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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim sec 0 6
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” Z+3 lim 2-3 z² + 4z +3
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” x³ lim x→1 ax² + ax - 1 x-1
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim x-b x³ - 6³ x-b
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” sin 4x lim x-0 sin 3x
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” 1 lim x-0 3x 1 x(x + 3)
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim 3tan (7) 01/201
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” [x] lim X- x→0- X
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim X-0+ X
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim 0 sec 0 0 k
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim sin y-3 n y -1/2
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim 0→0 cos - 2 0
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” 1 lim x 4.3 X - [x]
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” x-3 lim x-2 X-2
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” sin² t lim 1-0 1³
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim x-1+ 1 √x - 1 1 1 √x² - 1/
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim √2t(cost-1)
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim tan x 2 X-
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim cos 1-0 t
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim √t cos 1→0+ 1 t
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” x² - 24 lim x+5+ x² - 25
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim x→0 cos x 1 sin x
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim 0→0 tan - sin 0 sin³ 0
Find the left- and right-hand limits of the function ƒ in Figure 1 at x = 0, 2, 4. State whether ƒ is left- or right continuous (or both) at these points. 2 y 2 3 5 X
Sketch the graph of a function ƒ such that (a) lim f(x) = 1, x-2- lim f(x) = 3 X→2+ (b) lim f(x) exists but does not equal ƒ(4). X-4
Graph h and describe the discontinuity:Is h left- or right-continuous? h(x) = 2.x for x ≤ 0 (x1/2 for x>0
Sketch the graph of a function g such that lim g(x) = ∞0, x--3- lim g(x) = -00, X-3+ lim g(x) = ∞ X-4
Find the points of discontinuity ofDetermine the type of discontinuity and whether g is left- or right-continuous. g(x) = COS TX 2 |x-1| for |x| < 1 for x ≥ 1
Find a constant b such that h is continuous at x = 2, whereWith this choice of b, find all points of discontinuity. h(x) = √x+1 b-x² for x < 2 for x ≥ 2
In Exercises 57–64, find the horizontal asymptotes of the function by computing the limits at infinity. f(x) = 9x² - 4 2x² - x
In Exercises 57–64, find the horizontal asymptotes of the function by computing the limits at infinity. f(x) = x²-3x4 x-1
In Exercises 57–64, find the horizontal asymptotes of the function by computing the limits at infinity. f(u) = 2u² - 1 √6 + uª
In Exercises 57–64, find the horizontal asymptotes of the function by computing the limits at infinity. f(x) = 3x2/3 +9x3/7 7x4/5 - 4x-1/3
In Exercises 57–64, find the horizontal asymptotes of the function by computing the limits at infinity. f(t) = 1/3-1/3 (t-t-1)1/3
In Exercises 57–64, find the horizontal asymptotes of the function by computing the limits at infinity. f(t) = = 17 1+2¹
Calculate (a)–(d), assuming that (a) lim(f(x) - 2g(x)) X-3 f(x) x-3 g(x) + X (c) lim lim f(x) = 6, x-3 lim g(x) = 4 X-3 (b) lim x² f(x) x→3 (d) lim(2g(x)³ - g(x)³¹/2)
In Exercises 57–64, find the horizontal asymptotes of the function by computing the limits at infinity. g(x) = 6 1-32x
Assume that the following limits exist:Prove that if L = 1, then A = B. You cannot use the Quotient Law if B = 0, so apply the Product Law to L and B instead. A = lim f(x), x→a B = lim g(x), x→a f(x) x→a g(x) L = lim
In the notation of Exercise 66, give an example where L exists but neither A nor B exists.Data From Exercise 66Assume that the following limits exist:Prove that if L = 1, then A = B. You cannot use the Quotient Law if B = 0, so apply the Product Law to L and B instead. A = lim f(x), x→a B = lim
True or false? (a) If lim f(x) exists, then lim f(x) = f(3). x-3 X-3 f(x) (b) If lim: = 1, then f(0) = 0. X 1 1 = x→→7 f(x) 8 (c) If lim f(x) = 8, then lim X-7' (d) If lim f(x) = 4 and lim f(x) = 8, then lim f(x) = 6. X-5 x→5+ x→5- (e) If lim f(x) X-0 X = 1, then lim f(x) = 0. x→0 (f) If
Let r1 and r2 be the roots of ƒ(x) = ax2 − 2x + 20. Observe that ƒ “approaches” the linear function L(x) = −2x + 20 as a → 0. Because r = 10 is the unique root of L, we might expect one of the roots of ƒ to approach 10 as a → 0 (Figure 2). Prove that the roots can be labeled so
Use the IVT to prove that has a root in the interval [0, 2]. f(x) = x³- x² + 2 cos x + 2
Use the IVT to prove that the curves y = x2 and y = cos x intersect.
Give an example of a (discontinuous) function that does not satisfy the conclusion of the IVT on [−1, 1]. Then show that the functionsatisfies the conclusion of the IVT on every interval [−a, a]. f(x) = - sin- 0 X # 0 x = 0
Use the IVT to show that 2−x2 = x has a solution on (0, 1).
Let (a) Show that if |x − 2| Observe that if |x − 2| 12.(b) Find δ > 0 such that if |x − 2| (c) Prove rigorously that f(x) = 1 x + 2
Use the Bisection Method to locate a solution of x2 − 7 = 0 to two decimal places.
Plot the function ƒ(x) = x1/3. Use the zoom feature to find a δ > 0 such that if |x − 8| < δ, then |x1/3 − 2| < 0.05.
Prove rigorously that lim (4 + 8x) = -4. X→-1
Use the fact that ƒ(x) = 2x is increasing to find a value of δ such that |2x − 8| < 0.001 if |x − 2| < δ. Find c1 and c2 such that 7.999 < ƒ(c1) < ƒ (c2) < 8.001.
Prove rigorously that lim(x²-x) = 6. x→3
In Exercises 37–42, calculate the limit. lim 1-→-00 4+52t 5-53
Physicists have observed that Einstein’s theory of special relativity reduces to Newtonian mechanics in the limit as c → ∞, where c is the speed of light. This is illustrated by a stone tossed up vertically from ground level so that it returns to Earth 1 s later. Using Newton’s Laws, we
According to the Michaelis–Menten equation, when an enzyme is combined with a substrate of concentration s (in millimolars), the reaction rate (in micromolars/min) is(a) Show, by computing that A is the limiting reaction rate as the concentration s approaches ∞.(b) Show that the reaction rate
Every limit as x → ∞ can be rewritten as a one-sided limit as t → 0+, where t = x−1. Setting g(t) = ƒ(t−1), we haveShow that and evaluate using the Quotient Law. lim f(x) = lim g(t) X→∞0 1-0+
Rewrite the following as one-sided limits as in Exercise 46 and evaluate.Data From Exercise 46Every limit as x → ∞ can be rewritten as a one-sided limit as t → 0+, where t = x−1. Setting g(t) = ƒ(t−1), we haveShow that and evaluate using the Quotient Law. 3-12x³ x+00 4x³ + 3x + 1 (a)
Show that Observe that lim (√x² + 1 - x) = 0. X→∞
In Exercises 37–42, calculate the limit. lim(√4x4 + 9x - 2.x²) X-00
In Exercises 37–42, calculate the limit. lim (√9x³ + x - x³/2) X→∞0
In Exercises 37–42, calculate the limit. lim (2√x - √x + 2) X-00
In Exercises 37–42, calculate the limit. lim X 00+x 1 x + 2
In Exercises 37–42, calculate the limit. lim |x x + x 1 I + X ∞0-+x
In Exercises 7–16, prove using the IVT.For all positive integers k, cos x = xk has a solution.
Plot ƒ(x) = √2x − 1 together with the horizontal lines y = 2.9 and y = 3.1. Use this plot to find a value of δ > 0 such that if 0 < |x − 5| < δ, then |√2x − 1 − 3| < 0.1.
In Exercises 7–16, evaluate the limit. 9x²-2 lim X→∞06-29x
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” 5-x² lim x-1 4x + 7
In Exercises 7–16, prove using the IVT.2x = bx has a solution if b > 2.
Plot ƒ(x) = tan x together with the horizontal lines y = 0.99 and y = 1.01. Use this plot to find a value of δ > 0 such that if 0 < Ιx − π/4Ι < δ, then |tan x − 1| < 0.01.
In Exercises 7–16, evaluate the limit. lim x-00 7x²-9 4x +3
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” 4 lim x-2x³
In Exercises 7–16, prove using the IVT.2x + 3x = 4x has a solution.
The function Use a plot of ƒ to find a value of δ > 0 such that |ƒ(x) − 1| f(x) = 2x² satisfies lim f(x) = 1. X-0
Let and ∈ = 0.5. Using a plot of ƒ, find a value of δ > 0 such that if then Repeat for ∈ = 0.2 and 0.1. f(x) = 4 x² + 1
In Exercises 7–16, evaluate the limit. lim X→→∞0 5r-9 4x³ + 2x + 7
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim X-1 3x² + 4x + 1 x + 1
In Exercises 7–16, prove using the IVT.cos x = tan 2x has a solution in (0, 1).
In Exercises 7–16, evaluate the limit. lim X→-00 3x²³ - 10 x + 4
Consider (a) Show that if |x − 2| Then explain why this proves that 1 lim x→2 X
Verify each limit using the limit definition. For example, in Exercise 13, show that |3x − 12| can be made as small as desired by taking x close to 4. lim(5x + 2) = 17
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim 1-9 √t - 3 t-9
In Exercises 7–16, prove using the IVT.2x + 1/x = −4 has a solution.
Consider (a) Show that if |x − 1| 2.(b) Find δ > 0 such that if 0 (c) Prove rigorously that the limit is equal to 2. lim √x + 3. X→1
In Exercises 7–16, evaluate the limit. lim X--8 2x + 3x431x 8x431x² + 12
Verify each limit using the limit definition. For example, in Exercise 13, show that |3x − 12| can be made as small as desired by taking x close to 4. lim(7x - 4) = 10 x-2
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim x→3 √x + 1-2 x-3
In Exercises 7–16, prove using the IVT.x1/3 = 1/(x − 1) has a solution in (1, 2).
Let ƒ(x) = sin x. Using a calculator, we findUse these values and the fact that f is increasing on to justify the statementThen draw a figure like Figure 3 to illustrate this statement. ƒ (7 -0.1) ≈ 0 f 4 (元) 0.633, f 0.707, f ✓(+0.1) = 0.774
In Exercises 17–24, find the horizontal asymptotes. f(x) = 2x² - 3x 8x2 + 8
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” x - EX lim x-1 X-1 X-
Use the Intermediate Value Theorem to show that the equation x6 − 8x4 + 10x2 − 1 = 0 has at least six distinct solutions.
In Exercises 17–24, find the horizontal asymptotes. f(x) = 8x³x² 7+11x - 4x4
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim h→0 2(a+h)² - 2a² h
In Exercises 18–20, determine whether or not the IVT applies to show that the given function takes on all values between ƒ(a) and ƒ(b) for x ∈ (a, b). If it does not apply, determine any values between ƒ(a) and ƒ(b) that the function does not take on for x ∈ (a, b).for the interval [−1,
In Exercises 17–24, find the horizontal asymptotes. f(x) = √36x² +7 9x + 4
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” t-6 lim 1-9 √t-3
In Exercises 18–20, determine whether or not the IVT applies to show that the given function takes on all values between ƒ(a) and ƒ(b) for x ∈ (a, b). If it does not apply, determine any values between ƒ(a) and ƒ(b) that the function does not take on for x ∈ (a, b).for the interval [−1,
In Exercises 18–20, determine whether or not the IVT applies to show that the given function takes on all values between ƒ(a) and ƒ(b) for x ∈ (a, b). If it does not apply, determine any values between ƒ(a) and ƒ(b) that the function does not take on for x ∈ (a, b).for the interval [−2,
In Exercises 17–24, find the horizontal asymptotes. f(x) = √36x4 + 7 9x² + 4
In Exercises 11–50, evaluate the limit if it exists. If not, determine whether the one-sided limits exist. For limits that don’t exist indicate whether they can be expressed as “= −∞” or “= ∞” lim s→0 1- √s² + 1 $²
Carry out three steps of the Bisection Method for ƒ(x) = 2x − x3 as follows:(a) Show that ƒ has a zero in [1, 1.5].(b) Show that ƒ has a zero in [1.25, 1.5].(c) Determine whether [1.25, 1.375] or [1.375, 1.5] contains a zero.

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