AP Calculus AB One Last Mega Review Packet of Stuff. Take the derivative of the following. 1.) 3.) 5.) 7.) Determine the limit of the following.

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1 AP Calculus AB One Last Mega Review Packet of Stuff Name: Date: Block: Take the erivative of the following. 1.) x (sin (5x)).) x (etan(x) ) 3.) x (sin 1 ( x3 )) 4.) x (x3 5x) 4 5.) x ( ex sin(x) ) 6.) 6x 3 3x x (4x5 x ) 7.) x (5x +1 ) 8.) ln(3xe 5x ) Determine the limit of the following. ln(x) 9.) lim x x 10.) lim 1 sin(x) x π 1+cos(x) Use the Funamental Theorem of Calculus 11.) ( x x x tan(t) t) 1.) ( x e x (x 1) (t t) t)

2 13.) Determine the equation of the tangent line to the curve of y = 3 + tan 1 (t) t, when x = 0. Note: The omain of y is the interval [ π, π ]. e x 1 Integrate the following. 14.) 3x(x + 3) 4 x 15.) cos (πx)x 16.) x 81+x 6 x 17.) xx x 18.) sec (3x) tan(3x) x 19.) (x + ) 6x + 4x x ) ( x 1) x 3π x 1.) π e x cos(e x ) x e 6 (ln x)4.) x 3.) e x 1 x 4 8x x The following are two ifficult integration questions, choose "u" carefully. 4.) x 3 x + 1 x 5.) (x + 1) x x

3 Fin the following erivatives implicitly 6.) Determine y x, given x y + xy = 9 7.) Fin y, if x + sin y = xy 8.) Fin the equation of the tangent lines where x =, for the equation x + xy y = 1 9.) What is the point of intersection of the tangent lines foun in problem 8? 30.) Fin y x, given 4x + 3y = 4

4 Solve the following Relate Rates 31.) An observer is staning 1,500 feet from the launch point of moel rocket. The rocket launches with an upwar velocity of 88 feet per secon. At the instant the rocket is 440 feet off the groun what is the rate of change in the angle of elevation from the observer to the rocket? 3.) The formula for the volume of a regular oecaheron is V = 1 4 ( )e3, where e is the length of one ege of the shape. If the rate of change of e is 5 centimeters per secon, at approximately what length oes an ege of the shape nee to be to have a rate of change in volume equal to 10 centimeters per secon. Note: a calculator is necessary for this question. 33.) Consier a conical tank whose raius at the top is 4 feet an whose epth is 10 feet. It s being fille with water at the rate of cubic feet per minute. How fast is the water level rising when it is at epth 5 feet? Use the iagram below to answer the following questions. 34.) Given the graph of f(x) consisting of four line segments an a half circle on the interval [ 9, 9], let g(x) x be the function efine as g(x) = f(x) x. 3 a.) What is the average rate of change of g(x) on the interval from x = 5 to x = 5? b.) What is the instantaneous rate of change of g(x) at x = 5?

5 c.) Given that g( 5) = 7 etermine the value of g(0). 6.) Evaluate f (x) x. 4 e.) ln(x+6) Evaluate lim x 5 g(x) f.) Determine the interval(s) in which g(x) is concave own an/or concave up on the interval [ 9, 9]. Justify your answer. g.) Determine whether g(x) has any relative maximum or minimum values on the interval [ 9, 9]. h.) Determine the absolute maximum an minimum values of g(x) on the interval [ 9, 9]. i.) Determine the coorinates of the point of inflection for g(x). 35.) Given the ifferential equation y questions. = 1 x xe y an the accompanying slope fiel below, answer the following

6 a.) Determine the general solution to the ifferential equation. b.) Fin the particular solution through the point (e, ln 3). c.) Sketch a solution curve through the point (e, 0)..) Fin the equation of the tangent line to the curve at the point (e, ln 3). e.) Determine the concavity at the point (e, ln 3).

7 36.) The region R above is enclose by the functions f(x) = x an g(x) = x x. a.) Determine the area of the region R. b.) Determine the volume soli form by rotating the region R, aroun the line y = 4. c.) The region R is the base of soli where cross sections perpenicular to the x axis are semicircles. Determine the volume of the soli..) Determine the value k, such that vertical line x = k ivies the region R in half.