Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
AP Calculus Review for Test 1 Term 4 Name__________________________ Test One Wednesday April 4 2012 Determine the limit by substitution. x2 + 14x + 49 1) lim x 4 Answer: 11 Determine the limit algebraically, if it exists. 10 - x 2) lim x 10 10 - x Answer: Does not exist 3) x2 - 4 lim x -2 x + 2 Answer: -4 Determine the limit graphically, if it exists. 4) lim f(x) x 1+ Answer: 3 Find the indicated limit. 11x 5) lim x x 0Answer: -11 1 Find the limit. 6) Let x lim f(x) = 1 and lim g(x) = 5. Find lim [f(x) + g(x)]2. 10 x 10 x 10 Answer: 36 Find the limit, if it exists. x2 - 4x + 17 7) lim x3 + 9x2 + 8 x Answer: 0 6x + 1 8) lim 13x - 7 x Answer: 6 13 Find a value for a so that the function f(x) is continuous. x2 + x + a, x < -4 9) f(x) = x3, x -4 Answer: a = -76 Find dy/dx. 10) y = 2 tan4x Answer: 8 tan3x sec2x 11) y = 19x - x7 Answer: 19 - 7x6 2 19x - x7 Find dy/dx by implicit differentiation. If applicable, express the result in terms of x and y. 12) 2y + 9xy - 4 = 0 Answer: -9y 2 + 9x 13) cos xy + x5 = y5 Answer: 5x4 - y sin xy 5y4 + x sin xy 2 At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 14) x2 + y2 - 2x + 4y = 8, tangent at (4, 0) 3 Answer: y = - (x - 4) 2 15) x2 - y2 = 15, normal at (-8, 7) Answer: 7x - 8y + 112 = 0 Find dy/dx. 16) y = x7/5 Answer: dy 7 2/5 = x dx 5 Find the derivative of the given function. 17) y = 3 sin-1 (5x4) Answer: 60x3 1 - 25x8 Find dy/dx. 18) y = 7xex - 7ex Answer: 7xex Find the location of the absolute extremum for the function. 19) Answer: x = 1 3 Find the extreme values of the function on the interval and where they occur. 20) f(x) = 1x - 2; -3 x 4 Answer: Maximum value is 2 at x = 4; minimum value is - 5 at x = -3 Determine whether the function satisfies the hypotheses of the Mean Value Theorem on the given interval. 21) g(x) = x3/4 on [0, 3] Answer: Yes 22) f(x) = x1/3 on [-2, 5] Answer: No Solve the problem. 23) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph of f that passes through the point P. f' f'' A) B) 4 C) D) Answer: B 24) Using the following properties of a twice-differentiable function y = f(x), select a possible graph of f. x y x<2 -2 10 -2 < x < 0 0 -6 0<x<2 2 -22 x>2 Derivatives y > 0,y < 0 y = 0,y < 0 y < 0,y < 0 y < 0,y = 0 y < 0,y > 0 y = 0,y > 0 y > 0,y > 0 A) B) 5 C) D) Answer: C Use the First Derivative Test to determine the local extrema of the function, and identify any absolute extrema. 25) y = xe2x 1 1 Answer: Absolute minimum at - , 2 2e Use analytic methods to find those values of x for which the given function is increasing and those values of x for which it is decreasing. 26) f(x) = x4 - 8 Answer: Increasing on (-2, 0) and (2, ), decreasing on (- , -2) and (0, 2) Use the Concavity Test to find the intervals where the graph of the function is concave up. 4 27) y = x Answer: (0, ) Find the points of inflection. 4 28) y = x3 - 12x2 + 10x + 47 3 Answer: (3, 5) Give an appropriate answer. 29) Find the value or values of c that satisfy f(b) - f(a) = f (c) for the function f(x) = x2 + 3x + 3 on b-a the interval [-2, 1]. Answer: - 1 2 6 Use a finite approximation to estimate the area of the region enclosed between the graph of f and the x-axis for a x b. 1 30) f(x) = , a = 2, b = 6 x Use Left, Midpoint, and Right rectangular approximation with four rectangles of equal width. Answer: 352 435 Express the limit as a definite integral. n 2 (4 c k - 9ck + 16) 31) limn k=1 xk, [-4, 3] 3 (4x2 - 9x + 16) dx Answer: -4 Evaluate the definite integral. 6 ex dx 32) -4 1 Answer: e6 e4 9 sin x dx 33) 0 Answer: 18 -1 6x-4 dx 34) -2 Answer: 7 4 Find dy/dx. x 10t + 3 dt 35) 0 Answer: 10x + 3 7 x 36) 0 dt 2t + 6 Answer: 1 2x + 6 x10 37) cos t dt 0 Answer: 10x9 cos (x5) Find the general solution to the exact differential equation. dy = e4x - csc x cot x 38) dx Answer: y = 39) 1 4x e + csc x + C 4 dy = csc2x - 20x4 dx Answer: y = -cot x - 4x5 + C Solve the initial value problem explicitly. dy = 6x2 - 4x + 4; y = 15 when x = 1 40) dx Answer: y = 2x3 - 2x2 + 4x + 11 41) dy = sin (5x + ), y = 6 when x = 0 dx Answer: y = - 1 29 cos (5x + ) + 5 5 Solve the initial value problem using the Fundamental Theorem. Your answer will contain a definite integral. dy = cos(x2) and y = 2 when x = 7 42) dx x cos(t2) dt + 2 Answer: y = 7 8 Evaluate the integral. 1 x5 dx 43) x5 Answer: 44) 7 1 sin + 6 dt t t2 Answer: 45) x6 1 + +C 6 4x4 7 1 cos +6 +C 7 t t1/6 sin(t7/6 - 3) dt Answer: - 6 cos(t7/6 - 3) + C 7 Solve the initial value problem using the Fundamental Theorem. Your answer will contain a definite integral. dy = cos(x2) and y = 5 when x = 6 46) dx x cos(t2) dt + 5 Answer: y = 6 Evaluate the integral. cos(3 + 7) d 47) sin2(3 + 7) Answer: - 48) 1 +C 3 sin(3 + 7) sin t (3 + cos t)3 Answer: dt 1 2(3 + cos t)2 +C 9 Solve the problem. 49) A spherical balloon is inflated with helium at a rate of 90 ft3/min. How fast is the balloon's radius increasing when the radius is 3 ft? Answer: 2.50 ft/min 50) A man flies a kite at a height of 50 m. The wind carries the kite horizontally away from him at a rate of 10 m/sec. How fast is the distance between the man and the kite changing when the kite is 130 m away from him? Answer: 9.2 m/sec 51) A ladder is slipping down a vertical wall. If the ladder is 10 ft long and the top of it is slipping at the constant rate of 3 ft/s, how fast is the bottom of the ladder moving along the ground when the bottom is 8 ft from the wall? Answer: 2.3 ft/s Find the area of the shaded region. f(x) = x3 + x2 - 6x g(x) = 6x 52) Answer: 160 3 Find the area of the regions enclosed by the lines and curves. 53) y2 = x + 5 and x = y + 25 Answer: 1331 6 54) y = x4 - 10x2 + 36 and y = 3x2 Answer: 1436 15 10 55) A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions? Answer: 80,000 200 by 400 11