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mathematics
calculus 4th

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

Use the formal definition of the limit to prove the statement rigorously. lim √x = 2 X-4
In Exercises 17–24, find the horizontal asymptotes. f(t) = 3¹ 1+ 3-1
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 “= ∞” 1 lim x-1+ x + 1
Figure 6 shows that ƒ(x) = x3 − 8x − 1 has a root in the interval [2.75, 3]. Apply the Bisection Method twice to find an interval of length 1/16 containing this root. Aud 1 2 3 FIGURE 6 Graph of y = x³8x - 1. -X
Use the formal definition of the limit to prove the statement rigorously. lim(3x² + x) = 4 X-1
In Exercises 17–24, find the horizontal asymptotes. f(t) = 1/3 (64f² +9)1/6 A
Assume thatWhich of the following statements are correct?(a) x = L is a vertical asymptote of g.(b) y = L is a horizontal asymptote of g.(c) x = L is a vertical asymptote of ƒ.(d) y = L is a horizontal asymptote of ƒ. lim f(x) = L and lim g(x) = ∞o X-L
Match each quantity (a)–(d) with (i), (ii), or (iii) if possible, or state that no match exists. (a) 2ª3b (c) (2ª)b (i) 2ab (ii) 6a+b (b) (d) 2a-b3b-a (iii) (-)-b
Is there a definite way of choosing the optimal viewing rectangle, or is it best to experiment until you find a viewing rectangle appropriate to the problem at hand?
Calculate the values of the six standard trigonometric functions for the angle θ in Figure 20. 0 17 15 8
Describe the calculator screen produced when the function y = 3 + x2 is plotted with a viewing rectangle:(a) [−1, 1] × [0, 2](b) [0, 1] × [0, 4]
In Exercises, find the equation of the line.Line through (2, 1) perpendicular to the line given by y = 3x + 7
In Exercises, find the equation of the line.Line through (3, 4) perpendicular to the line given by y = 4x − 2
In Exercises, find the equation of the line.Horizontal line through (−3, 5)
In Exercises, find the equation of the line.Line through (2, 3) parallel to y = 4 − x
During a year, the length of a day, from sunrise to sunset, in Wolf Point, Montana, varies from a shortest day of approximately 8.1 hours to a longest day of approximately 15.9 hours, while in Mexico City, the day lengths vary from 10.7 hours to 13.3 hours. For each location, determine a function
Find functions ƒ and g such that the function f(g(t)) = (12t+ 9)4
In Exercises 47–56, derive the identity using the identities listed in this section.tan x = sin 2x/1 + cos 2x
In Exercises 45–46, solve for 0 ≤ θ < 2π.sin θ = sin 2θ. Use the double-angle formula for sine.
In Exercises 45–46, solve for 0 ≤ θ < 2π.sin θ = cos 2θ. Use appropriate identities to express cos 2θ in terms of the sine function.
In Exercises 47–56, derive the identity using the identities listed in this section.tan(π − θ) = − tan θ
Use Exercises 50 and 51 to show that tan θ and cot θ are periodic with period π.Data From Exercise 50 Derive the identity using the identities listed in this section.sin(θ + π) = − sin θData From Exercise 51Derive the identity using the identities listed in this section.cos(θ + π) = −
Use the double-angle formulas to show that sin2θ and cos2θ are periodic with period π.
Use Exercise 55 to compute tan π/8.Data From Exercise 55Derive the identity using the identities listed in this section.tan x = sin 2x/1 + cos 2x
Use the identity of Exercise 48 to show that cos π/8 is equal to √1/2 + √2/4.Data From Exercise 48Derive the identity using the identities listed in this section.cos2 θ/2 = 1 + cos θ/2
Use the Law of Cosines to find the distance from P to Q in Figure 26. 10 77/9 8 P
Use Figure 27 to derive the Law of Cosines from the Pythagorean Theorem. b. 0 с a a - b cos 0
Use the addition formulas for sine and cosine to prove tan(a + b) = cot(a - b) tan a + tan b 1 - tan a tan b cota cotb + 1 cot b - cot a
Use the addition formula to prove cos 30 = 4 cos³ 0 - 3 cos 0
Let θ be the angle between the line y = mx + b and the x-axis [Figure 28(A)]. Prove that m = tan θ. 0 S (A) y = mx +b T X
Let L1 and L2 be the lines of slope m1 and m2 [Figure 28(B)]. Show that the angle θ between L1 and L2 satisfies cot θ = m2m1 + 1/m2 −m1. L2 L₁ X
Use Exercise 66 to prove that two lines with nonzero slopes m1 and m2 are perpendicular if and only if m2 = −1/m1.Data From Exercise 66Let L1 and L2 be the lines of slope m1 and m2 [Figure 28(B)]. Show that the angle θ between L1 and L2 satisfies cot θ = m2m1 + 1/m2 −m1. L2 L₁ X
Apply the double-angle formula to prove:Guess the values of cos π/32 and of cos π/2n for all n. (a) cos=√√2+ √2 (b) cos = √2 + √2+ √2 + √2
Given that which of the following statements is true? lim cos x = 1, x-0
What are the horizontal asymptotes of the function in Figure 6? -20 y 2 y = f(x) 50 20 40 60 80
The position of a particle at time t (s) is s(t) = √t2 + 1 m. Compute its average velocity over [2, 5] and estimate its instantaneous velocity at t = 2.
Average velocity is equal to the slope of a secant line through two points on a graph. Which graph?
Based on the information conveyed in Figure 5(A), find values of L, ∈, and δ > 0 such that the following statement holds: If |x| -0.1 4.8+ 4 3.5+ (A) y = f(x) 0.1 X
Which of the following is indeterminate at x = 1? x² + 1 x-1' ²2²-1 x + 2 x² - 1 √x+3-2' x² + 1 √x+3-2
Prove that ƒ(x) = x2 takes on the value 0.5 in the interval [0, 1].
Use the IVT to show that ƒ(x) = x3 + x takes on the value 9 for some x in [1, 2].
Assume that −x4 ≤ ƒ(x) ≤ x2. What is ? Is there enough information to evaluate ? Explain. lim f(x) X-0
Which property of ƒ(x) = x3 allows us to conclude that lim x³ = 8? X→2 X
What are the following limits? (a) lim x³ 00-X (b) lim x³ 00-7x (c) lim x X118
Sketch the graph of a function ƒ that has both y = −1 and y = 5 as horizontal asymptotes.
Suppose it is known that for a given ∈ and δ, if 0 (a) If 0 < x - 3|< 26, then [f(x) - 2| < €. (b) If 0 < x - 3|< 6, then f(x) - 2| < 2€. (c) If 0
Can instantaneous velocity be defined as a ratio? If not, how is instantaneous velocity computed?
A wrench dropped from a state of rest at time t = 0 travels a distance s(t) = 4.9t2 m in t seconds. Estimate the instantaneous velocity at t = 3.
A rock dropped from a state of rest at time t = 0 on the planet Ginormon travels a distance s(t) = 15.2t2 m in t seconds. Estimate the instantaneous velocity at t = 5.
The temperature in Vancouver was 8°C at 6 AM and rose to 20°C at noon. Which assumption about temperature allows us to conclude that the temperature was 15°C at some moment of time between 6 AM and noon?
Based on the information conveyed in Figure 5(B), find values of c, L,∈, and δ > 0 such that the following statement holds: If 0 10.4 10 9.8 y = f(x) 2.9 3 3.1 (B) X
Give counterexamples to show that these statements are false:(a) If ƒ (c) is indeterminate, then the right- and left-hand limits as x → c are not equal.(b) If exists, then ƒ(c) is not indeterminate.(c) If ƒ(x) is undefined at x = c, then ƒ(x) has an indeterminate form at x = c. lim f(x) X→C
Show that takes on the value 0.499 for some t in [0, 1]. g(t) = t t +1
If you want to evaluate it is a good idea to rewrite the limit in terms of the variable (choose one):(a) θ = 5h (b) θ = 3h (c) θ = 5h/ 3 sin 5h lim h-0 3h h→0
What can be said about ƒ(3) if ƒ is continuous and lim f(x) = ? X-3
Suppose that ƒ(x) < 0 if x is positive and ƒ(x) > 1 if x is negative. Can ƒ be continuous at x = 0?
Is it possible to determine ƒ(7) if ƒ(x) = 3 for all x < 7 and f is right-continuous at x = 7? What if f is left continuous?
Which of the following is a verbal version of the Product Law (assuming the limits exist)?(a) The product of two functions has a limit.(b) The limit of the product is the product of the limits.(c) The product of a limit is a product of functions.(d) A limit produces a product of functions.
Which statement is correct? The Quotient Law does not hold if(a) The limit of the denominator is zero(b) The limit of the numerator is zero
What does the following table suggest about lim f(x) and lim f(x)?
What is the limit of ƒ(x) = 1 as x → π?
What is the limit of g(t) = t as t → π?
Can ƒ(x) approach a limit as x → c if ƒ(c) is undefined? If so, give an example.
Can you tell whether exists from a plot of ƒ for x > 5? Explain. lim f(x) X-5
If you know in advance that exists, can you determine its value from a plot of ƒ for all x > 5? lim f(x)
Estimate the slope of the tangent line at the point indicated.ƒ(x) = x2 + x; x = 0
Compute the ball’s average velocity over the time interval [3, 6] and estimate the instantaneous velocity at t = 3.
Consider the function ƒ(x) = (x − 1)1/3.(a) Compute the slope of the secant lines between 1 and x for x = 0.9, 0.99, 0.9999 and for x = 1.1, 1.01, 1.0001.(b) Discuss what the secant-line slopes in (a) suggest happens to the tangent line at 1.(c) Plot the graph of ƒ near x = 1 and verify your
Compute the stone’s average velocity over the time intervals [1, 1.01], [1, 1.001], [1, 1.0001] and [0.99, 1], [0.999, 1], [0.9999, 1], and then estimate the instantaneous velocity at t = 1.
Compute Δy/Δx for the interval [2, 5], where y = 4x − 9. What is the slope of the tangent line at x = 2?
Compute the ball’s average velocity over the time interval [5, 9] and estimate the instantaneous velocity at t = 5.
A stone is tossed vertically into the air from ground level with an initial velocity of 15 m/s. Its height at time t is h(t) = 15t − 4.9t2 m.Compute the stone’s average velocity over the time interval [0.5, 2.5] and indicate the corresponding secant line on a sketch of the graph of h.
On her bicycle ride Fabiana’s position (in km) as a function of time (in hours) is s(t) = 22t + 17. What was her average velocity between t = 2 and t = 3? What was her instantaneous velocity at t = 2.5?
Estimate the slope of the tangent line at the point indicated. y(t) = √3t+1; t=1
Estimate the slope of the tangent line at the point indicated. f(x) = tan x; X = π 4
The position of a particle at time t is s(t) = 2t3. Compute the average velocity over the time interval [2, 4] and estimate the instantaneous velocity at t = 2.
Estimate the slope of the tangent line at the point indicated. y(x) = 1 x + 2 x = 2
Estimate the slope of the tangent line at the point indicated. f(x) = sin x; X = 76
Estimate the slope of the tangent line at the point indicated.P(x) = 3x2 − 5; x = 2
Estimate the slope of the tangent line at the point indicated.ƒ(t) = 12t − 7; t = −4
Estimate the slope of the tangent line at the point indicated.ƒ(x) = tan x; x = 0
The height of a projectile fired in the air vertically with initial velocity 25 m/s is h(t) = 25t − 4.9t2 m(a) Compute h(1). Show that h(t) − h(1) can be factored with (t − 1) as a factor.(b) Using part (a), show that the average velocity over the interval [1, t] is 20.1 − 4.9t.(c) Use this
The height (in centimeters) at time t (in seconds) of a small mass oscillating at the end of a spring is h(t) = 3 sin(2πt). Estimate its instantaneous velocity at t = 4.
Consider the function ƒ(x) = √x.(a) Compute the slope of the secant lines from (0, 0) to (x,ƒ(x)) for x = 1, 0.1, 0.01, 0.001, 0.0001.(b) Discuss what the secant-line slopes in (a) suggest happens to the tangent line at 0.(c) Plot the graph of ƒ near x = 0 and verify your observation from (b).
Which graph in Figure 5 has the following property: For all x, the slope of the secant line over [0, x] is greater than the slope of the tangent line at x. Explain. y y V.K -X (A) (B)
Sketch the graph of ƒ(x) = x(1 − x) over [0, 1]. Refer to the graph and, without making any computations, find:(a) The slope of the secant line over [0, 1](b) The slope of the tangent line at x = 1/2(c) The values of x at which the slope of the tangent line is positive
Evaluate the limit. lim x X-21
(a) Figure 6(A) shows two rectangles whose combined area is an overestimate of the area A under the graph of y = x2 from x = 0 to x = 1. Compute the combined area of the rectangles.(b) We can improve the estimate by using three rectangles obtained by dividing [0, 1] into thirds, as shown in Figure
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. lim √x-1 x-1 I-X 1-x
For ƒ(x) = x3, show that the slope of the secant line over [−3, x] is x2 − 3x + 9, and use this to estimate the slope of the tangent line at x = −3.
Numerically investigate computing the values of sin x with x in degrees. Make an estimate of the limit accurate to 5 decimal places. lim sin.x X x-0
Let Q(t) = t2. Find a formula for the slope of the secant line over the interval [1, t] and use it to estimate the slope of the tangent line at t = 1. Repeat for the interval [2, t] and for the slope of the tangent line at t = 2.
For ƒ(x) = x3, show that the slope of the secant line over [1, x] is x2 + x + 1, and use this to estimate the slope of the tangent line at x = 1.
Show, via illustration, that the limits x and a are equal but the functions in each limit are different. lim. x-a
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. lim 9-0 cos - 1 0
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. 2x² - 32 lim x→→4 X+4
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 x² = 0 x-0
Determine for g as in Figure 11. lim_g(x) x-0.5
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(x³ + 12) = 12 x-0
Evaluate the limit. lim x X-21

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