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mathematics
calculus 4th

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

In Exercises 55–57, evaluate using the result of Exercise 54.Data From Exercise 54Investigate numerically or graphically. Then prove that the limit is equal to 1/2. See the proof of Theorem 2. lim h→0 1 - cos h
Investigate numerically or graphically. Then evaluate the limit using the double angle formula cos 2h = 1 − 2 sin2 h. lim h→0 1 - cos 2h h²
Explain why involves an indeterminate form, and then evaluate the limit. lim (2 tan 0 sec )
In Exercises 55–57, evaluate using the result of Exercise 54.Data From Exercise 54Investigate numerically or graphically. Then prove that the limit is equal to 1/2. See the proof of Theorem 2. lim h→0 1 - cos h
Use the Squeeze Theorem to prove that if lim |f(x)] = 0, then lim f(x) = 0.
(a) Investigate numerically for the five values (b) Can you guess the answer for general c?(c) Check numerically that your answer to (b) works for two other values of c. lim X-C sin x - sin c X-C
Investigate the asymptotic behavior of numerically and graphically:(a) Make a table of values of ƒ(x) for x = ±50, ±100, ±500, ±1000.(b) Plot the graph of ƒ.(c) What are the horizontal asymptotes of ƒ? f(x) = X x² + 1 X
Sketch the graphs of two functions ƒ and g that have both y = −2 and y = 4 as horizontal asymptotes but lim f(x) lim g(x). X-00 X→∞0
Sketch the graph of a function ƒ with a single horizontal asymptote y = 3.
Investigate numerically and graphically: lim X-too 12x + 1 √4x²+9
In Exercises 7–16, evaluate the limit. 6 + X 00x ալ X
In Exercises 7–16, evaluate the limit. lim X→∞0 3x² + 20x 4x² +9
In Exercises 7–16, evaluate the limit. lim 3x² + 20x 2x4 + 3x³ - 29
In Exercises 7–16, evaluate the limit. 7x-9 lim x→∞0 4x + 3
In Exercises 7–16, evaluate the limit. lim 4 5 S+X00+x
The following statement is incorrect: “If ƒ has a horizontal asymptote y = L at ∞, then the graph of ƒ approaches the line y = L as x gets greater and greater, but never touches it.” In determine and indicate how ƒ demonstrates that the statement is incorrect. lim f(x) 00-X
In Exercises 17–24, find the horizontal asymptotes. g(t) = 10 1+3-1
In Exercises 27–34, evaluate the limit. lim X--∞ 8x² + 7x¹/3 √16x4 +6
In Exercises 17–24, find the horizontal asymptotes.p(t) = 2−t2
In Exercises 27–34, evaluate the limit. lim X→∞0 √9x4 + 3x + 2 4x³ + 1
In Exercises 27–34, evaluate the limit. lim X→00 √x³ + 20x 10x - 2
In Exercises 27–34, evaluate the limit. 14/3 - 9t1/3 lim 1-00 (814 + 2)¹/3 = lim 1- || 1 1-00 (8+)¹/3 2
In Exercises 27–34, evaluate the limit. 14/3+1/3 lim 100 (412/3 + 1)2
In Exercises 27–34, evaluate the limit. lim X→-∞0 4x - 3 √25x² + 4x
In Exercises 27–34, evaluate the limit. lim X-00 [x] + x X + 1 x
In Exercises 27–34, evaluate the limit. 4+6e²t lim 1--005-9e³t
Make a sketch illustrating the following statement: To prove , given ∈ > 0, we can take δ = ∈, to have the gap be small enough. lim x = a, DEX De
Consider where ƒ(x) = 8x + 3. lim f(x), X-4
Considerwhere ƒ(x) = 4x − 1. lim f(x), X-2
Make a sketch illustrating the following statement: To prove we can choose any δ > 0 to have the gap be small enough. lima = a, given X-C € > 0,
Consider (refer to Example 2). lim x² = 4 X→2
Consider the limit lim x² = 25. X-5
Refer to Example 3 to find a value of δ > 0 such that If 0
Use Figure 6 to find a value of δ > 0 such that the following statement holds: If 0 2 − 14 0.30 0.25 0.20 0.15+ 0.10+ 0.05 1.9 2.0 2.1 ****************************** ************** ***************** 2 || x² X
Use the formal definition of the limit to prove the statement rigorously. 1 lim x sin = = 0 x-0 X
Use the formal definition of the limit to prove the statement rigorously. -2 lim x-² X-2 || 1
Use the formal definition of the limit to prove the statement rigorously. lim(x² + x³) = 0 x-0
Use the formal definition of the limit to prove the statement rigorously. lim x³ = 1 x→1
Use the identityto prove thatThen use the inequality Finally, prove rigorously that y y sin (+1) cos (**) 2 sin x + sin y = 2 sin
Prove rigorously that does not exist. lim sin x→0
Let ƒ(x) = min(x, x2), where min(a, b) is the minimum of a and b. Prove rigorously that lim f(x) = 1. x-1
Prove rigorously that lim |x| = 0. x-0
Prove that a function converges to at most one limiting value. In other words, use the limit definition to prove that if lim f(x) = L₁ and lim f(x) = L₂, then L₁ = L₂. X-C
Prove the statement using the formal limit definition.The Constant Multiple Law [Theorem 1, part (ii) in Section 2.3]
Prove the statement using the formal limit definition.The Squeeze Theorem (Theorem 1 in Section 2.6) THEOREM 1 Squeeze Theorem Assume that for xc (in some open interval con- taining c), 1(x) ≤ f(x) ≤ u(x) and Then lim f(x) exists and lim f(x) = L. X-C X-C lim 1(x) = lim u(x) = L X-C
In Exercises 7–16, prove using the IVT.A positive number c has an nth root for all positive integers n.
Evaluate in terms of the constant a. lim X-0 (x + a)³-a³ X
Evaluate in terms of the constant a. lim x→a √x - √a x - a
Evaluate in terms of the constant a. lim h→0 √a +2h-√a h
Evaluate Set x = 4√1 + h, express h as a function of x, and rewrite as a limit as x → 1. lim h→0 √1+h-1 h
Evaluate Set x = 6 √1 + h, express h as a function of x, and rewrite as a limit as x → 1. 1+ h-1 h-01+h-1 lim
ind all values of c such that the limit exists. lim r -5r-6 X-C
Find all values of c such that the limit exists. lim x→1 x² + 3x + c x-1
Evaluate in terms of the constant a. 1 h 1 a lim ha h-a
Find all values of c such that the limit exists. lim x-1 1 x-1 C x³ - 1,
For which sign, + or −, does the following limit exist? 1 1 lim - ± x→0 X x(x-1),
Find all values of c such that the limit exists. lim x-0 1+ cx² √1 + x² x4
Here is a function with strange continuity properties:(a) Show that ƒ is discontinuous at c if c is rational. There exist irrational numbers arbitrarily close to c.(b) Show that ƒ is continuous at c if c is irrational. Let I be the interval {x : |x − c| 0, I contains at most finitely many
Let ƒ(x) = 1 if x is rational and f (x) = 0 if x is irrational. Prove that lim does not exist for any c. There exist rational and irrational numbers arbitrarily close to any c. lim f(x) X→C
Prove the statement using the formal limit definition.The Product Law [Theorem 1, part (iii) in Section 2.3]. Use the identity. f(x)g(x) - LM = (f(x) - L) g(x) + L(g(x) - M)
Write a formal definition of the following: lim f(x) = L X→∞0
Write a formal definition of the following: lim f(x) = ∞ x-a
Show that takes on the value 0.4. X² f(x) = x²+1
Show that g(t) = t2 tan t takes on the value 1/2 for some t in [0, π/4].
Show that cos x = x has a solution in the interval [0, 1]. Show that f (x) = x − cos x has a zero in [0, 1].
Use the IVT to find an interval of length 12 containing a root of ƒ(x) = x3 + 2x + 1.
In Exercises 7–16, prove using the IVT.√c + √c + 2 = 3 has a solution.
In Exercises 7–16, prove using the IVT.For all integers n, sinnx = cos x for some x ∈ [0, π].
In Exercises 7–16, prove using the IVT.√2 exists. Hint: Consider (x) = x2.
Figure 8(A) shows a slice of ham. Prove that for any angle θ (0 ≤ θ ≤ π), it is possible to cut the slice in half with a cut of incline θ. The lines of inclination θ are given by the equations y = (tan θ)x + b, where b varies from −∞ to ∞. Each such line divides the slice into two
Use the IVT to show that if ƒ is continuous and one-to-one on an interval [a, b], then ƒ is either an increasing or a decreasing function.
What is the sign of a if ƒ(x) = ax3 + x + 1 satisfies lim f(x) = ∞o? 00? X→→∞0
Figure 8(B) shows a slice of ham on a piece of bread. Prove that it is possible to slice this open-faced sandwich so that each part has equal amounts of ham and bread. By Exercise 36, for all 0 ≤ θ ≤ π there is a line L(θ) of incline θ (which we assume is unique) that divides the ham into
Sketch the graph of a function that approaches a limit as x→∞but does not approach a limit (either finite or infinite) as x → −∞.
What is the sign of the coefficient multiplying x7 if ƒ is a polynomial of degree 7 such that lim f(x) = ∞o? X→-00
The 1-Dimensional Brouwer Fixed Point Theorem. It indicates that every continuous function ƒ mapping the closed interval [0, 1] to itself must have a fixed point; that is, a point c such that (c) = c.(a) Give an example showing that if ƒ is continuous and 0 < ƒ(x) < 1 for 0 < x < 1, then there
The 1-Dimensional Brouwer Fixed Point Theorem. It indicates that every continuous function ƒ mapping the closed interval [0, 1] to itself must have a fixed point; that is, a point c such that (c) = c.Show that if ƒ is continuous and 0 ≤ ƒ (x) ≤ 1 for 0 ≤ x ≤ 1, then ƒ(c) = c for some c
(a) Assume that g and h are continuous on [a, b]. Use Corollary 2 to show that if g(a) < h(a) and h(b) < g(b), then there exists c ∈ [a, b] such that g(c) = h(c).(b) Interpret the result of (a) in terms of the graphs of g and h, and show, by a graphical example, that the conclusion in (a) need
In Exercises 25–28, draw the graph of a function ƒ on [0, 4] with the given property.Infinite one-sided limits at x = 2 and satisfies the conclusion of the IVT on [0, 4].
In Exercises 25–28, draw the graph of a function ƒ on [0, 4] with the given property.Infinite one-sided limits at x = 2 and does not satisfy the conclusion of the IVT
In Exercises 25–28, draw the graph of a function ƒ on [0, 4] with the given property.Jump discontinuity at x = 2 and satisfies the conclusion of the IVT on [0, 4]
In Exercises 25–28, draw the graph of a function f on [0, 4] with the given property.Jump discontinuity at x = 2 and does not satisfy the conclusion of the IVT
Show that tan3 θ − 8 tan2 θ + 17 tan θ − 8 = 0 has a root in [0.5, 0.6]. Apply the Bisection Method twice to find an interval of length 0.025 containing this root.
Find an interval of length 1/4 in [1, 2] containing a root of the equation x7 + 3x − 10 = 0.
In Exercises 57–64, find the horizontal asymptotes of the function by computing the limits at infinity. f(u) = 8u3 √16u² +6
In Exercises 5–10, estimate the limit numerically to two decimal places or state that the limit does not exist. lim x→0 1 - cos³ (x) x²
For ƒ(x) = √2x compute the slopes of the secant lines from 16 to each of 16 ± 0.01, 16 ± 0.001, 16 ± 0.0001 and use those values to estimate the slope of the tangent line at x = 16.
Show that the slope of the secant line for ƒ(x) = x3 − 2x over [5, x] is equal to x2 + 5x + 23. Use this to estimate the slope of the tangent line at x = 5.
In Exercises 5–10, estimate the limit numerically to two decimal places or state that the limit does not exist. lim x¹/(x-1) X-
In Exercises 5–10, estimate the limit numerically to two decimal places or state that the limit does not exist. x-4 lim x2x² - 4 ܐ
In Exercises 5–10, estimate the limit numerically to two decimal places or state that the limit does not exist. x-2 lim x-2 2-4
In Exercises 5–10, estimate the limit numerically to two decimal places or state that the limit does not exist. 7 3 lim x-11-x7 1-x³
In Exercises 5–10, estimate the limit numerically to two decimal places or state that the limit does not exist. 3* - 9 lim x25x - 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(3 + x¹/2)
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 y 3y² + 5y-2 6y² - 5y + 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 “= ∞” x³ - 2x lim x→1 X1 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 a→b a² - 3ab + 2b² a-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 “= ∞” 43x - 4x lim x-0 4-1
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 “= ∞” sin 50 lim 0→0 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 “= ∞” lim X→1.5 X

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