Name Date Period. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

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1 AB Fall Final Exam Review Name Date Period MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ) The position of a particle moving along a coordinate line is s = 3 + 6t, with s in meters and t in seconds. Find the particle's velocity at t = sec. - 3 m/sec m/sec 2 m/sec m/sec 6 ) 2) At time t, the position of a body moving along the s-axis is s = t3-2t2 + 20t m. Find the body's acceleration each time the velocity is zero. a(0) = 8 m/sec2, a(4) = -8 m/sec2 a(0) = -8 m/sec2, a(4) = 8 m/sec2 a(0) = 0 m/sec2, a(4) = 0 m/sec2 a(20) = 20 m/sec2, a(8) = 20 m/sec2 2) 3) Assume that a watermelon dropped from a tall building falls y = 6t2 ft in t sec. Find the watermelon's speed at the instant t = 4 sec. 6 ft/sec 30 ft/sec 64 ft/sec 28 ft/sec 3) Find dy/dx. 4) y = log (x - 4) x - 4 ln 0 (x - 4) ln 0 ln 0 (x - 4) ln 0 4) ) y = x - x + ) x + x( x + ) 2 - x( x + ) 2 0 (x + ) x - 2 6) y = 2 secx 0 tan2x sec4x 0 tan x secx 0 sec4x 0 tan2x sec x 6) 7) y = 0xex - 0ex 0xex 0x 0ex 0xex + 20ex 7) 8) y = ln (ln 6x) x ln 6x 6x ln 6x x 8) 9) y = 24-x 24-x -24-x ln 24 (24-x) - ln 24 (24-x) 9) D-

2 0) y = (2-9x2)(9x2-324) -324x x -324x x2 8x x -324x ) ) y = 6x - 8 x2 + 6 ) -30x x + 84 (x2 + 6) 2 90x 2-80x + 36 (x2 + 6) 2-30x x + 36 (x2 + 6) 2 30x 3-60x2 + 6x (x2 + 6) 2 Determine the limit algebraically, if it exists. x ) lim x 0 x 2) Does not exist 3) lim x 2 x2 + 3x - 0 x - 2 Does not exist ) Find the indicated derivative. 4) Find y if y = 7x sin x. y = 4 cos x - 7x sin x y = - 4 cos x + 7x sin x y = - 7x sin x y = 7 cos x - 4x sin x 4) Use the First Derivative Test to determine the local extrema of the function, and identify any absolute extrema. ) y = xex ) Absolute minimum at, e Absolute minimum at -, - e Absolute maximum at, e Absolute maximum at -, - e 6) y = xe7x 6) Absolute minimum at 7, e 7 Absolute maximum at - 7, - e 7 Absolute maximum at 7, 7e Absolute minimum at - 7, - 7e Find the horizontal tangents of the curve. 7) y = x4-8x2-4 At x = 0 At x = 0, 3 At x = 3, -3, At x = 0, 3, -3 7) D-2

3 Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 8) ) f(x) + g(x) at x = Give an appropriate answer. 9) Find the value or values of c that satisfy interval [3, 6]. f(b) - f(a) b - a = f (c) for the function f(x) = x + 48 on the x 3, 6-4 3, 4 3 0, ) Find the limit. 20) Let lim f(x) = -0 and x -4 lim g(x) = -4. Find x -4 lim x -4 [f(x) + g(x)] ) Find the derivative at each critical point and determine the local extreme values. 2) y = x2/3(x2-4); x 0 x = 0 x = Undefined 0 local max minimum 0 x = 0 Undefined local max 0 x = 0 minimum -3 2) x = 0 0 maximum 0 x = 0 minimum -3 x = 0 Undefined local max 2 x = 0 minimum -3 Find the derivative of the given function. 22) y = sin- x - 2x2 - (sin- x)2 22) - - 2x2 (sin- x) x2 23) y = tan- x - x 2( + x) x 22 x( + x) + x 23) D-3

4 Determine the limit by substitution. 24) lim x 8 x x + 8 Does not exist ) Find dy/dx by implicit differentiation. If applicable, express the result in terms of x and y. 2) cos xy + x6 = y6 6x - y sin xy 6y + x sin xy 6x + y sin xy 6y - x sin xy 6x + x sin xy 6y 6x - x sin xy 6y 2) Find the indicated limit. 0x 26) lim x x Does not exist -0 26) Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 27) u(2) = 9, u (2) = 4, v(2) = -2, v (2) = -. 27) d (uv) at x = 2 dx Find y. 28) y = x x (x + 3)3/2 2 x + 3-4(x + 3)3/2 28) Find the extreme values of the function and where they occur. x + 29) y = x2 + 2x ) The maximum is 2 at x = 0; the minimum is - 2 at x = -2. The maximum is 2 at x = 0; the minimum is 2 at x = -2. The maximum is - 2 at x = 0; the minimum is 2 at x = -2. There are none. Use analytic methods to find the local extrema. x - 30) h(x) = x2 + x + 0 Local minimum at x = -4; local maximum at x = Local minimum at x = -3; no local maxima Local minimum at x = -3; local maximum at x = No local extrema 30) D-4

5 Determine the values of x for which the function is differentiable. 3) y = x + 3 All reals greater than 3 All reals greater than -3 All reals except -3 All reals greater than or equal to -3 3) Find the value of (f g) at the given value of x. 32) f(u) = u, u = g(x) = x - x 2, x = 32) Use logarithmic differentiation to find dy/dx. 33) y = (cos x)x ln cos x - x tan x (cos x)x (ln cos x + x cot x) (cos x)x (ln cos x - x tan x) ln x(cos x)x - 33) D-

6 Answer Key Testname: AB FALL FINAL ) D 2) A 3) D 4) D ) B 6) B 7) A 8) A 9) D 0) A ) C 2) A 3) C 4) A ) B 6) D 7) D 8) C 9) D 20) B 2) B 22) C 23) C 24) B 2) A 26) B 27) C 28) B 29) A 30) C 3) B 32) C 33) C D-6