MATH 1A.12: Review Assignment Test #1 Due: Friday, October 16 To receive full credit, this assignment should be completed neatly, in numerical order, with proper support of results. Decide whether the limit exists. If it exists, find its value. If it does not exist, explain why. 1) Find lim f(x). x 0 2) Find lim f(x). x 1 Graph the function and then find the specified limit. When necessary, state that the limit does not exist. 9) f(x) = 3 - x, for x 2, ; lim f(x) 1 + 2x, for x > 2. x 2+ In the exercise below, the initial substitution of x = a yields the form 0/0. Look for ways to simplify the function algebraically. When necessary, state that the limit does not exist. All answers should be supported with work. x2 - 64 10) lim x -8 x + 8 11) lim x 16 12) lim x -3 Graph the function and then find the specified limit. If necessary, state that the limit does not exist. 1 ; lim f(x) 3) f(x) = x + 2 x -2 x - 16 x-4 2x2 - 2x - 24 9 - x2 Determine the continuity of the function at the given points. 2, for x = -2 at x = -2 and x = -3 13) f(x) = 0.5, for x -2 Find the limit. Explain or show your reasoning. 1 4) lim x+3 x -3 5) 6) 7) 8) 1 lim (x - 7)2 x 7+ lim x 1- x2 - 4x + 3 x3 - x 5 lim 2 - 16 x -4 - x lim sec x x (- /2)- Provide an appropriate response. 14) Is the function given by f(x) = 16x + 3 continuous at x = 2? Why or why not? 15) Is the function given by f(x) = x+5 x2 - 11x + 30 continuous at x = 5? Why or why not? 16) Is the function given by f(x) = 1 , for x > 4 x-4 continuous at x = 4? Why x2 - 2x, for x 4 or why not? From the graph of f(x), indicate the values where f is discontinuous and indicate the TYPE of discontinuity. For each of these values, state whether f is continuous from the right, left, or neither. 23) Find the intervals on which the function is continuous. 1 continuous 17) Is the function given by f(x) = x+3 over the interval ( , 0)? Why or why not? 18) Is the function given by f(x) = 5 (x + 1)2 + 2 continuous on ? Why or why not? Provide an appropriate response. 19) Given f(x) = x + 4 and g(x) = x - 4, where is the function f(x)/g(x) continuous? Solve the problem. 20) The average price of a ticket to a minor league baseball game can be approximated by p(x) = 0.04x2 + 0.44x + 6.96, Use the graph to determine how close x must be to x 0 in order that f(x) is within of L. Your answer should be of the form "If x is within ___ of ___, then f(x) is within ____ of ___ ." 24) where x is the number of years after 1990 and p(x) is in dollars. (i) Find p(4). (ii) Find p(12). (iii) Find p(12) - p(4). p(12) - p(4) (iv) Find , and interpret this result. 12 - 4 (v) What is the instantaneous rate of change of the average price of ticket sales in 1994? How did you find it? State whether the function is continuous at the indicated point. If it is not continuous, tell why. 21) State whether r(t) is continuous at the point t = 8. t2 - 64 if t 8 r(t) = t-8 16 if t = 8 22) State whether r(t) is continuous at the point t = 4. t3 - 64 if t 4 r(t) = t-4 16 if t = 4 f(x) = 5x + 1 x0 = 2 L = 11 = .2 Find numbers a and b, or k, so that f is continuous at every point. 25) x < -4 -7, f(x) = ax + b, -4 x 3 21, x>3 26) f(x) = Solve the problem. 32) You are asked to make some circular cylinders, each with a cross-sectional area of 6 cm2 . To 3x + 8, if x < -1 kx + 4, if x -1 Determine whether the function is continuous at the given point c. If the function is not continuous, determine whether the discontinuity is removable or nonremovable. x2 - 36 ;c=6 27) f(x) = x-6 28) f(x) = 49 - x ; c = 49 7- x Use the graph to determine how close x must be to x 0 in order that f(x) is within of L. Your answer should be of the form "If x is within ___ of ___, then f(x) is within ____ of ___ ." 29) do this, you need to know how much deviation from the ideal cylinder diameter of x0 = 2.65 cm you can allow and still have the area come within 0.1 cm2 of the required 6 cm2. To find x 2 out, let A = and look for the interval in 2 which you must hold x to make A - 6 < 0.1. What interval do you find? Find the slope of a line tangent to the given curve at the point P by finding the limit of the slopes of the secant lines PQ as Q approaches P. 33) y = 3x2 - 3; P is (6, 105); let Q have x-values of 5.5, 5.9, 5.99, 5.999 34) y = x3 - 2x; P is (-2, -4); let Q have x-values of -1.5, -1.9, -1.99, -1.999 Find an equation for the line tangent to the given curve at the indicated point. 10x at (1, 5) 35) y = x2 + 1 36) y = 4 x - x + 1 at (16, 1) The function s gives the displacement (in feet) of an object as a function of time (in seconds). Calculate values of the average velocity for the given values of t and find the instantaneous velocity at the indicated value of t by noting the limit of the average velocities. 37) s = 6t2 - 2; when t = 3; f(x) = x2 + 2 x0 = 2 use values of t of 2.5, 2.9, 2.99, 2.999 L=6 =1 38) s = 80t - 16t2 ; when t = 0.5; use values of t of 0.45, 0.49, 0.499, 0.4999 Let L be the value of lim f(x). Use graphical methods, x x0 and the additional information provided, to find a that if 0 < x - x 0 < , then f(x) - L < . 30) Let f(x) = x - 1, x0 = 2, L = 1, and = 0.25 31) Let f(x) = x2 + 2, x0 = 2, L = 6, and = 1.0 > 0 so