AP Calculus AB. a.) Midpoint rule with 4 subintervals b.) Trapezoid rule with 4 subintervals

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Marianna Dickerson
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1 AP Calculus AB Unit 6 Review Name: Date: Block: Section : RAM and TRAP.) Evaluate using Riemann Sums L 4, R 4 for the following on the interval 8 with four subintervals. 4.) Approimate ( )d using a.) Midpoint rule with 4 subintervals b.) Trapezoid rule with 4 subintervals.) The function f() is increasing and concave down on the interval [, 6] using three subintervals of equal length: a.) Is the approimation L an overestimate or underestimate of f()d? Justify your answer. 6 b.) Is the approimation R an overestimate or underestimate of f()d? Justify your answer. 6

2 t (minutes) r(t) (gallons per minute) ) Water is flowing into a tank at the rate r(t), where r(t) is measured in gallons per minute and t is measured in minutes, the tank contains 5 gallons of water at time t =. Value of r(t) for selected values of t are given in the table above. Using a trapezoidal sum with intervals, indicated by the table what is the approimation of the number of gallons of water in the tank at time t = 9? Section : Convert from Riemann Sum to Integral Notation and Integral Notation to Riemann Sum 5.) ( 5 + ) d.) 9 sec ( π + ) d 4 n n.) lim ( + 5i i= ) n (5) 4.) lim n n n i= tan ( π + 5πi 4n ) (5π 4n ) Section : Determine the following antiderivatives.) ( 5) d.) ( 4 ) d.) 6 5 d 4.) ( ) d 4

3 5.) sin d 6.) cos d 7.) sec tan d 8.) sec d 9.) csc cot d.) csc d.) d.) 4 d.) e d 4.) e +5 d 5.) tan cos csc d 6.) (sec ) cot cos d

4 Section 4: FTOC Part : Determine the value of the following definite integrals.) ( 5 + ) d.) 4 ( ) d.) ( ) d 9 4.) 4 d π 5.) 4 sec tan π d 6.) π cos d 6 7π 6 7.) π sin 4 π d 8.) π csc cot d ) 5 d.) 6 8 d

5 .) (e ) π d.) (sin e ) d.) e +4 d 4.) 8 (e e) d 8 5.) 4 d 6.) cos d π π 4 sin 4 Section 5: FTOC Part : Determine the derivative of the following..) Find d d ( t )dt.) d d + t dt.) d d e+ dt 4.) d d t sec t dt

6 Section 6: Eliminate the parameter.) Given a(t) = t, v() = 5, and s() = 4, write an epression for s(t)..) Give f () = +, f () = and f() = 4, write an epression for f() Section 7: Properties of Definite Integrals.) Given f()d 5 = 7 and f()d = a.) 5 (f() + )d b.) (4f() + )d c.) f( 5)d 5 d.) f()d.) Given f() d =, 4 f() d = 5, 6 f() d = a.) If F() = determine F()

7 b.) If F(6) = 5 determine F() c.) If F(4) = determine F(6) Section 8: AP Style FTOC Questions.) Let h be the function defined by h() = π cos (t) dt. What is the equation of the tangent to the graph of h at the point where = π 4. 4.) Find the interval on which f() = (t + t + )dt is concave up..) Given the graph of g below consisting of one line segment and a semi-circle from [, 7] f() if f() = g(t)dt determine lim. 9

8 4.) The functions f and g are differentiable for all real numbers and g is strictly decreasing. The table below gives values of the functions and their first derivatives at selected values of. Let h be the function given by h() = g() f(t)dt. Find the value of h ( ). f() f () g() g () ) Given the graph of f() below, defined by three line segments on [ 5, 5], let F() = f(t)dt F( ) =. and a.) Place F(4), F (4) and F (4) in order from least to greatest. b.) Find the equation of the tangent line to F() at = 4.

9 c.) Use the results of b.) to approimate the value of F() at = 4.. Does this value overestimate or underestimate F(4.)? Justify your answer. d.) Find the value(s) where F() has a maimum. Justify your answer. 6.) Given the graph of g below consisting of four line segments and a semi-circle on [ 5, 5], if f() is defined as f() = g(t)dt determine the following. a.) f( ) b.) f(5) c.) f( ) d.) f ( ) e.) f () f.) f ( 5) g.) f ( 4) h.) f () i.) On what intervals is f() increasing? j.) On what intervals is f() decreasing? k.) On what intervals is f() concave up?

10 l.) On what intervals is f() concave down? m.) At what ordered pairs does f() have a relative maimum? n.) At what ordered pairs does f() have a relative minimum? o.) At what ordered pairs does f() have an absolute maimum? p.) At what ordered pairs does f() have an absolute minimum? q.) At what ordered pairs does f() have a point of inflection? r.) Evaluate lim f()

PRACTICE FINAL - MATH 1210, Spring 2012 CHAPTER 1

PRACTICE FINAL - MATH 1210, Spring 2012 CHAPTER 1 PRACTICE FINAL - MATH 2, Spring 22 The Final will have more material from Chapter 4 than other chapters. To study for chapters -3 you should review the old practice eams IN ADDITION TO what appears here.