Find all points where the function is discontinuous. 1) Find all vertical asymptotes of the given function. x(x - 1) 2) f(x) =

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1 Math 90 Final Review Find all points where the function is discontinuous. ) Find all vertical asymptotes of the given function. x(x - ) 2) f(x) = x3 + 4x Provide an appropriate response. 3) If x 3 f(x) x for x in [-,], find lim f(x) if it exists. Find the limit, if it exists. 4) lim x 2 x2-5x + 6 x2 + 3x - 0 5) lim x 8 2 Find all horizontal asymptotes of the given function, if any. 6) f(x) = x 2-2 4x - x4

2 Use the graph to evaluate the limit. 7) lim f(x) Solve the problem. 8) Given lim - f(x) = L l, I. lim f(x) = Ll II. III. lim f(x) = Lr lim f(x) does not exist. lim + f(x) = L r, and Ll Lr, which of the following statements is true? For the function f whose graph is given, determine the limit. 9) Find lim f(x) and x -- x -+ lim f(x). 0) lim (x3 + 5x2-7x + ) x 2 2

3 Find the intervals on which the function is continuous. sin (2 ) ) y = 2 Compute the values of f(x) and use them to determine the indicated limit. 2) If f(x) = x 4 - x -, find x lim f(x). x f(x) 3) lim x (9/x2) Find all vertical asymptotes of the given function. x - 7 4) f(x) = 49x - x3 For the function f whose graph is given, determine the limit. 5) Find lim f(x). Find the limit, if it exists. 6) lim x 7 x2 + 2x + Find all points where the function is discontinuous. 7) 3

4 Use the table of values of f to estimate the limit. 8) Let f(x) = x - 4 x - 2, find x 4 lim f(x). x f(x) 9) lim x ( /2)+ tan x A function f(x), a point x0, the limit of f(x) as x approaches x0, and a positive number is given. Find a number > 0 such that for all x, 0 < x - x0 < f(x) - L <. 20) f(x) = 3x, L =48, x0 = 6, and = 0. Use the graph to evaluate the limit. 2) lim f(x) 4

5 For the function f whose graph is given, determine the limit. 22) Find lim x 2+ f(x). Find the limit, if it exists. 23) lim x 4 x - 9 Find all points where the function is discontinuous. 24) 25) lim x -2 x + 2 Give an appropriate answer. 26) Let lim f(x) = 2 and lim x -7 g(x) = 3. Find lim x -7 x -7 8f(x) - 5g(x) 4 + g(x). Find all points where the function is discontinuous. 27) 5

6 Find all horizontal asymptotes of the given function, if any. 28) h(x) = 3x 4-3x2-4 5x5-7x ) h(x) = 9x 3-8x 3x3-5x + 2 Find all points where the function is discontinuous. 30) A function f(x), a point x0, the limit of f(x) as x approaches x0, and a positive number is given. Find a number > 0 such that for all x, 0 < x - x0 < f(x) - L <. 3) f(x) = -7x + 0, L = -, x0 = 3, and = ) lim x - -4x2-3x + 6-8x2-4x + 9 Use the graph to evaluate the limit. 33) lim f(x) Give an appropriate answer. 34) Let lim f(x) = 32. Find lim x 7 x 7 5 f(x). 6

7 Find the intervals on which the function is continuous. 3 35) y = x2-25 Use the graph to evaluate the limit. 36) Find lim f(x) and x (-)- x (-)+ lim f(x) 37) lim x 3+ 2 x2-9 Give an appropriate answer. 38) Let lim f(x) = 4 and x 7 lim g(x) = 5. Find x 7 lim [f(x) g(x)]. x 7 Find all points where the function is discontinuous. 39) A function f(x), a point x0, the limit of f(x) as x approaches x0, and a positive number is given. Find a number > 0 such that for all x, 0 < x - x0 < f(x) - L <. 40) f(x) = 6x + 7, L = 9, x0 = 2, and = 0.0 Find all points where the function is discontinuous. 4) 7

8 Find the derivative of the function. 42) y = log 9 (3x2-2x) 5/2 Find the derivative. 43) r = 3-7 cos The graph of a function is given. Descripe the form of the graph of its derivative. 44) Calculate the derivative of the function. Then find the value of the derivative as specified. 45) Find s' if s = t2 - t. Then find s'(4). Find the second derivative of the function. 46) s = t 7 + 6t + 5 t2 Find the indicated tangent line. 47) Find the tangent line to the graph of f(x) = 6e7x at the point (0, 6). Find dy/dx as a function of x. 48) y = e7x/2 Find the derivative of the function. 49) y = x3cos x - 0x sin x - 0 cos x 50) s = sin t 2 - cos t 2 Find an equation of the tangent line at x = a. 5) y = x 2 4 ; a = -3 Find the derivative of the function. 52) y = log -9x 8

9 Find the derivative of y with respect to x. 53) y = 5x3 sin- x Find y. 54) y = (3x - 5)(2x + ) Find the derivative of y with respect to x. 55) y = -sin- (3x2-4) Use implicit differentiation to find dy/dx. 56) xy + x + y = x2y2 Use differentiation to determine whether the integral formula is correct. 57) (3x - 3)4 dx = (3x - 3) 5 + C 5 Find the largest open interval where the function is changing as requested. 58) Increasing f(x) = x2 + Find the absolute extreme values of the function on the interval. 59) f(x) = 5x4/3, -27 x Find the extreme values of the function and where they occur. 60) y = x2e-x + 2xe-x Find the most general antiderivative. 6) (3x3-0x + 4) dx 62) lim + 2 x3 x 9

10 Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. 63) Find an antiderivative of the given function. 64) 7 4 x 5/4 Find the most general antiderivative. 65) 5t2 + t 7 dt Find an antiderivative of the given function. 66) 8 cos 9x Find the value or values of c that satisfy the equation the function and interval. 67) f(x) = x2 + 3x + 4, [-3, 2] f(b) - f(a) b - a = f (c) in the conclusion of the Mean Value Theorem for Use differentiation to determine whether the integral formula is correct. 68) sec (2x - 5) sec (2x - 5) tan (2x - 5) dx = + C 2 Find the most general antiderivative. 69) (-2 cos t) dt 0

11 Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. 70) Solve the problem. 7 7) Suppose that g is continuous and that g(x) dx = 9 and g(x) dx = 6. Find 0 7 g(x) dx. 0 Use the substitution formula to evaluate the integral. /8 72) ( + etan 2x) sec2 2x dx 0 Find the average value of the function over the given interval. 73) y = 3 - x2; [-2, 2] Evaluate the sum. 74) 7 k = k Find the total area of the region between the curve and the x-axis. 75) y = x2(x - 2)2; 0 x 2 Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 76) f(x) = x2 between x = 0 and x = using a right sum with two rectangles of equal width. Evaluate the integral. 77) ex dx - e2x Use the substitution formula to evaluate the integral. 78) 4 r dr r2

12 Find the total area of the region between the curve and the x-axis. 79) y = x2 + ; 0 x Evaluate the integral. 80) 5x x3 dx Evaluate the sum. 8) 4 k = k3 Find the derivative. tan x 82) y = 0 t dt Evaluate the integral. 4 83) 4 - x x dx Find the derivative. d x 84) dx 4t5 dt Evaluate the integral. 6 85) 6x5 dx -2 Use the substitution formula to evaluate the integral. /2 86) cos x (4 + 2 sin x)3 dx 0 Find the average value of the function over the given interval. 87) f(x) = 0x on [, 3] Evaluate the integral. 3 /4 88) 7 sec tan d - /4 Evaluate the sum. 89) 6 k = k2-8 2

13 Find the area of the shaded region. 90) Compute the definite integral. 3 9) x3 dx -3 Evaluate the integral. 92) x3 x4 + 2 dx Find the total area of the region between the curve and the x-axis. 93) y = x2-6x + 9; 2 x 4 Solve the problem. -2 g(x) 94) Suppose that g(t) dt = -3. Find dx and - g(t) dt Find the area of the shaded region. 95) Approximate the area under the graph of f(x) over the specified interval by dividing the interval into the indicated number of subintervals and using the left endpoint of each subinterval. 96) f(x) = ; interval [, 5]; 4 subintervals x2 97) f(x) = x2 + 2; interval [0, 5]; 5 subintervals 98) f(x) = x3 + x2 + ; interval [0, 4]; 4 subintervals 3

14 Solve the problem. 99) Water is falling on a surface, wetting a circular area that is expanding at a rate of 6 mm2/s. How fast is the radius of the wetted area expanding when the radius is 70 mm? (Round your answer to four decimal places.) 00) Assume that the profit generated by a product is given by P(x) = 3 x, where x is the number of units sold. If the profit keeps changing at a rate of $600 per month, then how fast are the sales changing when the number of units sold is 900? (Round your answer to the nearest dollar per month.) 0) A piece of land is shaped like a right triangle. Two people start at the right angle of the triangle at the same time, and walk at the same speed along different legs of the triangle. If the area formed by the positions of the two people and their starting point (the right angle) is changing at 2 m2/s, then how fast are the people moving when they are 3 m from the right angle? (Round your answer to two decimal places.) 02) A manufacturer estimates that the profit from producing x units of a commodity is -x2 + 40x - 00 dollars per week. What is the maximum profit he can realize in one week? 03) Suppose a ball is thrown into the air and after t seconds has a height of h(t) = -6 t2 + 80t feet. When will it reach its maximum height? 04) A rectangular corral with a total area of 60 square meters is to be fenced off and then divided into 2 rectangular sections by a fence down the middle. The fencing for the outside costs $9 per running meter, whereas that for the interior dividing fence costs $2 per running meter. Which of the following statements hold, if the cost (c) of the fencing is to be maximized? (I) The constraint equation is 3w + 2 l = 60. (II) The objective equation is 2l w = 60. (III) The constraint equation is w l = 60. (IV) The objective equation is C = 30w + 8l. (V) The constraint equation is C = 2w + 9wl. (VI) The objective equation is C = 60 - lw. 05) What is the maximum area that can be enclosed in a rectangular shape with 00 feet of fence if one of the two long sides is not fenced (there is a natural boundary there)? Solve the problem. 06) Find two numbers whose sum is 350 and whose product is as large as possible. 4