Calculus BC AP/Dual Fall Semester Review Sheet REVISED 1 Name Date. 3) Explain why f(x) = x 2 7x 8 is a guarantee zero in between [ 3, 0] g) lim x
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1 Calculus BC AP/Dual Fall Semester Review Sheet REVISED Name Date Eam Date and Time: Read and answer all questions accordingly. All work and problems must be done on your own paper and work must be shown. No work = No Credit = NO EXCEPTIONS. It is worth.5 quiz grades. Note: There are two parts of the eam. The NON CALC test has questions (5%) and FRQ s (5%). Chapter Limits ) Find the limit of the following: a) lim b) lim 9 d) lim 5 5 e) lim g) lim h) lim cos ) Solve. If it does not eist, state DNE. c) f) i) a) lim f b) lim f c) lim f lim lim lim sin 5 9 d) f e) f ) Eplain why f() = 7 8 is a guarantee zero in between [, ] ) Define the Intermediate Value Theorem. 5) State the definition of continuity of a function f at which = c. Given: g lim f ) Find the limits as they eist. lim 5 a) lim f 5 g b) lim g c) lim f g h d) lim f limh 7) Given the following function, find k so that the function is k, continuous at =. f k, Chapter Differentiation ) Find the equation of the tangent line to the graph, f() = 5 + at the point (, ). ) Using the difference quotient, find f (). Show all work. 5, f,, (a) Is f continuous at =? Use the definition of continuity to eplain answer. (b) Is f differentiable at =? Eplain. (c) Sketch a graph of f. ) Given: ) Given the f graph, graph f (). Show all work.
2 Calculus BC AP/Dual Fall Semester Review Sheet REVISED ) Find dy d if (a) y + y = (b) y + y = 8 (c) e y cos = + sin(y) ) The radius of a circle is increasing at a rate of cm/sec. Find the rate of change of the area of the circle when the radius is 5 cm. 5) Find dy of the following: d / / (a) f (b) f (d) y (e) y (g) 7 (h) (c) e (f) y y e 5 (i) sin 5 (j) sin 5 ) Suppose that f(5) =, f (5) =, g(5) =, and g (5) =. Find the following values. a) (fg) (5) = b) ( f g ) (5) = c) ( g f ) (5) = 7) f () f ' () g() g '() = 5 = 5 7, solve (b) If h g, solve (a) If h f g h '5 h ' 8) If F() = f(g()), where f( ) =, f ( ) =, f () = 5, g() =, and g () = 7, find F (). ) A 5 ft ladder is leaning against a wall. The bottom of the ladder begins to slide away from the wall at a rate of ft/s. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 9 feet away from the wall? ) Water is dripping through the top of a conical shaped cup at the rate of.5 cubic inches per minute. If the cup s original size is inches high with a diameter of inches across the top, how fast is the height of water decreasing when the water is two inches deep in the cone? ) If a tank holds 5 gallons of water, which drains from the bottom of the tank in 5 minutes, then Toricelli's Law gives the volume V of water remaining in the tank after t minutes as V 5 t t 5. 5 Find the rate at which water is draining from the tank after the following amounts of time. (Remember that the rate must be negative because the amount of water in the tank is decreasing.) (a) 5 min (b) 5 minutes (c) At what time is the water flowing out the fastest? (d) At what time is the water flowing out the slowest? 9) Solve for f (): (a) 8 (b) sec (c) y = cos sin
3 Calculus BC AP/Dual Fall Semester Review Sheet REVISED Chapter Applications of Differentiation ) Given f() = +, identify all absolute etrema on the interval [, ]. ) Find the absolute maimum and absolute minimum values of f on the given interval, f(t) = cos t + sin(t) at I [, π ] (a) Find the velocity of the object when t = 5 (b) Find the acceleration at time t (c) Establish when the particle is at rest (d) When is the particle slowing down? ) A particle starts by moving to the right along a horizontal line; the graph of its position function is shown. ) Define the Etreme Value Theorem and Mean Value Theorem and identify the difference of the two theorems. ) What is the number for c which satisfies the conditions of the Mean Value Theorem of differentiable calculus for f() = on [, ]. 5) Find the relative etrema of f() = + using the First Derivative Test. ) Use the second derivative test to determine the relative etrema for f() = 8. 7) Determine the f() =, determine the following: (a) and y intercepts (b) horizontal and vertical asymptotes (c) intervals of increasing and decreasing 8) Determine the open intervals in which the graph f() = is concave up or down. 9) Find the relative etrema of f() = 9 on the I[, ]. ) Let s(t) = t t + be the position function of a particle moving along the ais. (a) When is the particle moving to the right? (b) When is the particle moving to the left? (c) When is the particle standing still? ) A particle is moving along a horizontal line according to the equation, s(t) = t 7 t + 5t where [, ). Determine the interval of time when the particle is moving to the right. ) Find the shortest distance from f() = to the point (, ½). ) Find two positive numbers whose product is 9 and whose sum is a minimum. 5) Find the dimensions of a rectangle with perimeter 9 m whose area is as large as possible. ) The top and bottom margins of a poster are each 9 cm and the side margins are each cm. If the area of printed material on the poster is fied at 8 cm, find the dimensions of the poster with the smallest area.
4 Calculus BC AP/Dual Fall Semester Review Sheet REVISED 7) A rancher has feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area will be a maimum? Chapter Integration ) Solve for these indefinite integrals. (a) d (b) cos ) Find y = f() if f ", f ', and ) Given d f. f d for the functions below, use the left, right, and trapezoidal Riemann sum with subintervals. f().5.8 ) If f() =,, evaluate the Riemann sum with n =, taking the sample points to be right endpoints. Then, determine if the graph is an underestimation or an overestimation. 5) Sketch the region whose is area is given by the definite integral and then use a geometric formula to evaluate the integral of 7 9 d ) The graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral. A) g d 8 B) g d C) g d y 9 9 7) Given f d and f d, solve for 9, and 7 f d 9 g d 8) Evaluate: (a) d (b) (d) sin cos f d, 5 d (c) d (e) d 9) Use the Fundamental Theorem of Calculus to solve: 5 (a) d (b) csc d (c) (d) d (e) d (f) 5cos 5 d d 5 d ) Use the Mean Value Theorem of Integration to identify c of f() = 9, [, ] ) Find the average value of f() = on [, ] ) Solve (a) d sin t dt d d and (b) sin t dt d ) Accumulation/Riemann s Sum: Let y(t) represent the population of Sugar Mill over a -year period, where y is differentiable function of t. The table shows the population recorded every two years. t (years) y (people)
5 Calculus BC AP/Dual Fall Semester Review Sheet REVISED 5 (a) Use the data from the table to find an approimation for y (7) and eplain the meaning of y (7) in terms of the population of Sugar Mill. Show the computations that lead to the answer. (b) Use the data from the table to approimate the average population of Sugar Mill over the time interval, < t < by using a LEFT Riemann s Sum with five equal intervals. Show all work. (c) A model for the population of another town, Pine Grove, over the same -year period is given by the function t 5 P t where t is measured in years and P(t) is measured in people. Use the model to find the value of (d) Use the model given in part (c) to find the value of P '7. P(t)dt. Eplain the meaning of this integral epression in terms of the population of Pine Grove. Chapter 5: Logarithms ) Write the following epression as a log of a single quantity, ln ln( + ). ) Find the derivative: (a) y y ln 7 log ln 7 (b) (d) 7 7 f t t 5 t (e) ) Find (f ) = 5 if f() = 5 ) Given f 5, g are inverses, solve for f ' 7,, g '. f and (c) 5 y e f ', and f and 5) Solve for g ' : 8 f() f () / 5 5 ) Solve d 9) d d cos arcsin 7 7) tan d ) ) 5 Chapter - Differential Equations ) Solve for the differential equation: dy dy (a) cos (b) d d y (c) y' 8 (d) dy y d 8) ) Write the function y = f(t) passing through the point (, ) dy with the given first derivative of t dt ) The number of bacteria in a culture is increasing according to the law of eponential growth. There are 5 bacteria in the culture after hours and 55 bacteria after hours. (calc) e (a) Find the initial population. (b) Write an eponential growth model for the bacteria population. Let t represent time in hours. (c) Use the model to determine the number of bacteria after 8 hours. (d) After how many hours will the bacteria count be,? d d
6 Calculus BC AP/Dual Fall Semester Review Sheet REVISED ) At time t =, bacterium weighs grams. Three hours later, the culture weighs 5 grams. The maimum weight of the culture is grams. (calc.) (a) Write a logistic equation that models the weight of the bacterial culture. (b) When will the culture's weight reach grams? (c) At what time is the culture's weight increasing most rapidly? 5) In a town of population, twenty thousand residents heard a radio announcement about a local political scandal. The rate of growth of the spread of information was jointly proportional to the amount of people who had not heard it. If 5% heard the scandal after one hour, how long until 8% of the population has heard the rumor? (calc) ) Consider the differential equation dy d = y ( + ) Let y = f() be the particular solution to the differential equation with initial condition f() =. Use Euler s method, starting at = with two steps of equal size, to approimate f ( ). ) AB- Free Response Questions ) AB/BC-
7 Calculus BC AP/Dual Fall Semester Review Sheet REVISED 7 ) AB/BC5- At time t =, boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is 9 degrees Celsius C at time t, and the internal temperature of the potato is greater than 7 C for all times t. The internal temperature of the potato at time t minutes can be modeled by the function H that satisfies the differential equation dh dt H 7, where H 9. H t is measured in degrees Celsius and (a) Write an equation for the line tangent to the graph of H at t =. Use this equation to approimate the internal temperature of the potato at time t =. d H (b) Use to determine whether your answer in part (a) is an dt underestimate or an overestimate of the internal temperature of the potato at time t =. (c) For t, an alternate model for the internal temperature of the potato at time t minutes is the function G that satisfies the dg differential equation G 7 /, where G(t) is measured dt in degrees Celsius and G. Find an epression for G(t). 9 Based on this model, what is the internal temperature of the potato at time t =? ) AB5- The function g is defined and differentiable on the closed interval [ 7,5]and satisfies g() = 5. The graph of y = g (), the derivative of g, consists of a semicircle and three-line segments, as shown in the figure above. (a) Find the -coordinate of each point of inflection of the graph of y = g()on the interval 7 < < 5. Eplain your reasoning. (b) The function h is defined by h() = g(). Find the coordinate of each critical point of h, where 7 < < 5, and classify each critical point as the location of a relative minimum, relative maimum, or neither a minimum nor a maimum. Eplain your reasoning. 5) List ways you will do to help you study for the midterm eam.
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