sine = 4.. d d ( + = 5.. d d (cos4 = 6.. d d sin = C. Finding anti-derivatives = thinking backwards. 5 ( d Practice: 7. tan sec e d 8. cos d 4 cos sin d. 9. cos sin d D. Activity: In groups of 4, take turns making up problems for others to solve.
15 FTC Part : Worksheet 9 Homework dy. If e and y ( 4, find a rule for y in terms of. d. Evaluate: a. d b. 7 6 d 4 c. 5(sin cos d d. csc ( d e. sin d f. 4 tan4 sec 4d g. cos d h. e d i. 4 sin d j. 5sin cos d cos k. cos sin d sec l. d tan
16 . (no calculators Find the definite integral e a. d sin b. e cos d c. 9 d d. cos d e. sin d f. sec 4 d g. 6 sin 5cos d
17 FTC Part Worksheet : Substitution Method. You can always think backwards, but sometimes it s helpful to have a method. Practice:. d. cos e sin d d 4. sin t dt. cos 5. d d d 8. sec tan 5 d 7. sin
18 FTC Part : Worksheet Homework. Find the anti-derivative. Use substitution if it helps. a. d b. ( d c. d d. sin cos d e. e d f. cos sin d
19 . Find the anti-derivative. cos a. d b. d sin( 6 c. d 5 d. 5 d dy dt. If t t 8 and y when t, then find an equation for y in terms of t.
20 FTC Part : Worksheet Changing the Limits of Integration after Substituting. If you make a substitution to find the antiderivative, how can you change the limits of integration (a & b so that you are finding the same area? 4 d Let u Then du d and 4 du d, so d u du Find the values of a & b then evaluate the definite integral without changing back to. b a. Evaluate by making a substitution and changing the limits of integration. 4 tan sec d. Find the area under the curve of the given functions over the given interval. t a. y e [, ] b. y [,] 4. A cubic polynomial function f is defined by f ( 4 a b k where a, b, and k are constants. The function f has a local minimum at, and the graph of f has a point of inflection at. a. Find the values of a and b. b. If f ( d, what is the value of k?
21 5. Find the average value of each of the following functions over the given interval. a. y = on [,4] b. y sin, 6 6. Let R be the region under the graph of y 4 on the interval 6. a. Find the area of R b. (calculator If the line k divides R into two regions of equal area, what is the value of k? 7. What is the average value of y on the interval 6? 8. Find the values of a and b that make the following true. a. b cos cos u d du b. a d 5 u a b du 9. If f( is continuous and 4 f ( d, find f d. If f( is continuous and 9 f ( d 6, find f d
22 FTC Part Review (show work on a separate piece of paper if you need more space ( and let g( cos on the interval, f (, g (. Let f sin a. Sketch a graph of the region R between bounded by, and the -ais on this interval. b. Write an integral epression which stands for the area of R. c. Use your graphing calculator to evaluate your integral epression to decimal places.. (no calculator Let f ( 4. a. make a table of values of f ( for =, -,,, b. make a rough sketch of f ( on [ -, ] c. Find the eact value of the total area of the region formed by f ( and the -ais. (no calculator. Find the anti-derivative: a. tan sec d b. d e 4. (no calculator. Find the area of the closed region bounded by a. the graph of y and the -ais from = to =. b. the curve y e and the lines y and c. the graph of y from = to = 5. Use the identity cos cos to find cos d 6. (no calculator Sketch the area between the functions. Write a definite integral which stands for the area. Evaluate your definite integral. 4 a. f ( g( b. f ( g( 7. Let R be the region enclosed by the graph of y e and the -ais between and. a. Write a definite integral which stands for the area of the region R. b. Use your calculator to evaluate your definite integral to decimal places.
23 8. A particle moves along a line so that at any time t its velocity is given by At time t = the position of the particle is s ( 5 a. Determine (by graphing the maimum velocity of the particle. b. Determine the position of the particle at t =. c. What is the total distance traveled by the particle from t = to t =? d. As t approaches infinity, what is the limit of the velocity? t v( t t 9. (no calculator Let the velocity of a particle moving along the -ais be given by v( t 4t t a. Find v ( b. Find an equation for the position (t if sin. (no calculator Let f (. Find the area under the curve of f on the interval,.. Find the anti-derivative a. 5r r dr b. 4 d 8. (no calculator Let,.5 p ( w (4w. Find the average value of p on the interval. Let the velocity of a particle moving along the -ais be given by v( t t cos( t a. On the interval, when is the particle moving to the right? b. If the particle starts at the origin when t =, find an epression for (t 4. Find the constant k so that the area of the region bounded by y k is eactly 7. Hint: Make a sketch of both graphs. y k and
24 5. Use the FTC to find the definite integral: a. sin t dt b. 6 d c. d d. d e. sin.5 d f. d 6. Find the anti-derivative. e 4 a. cos d b. d c. d d. sec d e. e ( e d f. d
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