Problem 1QCE: What three conditions are satisfied if f is continuous at x=c ? Problem 2QCE: Suppose that f and g are continuous functions such that f2=1 and limx2fx+4gx=13 . Find (a) g2 (b)... Problem 3QCE: Suppose that f and g are continuous functions such that limx3gx=5 and f3=2 . Find limx3fx/gx . Problem 4QCE: For what values of x, if any, is the function fx=x216x25x+4 Discontinuous? Problem 5QCE: Suppose that function f is continuous everywhere and that f2=3,f1=1,f0=4,f1=1 , and f2=5 . Does the... Problem 1ES: Let f be the function whose graph is shown. On which of the following intervals, if any, is f... Problem 2ES: 1-4 Let f be the function whose graph is shown. On which of the following intervals, if any, is f... Problem 3ES: 1-4 Let f be the function whose graph is shown. On which of the following intervals, if any, is f... Problem 4ES: 1-4 Let f be the function whose graph is shown. On which of the following intervals, if any, is f... Problem 5ES: Consider the functions fx=1,x41,x=4 and gx=4x10,x46,x=4 In each part, is the given function... Problem 6ES: Consider the functions fx=1,0x0,x0 and gx=0,0x1,x0 In each part, is the given function continuous at... Problem 7ES: In each part sketch the graph of a function f that satisfies the stated conditions. (a) f is... Problem 8ES: The accompanying figure models the concentration C of medication in the bloodstream of a patient... Problem 9ES: A student parking lot at a university charges $2.00 for the first half hour (or any part) and $1.00... Problem 10ES: In each part determine whether the function is continuous or not, and explain your reasoning. (a)... Problem 11ES Problem 12ES: Find values of x, if any, at which f is not continuous. fx=x83 Problem 13ES: Find values of x, if any, at which f is not continuous. fx=x+2x2+4 Problem 14ES: Find values of x, if any, at which f is not continuous. fx=x+2x24 Problem 15ES: Find values of x, if any, at which f is not continuous. fx=x2x2+x Problem 16ES: Find values of x, if any, at which f is not continuous. fx=2x+14x2+4x+5 Problem 17ES: Find values of x, if any, at which f is not continuous. fx=3x+x1x21 Problem 18ES: Find values of x, if any, at which f is not continuous. fx=5x+2xx+4 Problem 19ES: Find values of x, if any, at which f is not continuous. fx=x2+6x+9x+3 Problem 20ES: Find values of x, if any, at which f is not continuous. fx=48x4+x Problem 21ES: Find values of x, if any, at which f is not continuous. fx=2x+3,x47+16x,x4 Problem 22ES: Find values of x, if any, at which f is not continuous. fx=3x1,x13,x=1 Problem 23ES: Determine whether the statement is true or false. Explain your answer. If fx us continuous at x=c,... Problem 24ES: Determine whether the statement is true or false. Explain your answer. If fx us continuous at x=c,... Problem 25ES: Determine whether the statement is true or false. Explain your answer. If f and g are continuous at... Problem 26ES: Determine whether the statement is true or false. Explain your answer. If f and g are discontinuous... Problem 27ES: Determine whether the statement is true or false. Explain your answer. If fx us continuous at x=c,... Problem 28ES: Determine whether the statement is true or false. Explain your answer. If fx us continuous at x=c,... Problem 29ES: Find a value of the constant k, if possible, that will make the function continuous everywhere. (a)... Problem 30ES: Find a value of the constant k, if possible, that will make the function continuous everywhere. (a)... Problem 31ES: Find values of the constants k and m, if possible, that will make the function f continuous... Problem 32ES: On which of the following intervals is fx=1x2 continuous? (a) 2,+ (b) ,+ (c) 2,+ (d) 1,2 Problem 33ES: A function is said to have a removable discontinuity at if exists but is not continuous at ... Problem 34ES: A function f is said to have a removable discontinuity at x=c if limxcfx exists but f is not... Problem 35ES: A function is said to have a removable discontinuity at if exists but is not continuous at ... Problem 36ES: A function is said to have a removable discontinuity at if exists but is not continuous at ... Problem 37ES: (a) Use a graphing utility to generate the graph of the function fx=x+3/2x2+5x3, and then use the... Problem 38ES: (a) Use a graphing utility to generate the graph of the function fx=x/x3x+2, and then use the graph... Problem 39ES: Prove that fx=x3/5 is continuous everywhere, carefully justifying each step. Problem 40ES: Prove that is continuous everywhere, carefully justifying each step.
Problem 41ES: Prove: (a) Part a of Theorem 1.5.3 (b) Part b of Theorem 1.5.3 (c) Part c of Theorem 1.5.3 . Problem 42ES: Prove part of Theorem .
Problem 43ES: (a) Use Theorem 1.5.5 to prove that if f is continuous at x=c , that limh0fc+h=fc . (b) Prove that... Problem 44ES: Prove: If f and g are continuous on a,b, and faga,fbgb, then there is at least one solution of the... Problem 45ES: Give an example of a function that is defined on a closed interval, and whose values at the... Problem 46ES: Let f be the function whose graph is shown in Exercise 2. For each interval, determine (i) whether... Problem 47ES: Show that the equation x3+x22x=1 has at least one solution in the interval 1,1 . Problem 48ES: Prove: If is a polynomial of odd degree, then the equation has at least one real solution.
Problem 49ES: The accompanying figure on the next page shows that graph of the equation y=x4+x1 . Use the method... Problem 50ES: The accompanying figure shows that graph of the equation y=5xx4 . Use the method of Example 6 to... Problem 51ES: Use the fact that is a solution of to approximate with an error of at most .
Problem 52ES: A sprinter, who is timed with a stopwatch, runs a hundred yard dash in 10s . The stopwatch is reset... Problem 53ES: Prove that there exist points on opposite sides of the equator that are at the same temperature. Problem 54ES: Let R denote an elliptical region in the xy-plane, and define fz to be the area within R that is on,... Problem 55ES: Let R denote an elliptical region in the plane. For any line L , prove there is a line perpendicular... Problem 56ES: Suppose that f is continuous on the interval 0,1 and that 0fx1 for all x in this interval. (a)... Problem 57ES: Let fx=x6+3x+5,x0 . Show that f is an invertible function and that f1 is continuous on 5,+ . Problem 58ES: Suppose that f is an invertible function, f0=0,f is continuous at 0, and limx0fx/x exists. Given... format_list_bulleted
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