with the initial condition y 2 1. Find y 3. the particular solution, and use your particular solution to find y 3.
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1 FUNDAMENTAL THEOREM OF CALCULUS Given d d 4 Method : Integrate with the initial condition. Find. 4 d, and use the initial condition to find C. Then write the particular solution, and use our particular solution to find. Method : Use the Fundamental Theorem of Calculus: b a f d f b f a Sometimes there is no antiderivative so we must use Method and our graphing calculator. E. f sin and f. Find f.
2 E. The graph of f consists of two line segments and a semicircle as shown on the right. Given that f, find: (a) f 0 (b) f Graph of f (c) f 6 E. The graph of f is shown. Use the figure and the fact that f to find: (a) f 0 Area = 4 Area = (b) f 7 (c) f 9 Area = 9 Then sketch the graph of f. E. A pizza with a temperature of 9 C is put into a C room when t = 0. The pizza s temperature is decreasing at a rate of r t 6e 0.t C per minute. Estimate the pizza s temperature when t = minutes.
3 CALCULUS WORKSHEET ON FUNDAMENTAL THEOREM OF CALCULUS Work problems - b both methods. Do not use our calculator.. and 6. Find.. f cos and f 0. Find f. 4 Work problems 7 using the Fundamental Theorem of Calculus and our calculator.. f cos and f 0. Find f. 4. f e and f. Find f.. A particle moving along the -ais has position t at time t with the velocit of the particle v t sin t. At time t = 6, the particle s position is (4, 0). Find the position of the particle when t = 7.
4 6. Let Ft represent a bacteria population which is 4 million at time t = 0. After t hours, the population is growing at an instantaneous rate of t million bacteria per hour. Find the total increase in the bacteria population during the first three hours, and find the population at t = hours. 7. A particle moves along a line so that at an time t 0 its velocit is given b t vt. At time t = 0, the position of the particle is s 0. Determine t the position of the particle at t =. Use the Fundamental Theorem of Calculus and the given graph. 8. The graph of f is shown on the right. 4 f d 6. and f. Find f The graph of f is the semicircle shown on the right. Find f 4 given that f The graph of f, consisting of two line segments and a semicircle, is shown on the right. Given that f, find: (a) f (b) f 4 (c) f 8
5 . Region A has an area of., and (a) 6 f d 6 0 f d.. Find: (b) 6 0 f d. The graph on the right shows the rate of change of the quantit of water in a water tower, in liters per da, during the month of April. If the tower has,000 liters of water in it on April, estimate the quantit of water in the tower on April 0.. A cup of coffee at 90 C is put into a 0 C room when t = 0. The coffee s temperature 0.t is changing at a rate of r t 7e C per minute, with t in minutes. Estimate the coffee s temperature when t = Use the figure on the right and the fact that F to sketch the graph of F. Label the values of at least four points.
6 Answers million, million (a) 9. (b) 6. (c) 6.. (a) (b) 6.. Answers will var. With two triangles and a trapezoid:,0 liters; with two triangles and two trapezoids:,00 liters C
7 CALCULUS WORKSHEET ON FUNDAMENTAL THEOREM OF CALCULUS Use our calculator on problems, 8, and.. If 4 f, f is continuous, and f d 7, what is the value of f 4?. If f d 7, find f d. 0.t. 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 000 liters of water when the pump is started, how much water does it hold one hour later? 4. Given the values of the derivative f in the table and that f 0 00, estimate f for =, 4, 6. Use a right Riemann sum f 0 8. Consider the function f that is continuous on the interval [,] and for which 0 (a) f d 4. Evaluate: 0 f d (c) f d ( f is even) (b) f d (d) f d ( f is odd)
8 6. Use the figure on the right and the fact that P 0 to find values of P when t =,,, 4, and. 7. Using the figure on the right, sketch a graph of an antiderivatives G t of g t satisfing G 0. Label each critical point of Gt with its coordinates. 8. Find the value of F, where F e and F 0., 9. Given f. Evaluate: f d., 0. A bowl of soup is placed on the kitchen counter to cool. The temperature of the soup is given in the table below. Time t (minutes) 0 8 Temperature T ( F) (a) Find 0 T d. (b) Find the average rate of change of T over the time interval t = to t = 8 minutes.
9 . The graph of f which consists of a line segment and a semicircle, is shown on the right. Given that f 4, find: (a) f (b) f. (Multiple Choice) If f and g are continuous functions such that g f for all, then f d (A) g g (B) g g (C) g g (D) f f (E) f f. (Multiple Choice) If the function f is defined b of f such that g, then g f and g is an antiderivatives (A) -.68 (B) -.8 (C).7 (D) 6.8 (E).8 4. (Multiple Choice) The graph of f is shown in the figure at right. If f d. and F f, then F F 0 (A) 0. (B). (C). (D) 4. (E).
10 Answers liters 4. 6, 8,. (a) 4 (b) 4 (c) 8 (d) 0 6., 0,, 0, 7.,, (a) F (b) F/min.. (a) 7 (b). C. B 4. D
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MA1 Elem. Calculus Spring 017 Final Exam 017-0-0 Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACT-approved calculator during
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We can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).
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means Name a function whose derivative is 2x. x 4, and so forth.
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dy x a. Sketch the slope field for the points: (1,±1), (2,±1), ( 1, ±1), and (0,±1).
Chapter 6. d x Given the differential equation: dx a. Sketch the slope field for the points: (,±), (,±), (, ±), and (0,±). b. Find the general solution for the given differential equation. c. Find the