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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

In Exercises, find the one-sided limit (if it exists). X lim x-(1/2) + 2x - 1
In Exercises, find the one-sided limit (if it exists). x + 1 lim x→-1-x² - 1
In Exercises, find the one-sided limit (if it exists). lim x x-0+ EX
In Exercises find the vertical asymptotes (if any) of the graph of the function.ƒ(x) = csc πX
In Exercises describe the interval(s) on which the function is continuous. f(x): = X x² + x + 2
In Exercises describe the interval(s) on which the function is continuous. x^ I + x f(x) =
In Exercises, find the one-sided limit (if it exists). lim x→0+ sin 4x 5x
In Exercises describe the interval(s) on which the function is continuous. f(x)=x√√√x + 3
In Exercises describe the interval(s) on which the function is continuous.ƒ(x) = 3 - √x
In Exercises, find the limit (if it exists). If it does not exist, explain why. lim h(t), where h(t) t→1 = [1³ + 1, t < 1 (t + 1), t > 1
In Exercises find the one-sided limit (if it exists). lim XT X CSC X
In Exercises, find the limit (if it exists). If it does not exist, explain why. lim f(s), where f(s) s-2 = [-s²4s2, s ≤ -2 s² + 4s + 6, $2 S> 2
In Exercises find the one-sided limit (if it exists). x + 2 lim x→0-cot x
In Exercises, find the limit (if it exists). If it does not exist, explain why. lim (2[x] + 1) x-2-
In Exercises find the one-sided limit (if it exists). lim x-(1/2)- x X sec TX
In Exercises, find the limit (if it exists). If it does not exist, explain why. lim [x - 1] x-4
In Exercises find the one-sided limit (if it exists). lim x² tan TTX x→(1/2)+
In Exercises use a graphing utility to graph the function and determine the one-sided limit. f(x) I + x + zx lim f(x) x→1+ x³ - 1
In Exercises use a graphing utility to graph the function and determine the one-sided limit. f(x) I - EX x² + x + 1 lim f(x) x-1-
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable?ƒ(x) = x² - x + 20
In Exercises, find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = 4 x - 5
In Exercises use a graphing utility to graph the function and determine the one-sided limit. f(x) 1 x² - 25 lim f(x) x-5-
In Exercises , find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x): = 1 6-zx
In Exercises, find the one-sided limit (if it exists). sec x lim x→0+ X
In Exercises describe the interval(s) on which the function is continuous. f(x) = = sec TTX 4
In Exercises, find the one-sided limit (if it exists). lim x→0+ csc 2x X
In Exercises explain why the function has a zero in the given interval. Function f(x)=x²2 cos x - Interval [0, π]
In Exercises explain why the function has a zero in the given interval. Function f(x) = x³ + 5x - 3 Interval [0, 1]
In Exercises explain why the function has a zero in the given interval. Function f(x) = 1/2x4x³ + 4 Interval [1, 2]
In Exercises use a graphing utility to graph the function on the interval [-4, 4]. Does the graph of the function appear to be continuous on this interval? Is the function continuous on [-4, 4]? Write a short paragraph about the importance of examining a function analytically as well as
In Exercises use a graphing utility to graph the function on the interval [-4, 4]. Does the graph of the function appear to be continuous on this interval? Is the function continuous on [-4, 4]? Write a short paragraph about the importance of examining a function analytically as well as
The function is defined as ƒ shown.(a) Find(if it exists).(b) Can the function ƒ be defined at x = 0 such that it iscontinuous at x = 0? f(x) = tan 2x X x = 0
In Exercises describe the interval(s) on which the function is continuous. f(x) = [2x - 4, x = 3 (1, x = 3
A utility company burns coal to generate electricity. The cost C in dollars of removing p% of the air pollutants in the stack emissions is(a) Find the cost of removing 15% of the pollutants.(b) Find the cost of removing 50% of the pollutants.(c) Find the cost of removing 90% of the pollutants.(d)
In Exercises describe the interval(s) on which the function is continuous. f(x) = x² - 1 x - 1' 2. x 1 x = 1
In Exercises, find the one-sided limit (if it exists). lim x-0- cos² x
In Exercises explain why the function has a zero in the given interval. Function f(x) 5 X + tan TTX 10 Interval [1,4]
In Exercises use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to
In Exercises use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to
In Exercises use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to
In Exercises use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to
In Exercises verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. ƒ(x) = x² + x − 1, [0,5], f(c) = 11
In Exercises verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. ƒ(x) = x² − 6x + 8, [0, 3], f(c) = 0 -
In Exercises verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. f(x) = x³ − x² + x2, [0, 3], f(c) = 4
In Exercises verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. f(x) = x² + x x - 1' 5 4 2² f(c) = 6
State how continuity is destroyed at x = c for each of the following graphs.a.b.c.d. y C X
Sketch the graph of any function such thatIs the function continuous at x = 3? Explain. lim f(x) = 1 and X-3+ lim f(x) = 0. x-37
If the functions ƒ and g are continuous for all real x, is ƒ + g always continuous for all real x? Is ƒ/g always continuous for all real x? If either is not continuous, give an example to verify your conclusion.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The functionis continuous on (-∞, ∞). f(x) = |x − 1| X - 1
Describe the difference between a discontinuity that is removable and one that is nonremovable. In your explanation, give examples of the following descriptions.(a) A function with a nonremovable discontinuity at x = 4(b) A function with a removable discontinuity at x = −4(c) A function that has
Describe how the functionsdiffer. f(x) = 3+ [x] and g(x) = 3− [−x]
Every day you dissolve 28 ounces of chlorine in a swimming pool. The graph shows the amount of chlorine ƒ(t) in the pool after t days. Estimate and interpret lim f(t) and lim f(t).
The number of units in inventory in a small company is given bywhere is the time in months. Sketch the graph of this function and discuss its continuity. How often must this company replenish its inventory? +2 N(t) = 25 ( 2 [¹ + ²1-₁) 2
A long distance phone service charges $0.40 for the first 10 minutes and $0.05 for each additional minute or fraction thereof. Use the greatest integer function to write the cost C of a call in terms of time t (in minutes). Sketch the graph of this function and discuss its continuity.
At 8:00 A.M. on Saturday, a man begins running up the side of a mountain to his weekend campsite (see figure). On Sunday morning at 8:00 A.M., he runs back down the mountain. It takes him 20 minutes to run up, but only 10 minutes to run down. At some point on the way down, he realizes that he
Use the Intermediate Value Theorem to show that for all spheres with radii in the interval [5, 8], there is one with a volume of 1500 cubic centimeters.
Prove that if ƒ is continuous and has no zeros on[a, b], then either f(x) > 0 for all x in [a, b] or f(x) < 0 for all x in [a, b].
Show that the Dirichlet functionis not continuous at any real number. f(x): = 0, 1, if x is rational if x is irrational
The table lists the speeds S (in feet per second) of a falling object at various times t (in seconds).(a) Create a line graph of the data.(b) Does there appear to be a limiting speed of the object? If there is a limiting speed, identify a possible cause. t 0 S 0 5 48.2 10 53.5 15 20 25 55.2 55.9
A swimmer crosses a pool of width b by swimming in a straight line from (0, 0) to (2b, b). (See figure.)(a) Let ƒ be a function defined as the y-coordinate of the point on the long side of the pool that is nearest the swimmer at any given time during the swimmer's crossing of the pool. Determine
Find all values of c such that ƒ is continuous on (-∞, ∞). f(x) = 1 - x2, x = c Χ. X > C
LetWhat is the domain of ƒ? How can you define ƒ at x = 0 in order for ƒ to be continuous there? f(x) = √x + c²-c X C > 0.
Prove that for any real number y there exists x in (- π/2, π/2) such that tan x = y.
Continuity of a Function Discuss the continuity of the function ℎ(x) = x[[x]].
Proof(a) Let ƒ1(x) and ƒ2(x) be continuous on the closed interval [a, b]. If ƒ1(a) < ƒ2(a) and ƒ1(b) > ƒ2(b), prove that there exists c between a and b such that ƒ1(c) = ƒ2(c).(b) Show that there exists c in [0, π/2] such that cos x = x. Use a graphing utility to approximate c to
If x and y are real numbers with y ≥ 0 and y(y + 1) ≤ (x + 1)², then y(y − 1) ≤ x².
Determine all polynomials P(x) such thatP(x² + 1) = (P(x))² + 1 and P(0) = 0.
In Exercises find the limit (if it exists). If it does not exist, explain why. 1 lim x8+ x + 8
In Exercises find the limit (if it exists). If it does not exist, explain why. 2 lim x 2 x + 2
In Exercises find the limit (if it exists). If it does not exist, explain why. 4- x lim x4+x² - 16
Use the rectangles in each graph to approximate the area of the region bounded by y = 5/ x, y = 0, x = 1, and x = 5. Describe how you could continue this process to obtain a more accurate approximation of the area. 5 4 3 ترا 2 1 y 1 2 3 4 5 x 5 4 3 دیا 2 y 1 2 3 4 5
In Exercises find the limit (if it exists). If it does not exist, explain why. x-5 lim x+5+ x² - 25
In Exercises find the limit (if it exists). If it does not exist, explain why. 4- x lim x4+x² - 16
In Exercises discuss the continuity of each function. f(x) = -3-2 X, 2, (2x نیا 2 1 -2. -3+ x < 1 x = 1 1, x>1 1 2 3 ➤X
In Exercises discuss the continuity of each function. f(x) = /[x] + x + -3-2-1 3 2 1 y -3+ 1 2 3 X
In Exercises discuss the continuity of the function on the closed interval. Function f(t) = 3√√√9 - 1² Interval [-3,3]
In Exercises discuss the continuity of each function. f(x) 2 – 1 x + 1 y 3 2 1 -3-2-1 -3 11 1 2 3
In Exercises discuss the continuity of the function on the closed interval. Function g(x) = √√√49 - x² Interval [-7,7]
In Exercises discuss the continuity of the function on the closed interval. Function f(x) = 3x, x≤0 (3 + ²/1 x ₂ x > 0 Interval [-1,4]
In Exercises discuss the continuity of the function on the closed interval. Function g(x) = 1 x² - 4 Interval [-1,2]
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? X 9 61 (x)ƒ
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = 4 x-6
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? ƒ(x) = x² - 9
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable?ƒ(x) = x² − 4x + 4
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = 1 I + zx
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = = COS TTX | 2
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable?ƒ(x) = 3x - cos x
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? X- z.X = f(x) X
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = x-5 -2 X 25
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = X x² - 4 12
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = X x² + 1 2
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = x + 2 x²-3x - 10
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = = x + 2 9-x-zx
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = |x + 71 x + 7
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) _ ]x - 5] x - 5
In Exercise use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph about
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = [x, x ≤ 1 x², x > 1
In Exercise use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph about
In Exercise use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph about
In Exercises find the x-values (if any) at which ƒ is not continuous. Which of the discontinuities are removable? f(x) = √√√x + 1, x ≤ 2 2x 3 - x, x > 2

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