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mathematics
calculus 4th

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

Climate scientists use the Stefan–Boltzmann Law R = σT4 to estimate the change in the earth’s average temperature T (in kelvins) caused by a change in the radiation R (in joules per square meter per second) that the earth receives from the sun. Here, σ = 5.67 × 10−8 Js−1m−2K−4.
In Exercises 3–8, compute ƒ(a) in two ways, using Eq. (1) and Eq. (2).ƒ(x) = x2 + 9x, a = 0 f'(a) = lim h→0 f(a+h)-f(a) h
In Exercises 9–12, refer to Figure 13.Estimate ƒ(2 + h) − ƒ(2)/h for h = −0.5. What does this quantity represent? Is it larger or smaller than ƒ'(2)? Explain. 3.0 2.5 2.0 1.5 1.0 0.5 y 0.5 1.0 f(x). 1.5 2.0 2.5 3.0 FIGURE 13 ·x
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = 1/√2x + 1, a = 4
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(t) =√t2 + 1, a = 3
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = x−2, a = −1
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = 1/x2 + 1, a = 0
Show that ƒ is not differentiable and x = 1 and has a corner in its graph there. f(x) = (1 (+2 x ≤ 1 x² x ≥ 1
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(t) = t−3, a = 1
Show that ƒ is not differentiable and x = 0 and has a corner in its graph there. f(x) = x ≤0 x≥0
In Exercises 49–51, sketch a graph of ƒ and identify the points c such that ƒ'(c) does not exist. In which cases is there a corner at c?ƒ(x) = |x + 3|
Figure 15(A) shows the graph of ƒ(x) = √x. The close-up in Figure 15(B) shows that the graph is nearly a straight line near x = 16. Estimate the slope of this line and take it as an estimate for ƒ'(16). Then compute ƒ'(16) with the limit definition and compare with your estimate. 5 4 3 N P 10
In Exercises 49–51, sketch a graph of ƒ and identify the points c such that ƒ'(c) does not exist. In which cases is there a corner at c?ƒ(x) = x2/5
In Exercises 49–51, sketch a graph of ƒ and identify the points c such that ƒ'(c) does not exist. In which cases is there a corner at c?ƒ(x) = |x2 − 4|
Determine the intervals along the x-axis on which the derivative in Figure 16 is positive. y 4.0 3.5 3.0 2.5 2.0- 1.5 1.0 0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 →Х
Let ƒ(x) = 4/1 + 2x . Plot ƒ over [−2, 2]. Then zoom in near x = 0 until the graph appears straight, and estimate the slope ƒ'(0).
Let ƒ(x) = cot x. Estimate ƒ'(π/2) graphically by zooming in on a plot of ƒ near x = π/2.
In Exercises 57–62, each limit represents a derivative ƒ'(a). Find ƒ(x) and a. lim h→0 (5 + h)³ - 125 h
Sketch the graph of ƒ(x) = sin x on [0, π] and guess the value of ƒ'(π/2). Then calculate the difference quotient at x = π/2 for two small positive and negative values of h. Are these calculations consistent with your guess?
In Exercises 57–62, each limit represents a derivative ƒ'(a). Find ƒ(x) and a. ²³-125 lim x+5 x-5
In Exercises 57–62, each limit represents a derivative ƒ'(a). Find ƒ(x) and a. lim h→0 sin (+h)-0.5 h
In Exercises 57–62, each limit represents a derivative ƒ'(a). Find ƒ(x) and a. lim h→0 52th - 25 h
In Exercises 57–62, each limit represents a derivative ƒ'(a). Find ƒ(x) and a. lim h→0 5h-1 h
Apply the method of Example 6 to ƒ'(x) = sin x to determine ƒ'(π/4) accurately to four decimal places. EXAMPLE 6 Estimate the derivative of f(x) = sin x at x =.
For each graph in Figure 17, determine whether ƒ'(1) is larger or smaller than the slope of the secant line between x = 1 and x = 1 + h for h > 0. Explain. y y= f(x) 1 (A) X y y = f(x), 1 (B) X
Refer to the graph of ƒ(x) = 2x in Figure 18.(a) Explain graphically why, for h > 0,(b) Use (a) to show that 0.69314 ≤ ƒ'(0) ≤ 0.69315.(c) Similarly, compute ƒ'(x) to four decimal places for x = 1, 2, 3, 4.(d) Now compute the ratios ƒ'(x)/ƒ'(0) for x = 1, 2, 3, 4. Can you guess an
The vapor pressure of water at temperature T (in kelvins) is the atmospheric pressure P at which no net evaporation takes place. Use the following table to estimate the indicated derivatives using the difference quotient approximation.Estimate P'(T) for T = 303, 323, 343. (Include proper units on
Verify that P = (1, 1/2) lies on the graphs of both ƒ(x) = 1/(1 + x2) and L(x) = 1/2 + m(x − 1) for every slope m. Plot y = ƒ(x) and y = L(x) on the same axes for several values of m until you find a value of m for which y = L(x) appears tangent to the graph of ƒ. What is your estimate for
Plot ƒ(x) = xx and y = 2x + a on the same set of axes for several values of a until the line becomes tangent to the graph. Then estimate the value c such that ƒ'(c) = 2.
Estimate P'(t) for t = 1997, 2001, 2005, 2009. (Include proper units on the derivative.)
Estimate P'(t) for t = 1999, 2003, 2007, 2011. (Include proper units on the derivative.)
Which of the two functions in Figure 21 satisfies the inequalityfor h > 0? Explain in terms of secant lines. f(a+h)-f(a-h) f(a+h)-f(a) 2h h
(a) Show that the symmetric difference quotient ƒ(a + h) − ƒ(a − h)/2h is the slope of the secant line to the graph of ƒ from x = a − h to x = a + h. (Include an illustration.)(b) Prove that the symmetric difference quotient is the average of the slopes of the secant lines from x to x + h
With P(T) as in Exercises 71 and 72, estimate P(T) for T = 303, 313, 333, 343, now using the SDQ.Data From Exercises 71The vapor pressure of water at temperature T (in kelvins) is the atmospheric pressure P at which no net evaporation takes place. Use the following table to estimate the indicated
Use Eq. (1) to estimate Δƒ = ƒ(3.02) − ƒ(3).ƒ(x) = x2 Δf ~ f'(α)Δ.χ
Use Eq. (1) to estimate Δƒ = ƒ(3.02) − ƒ(3).ƒ(x) = x4 Δf ~ f'(α)Δ.χ
Use Eq. (1) to estimate Δƒ = ƒ(3.02) − ƒ(3).ƒ(x) = x−1 Δf ~ f'(α)Δ.χ
Use Eq. (1) to estimate Δƒ = ƒ(3.02) − ƒ(3). Δf ~ f'(α)Δ.χ
Use Eq. (1) to estimate Δƒ = ƒ(3.02) − ƒ(3).ƒ(x) = √x + 6 Δf ~ f'(α)Δ.χ
Use Eq. (1) to estimate Δƒ = ƒ(3.02) − ƒ(3).ƒ(x) = tan πx/3 Δf ~ f'(α)Δ.χ
Use Eq. (1) to estimate Δƒ. Use a calculator to compute both the error and the percentage error.ƒ(x) = √1 + x, a = 3, Δx = 0.2 Δf ~ f'(α)Δ.x
The cube root of 64 is 4. How much smaller is the cube root of 63.6? Estimate using the Linear Approximation.
Use Eq. (1) to estimate Δƒ. Use a calculator to compute both the error and the percentage error.ƒ(x) = 2x2 − x, a = 5, Δx = −0.4 Δf ~ f'(α)Δ.x
Use Eq. (1) to estimate Δƒ. Use a calculator to compute both the error and the percentage error. Δf ~ f'(α)Δ.x
Use Eq. (1) to estimate Δƒ. Use a calculator to compute both the error and the percentage error. Δf ~ f'(α)Δ.x
Using Linear Approximation, estimate Δƒ for a change in x from x = a to x = b. Use the estimate to approximate ƒ(b), and find the error using a calculator.ƒ(x) = √x, a = 25, b = 26
Using Linear Approximation, estimate Δƒ for a change in x from x = a to x = b. Use the estimate to approximate ƒ(b), and find the error using a calculator.ƒ(x) = x1/4, a = 16, b = 16.5
Using Linear Approximation, estimate Δƒ for a change in x from x = a to x = b. Use the estimate to approximate ƒ(b), and find the error using a calculator.ƒ(x) = 1/√x , a = 100, b = 101
Using Linear Approximation, estimate Δƒ for a change in x from x = a to x = b. Use the estimate to approximate ƒ(b), and find the error using a calculator.ƒ(x) = 1/√x , a = 100, b = 98
Using Linear Approximation, estimate Δƒ for a change in x from x = a to x = b. Use the estimate to approximate ƒ(b), and find the error using a calculator.ƒ(x) = x1/3, a = 8, b = 9
Using Linear Approximation, estimate Δƒ for a change in x from x = a to x = b. Use the estimate to approximate ƒ(b), and find the error using a calculator.ƒ(x) = x1/3, a = 27, b = 30
Find the linearization at x = a and then use it to approximate ƒ(b). f(x) = 1 X a = 2, b = 2.02
Find the linearization at x = a and then use it to approximate ƒ(b).ƒ(x) = x4, a = 1, b = 0.96
Find the linearization at x = a and then use it to approximate ƒ(b). f(x) = x² x-3' a = 4, b = 4.1
Find the linearization at x = a and then use it to approximate ƒ(b).ƒ(x) = sin2 x, a = π/4, b = 1.1π/4
Find the linearization at x = a and then use it to approximate ƒ(b).ƒ(x) = (1 + x)−1/2, a = 0, b = 0.08
Find the linearization at x = a and then use it to approximate ƒ(b).ƒ(x) = (1 + x)−1/2, a = 3, b = 2.88
Find the linearization at x = a and then use it to approximate ƒ(b).ƒ(x) = e√x, a = 1, b = 0.85
Estimate Δy using differentials [Eq. (3)].y = cos x, a = π/6, dx = 0.014 Ay dy = f'(a)dx ~
Find the linearization at x = a and then use it to approximate ƒ(b).ƒ(x) = ex ln x, a = 1, b = 1.02
Estimate Δy using differentials [Eq. (3)].y = tan2 x, a = π/4, dx = −0.02 Ay dy= f'(a)dx ~
Estimate Δy using differentials [Eq. (3)]. Ay dy= f'(a)dx ~
Estimate Δy using differentials [Eq. (3)]. Ay dy= f'(a)dx ~
Estimate ƒ(4.03) for ƒ(x) as in Figure 9. y = f(x) (4,2) (10,4) Tangent line -X
At a certain moment, an object in linear motion has velocity 100 m/s. Estimate the distance traveled over the next quarter-second, and explain how this is an application of the Linear Approximation.
Estimate sin 61° − sin 60° using the Linear Approximation. Hint: Express Δθ in radians.
Box office revenue at a multiplex cinema in Paris is R(p) = 3600p − 10p3 euros per showing when the ticket price is p euros. Calculate R(p) for p = 9 and use the Linear Approximation to estimate ΔR if p is raised or lowered by 0.5 euro.
The stopping distance for an automobile is F(s) = 1.1s + 0.054s2 ft, where s is the speed in mph. Use the Linear Approximation to estimate the change in stopping distance per additional mph when s = 35 and when s = 55.
The atmospheric pressure P at altitude h = 20 km is P = 5.5 kilopascals. Estimate P at altitude h = 20.5 km assuming that dP dh = -0.87
At a certain moment, the temperature in a snake cage satisfies dT/dt = 0.008°C/s. Estimate the rise in temperature over the next 10 s.
Newton’s Law of Gravitation shows that if a person weighs w pounds on the surface of the earth, then his or her weight at distance x from the center of the earth iswhere R = 3960 miles is the radius of the earth (Figure 10).(a) Show that the weight lost at altitude h miles above the earth’s
The resistance R of a copper wire at temperature T = 20°C is R = 15 Ω. Estimate the resistance at T = 22°C, assuming that dR/dTΙT=20 = 0.06 Ω/°C.
Using Exercise 41(a), estimate the altitude at which a 130-lb pilot would weigh 129.5 lb.Data From Exercise 41 (a)(a) Show that the weight lost at altitude h miles above the earth’s surface is approximately ΔW ≈ −(0.0005w)h. Use the Linear Approximation with dx = h.
A stone tossed vertically into the air with initial velocity v cm/s reaches a maximum height of h = v2/1960 cm.(a) Estimate Δh if v = 700 cm/s and Δv = 1 cm/s.(b) Estimate Δh if v = 1000 cm/s and Δv = 1 cm/s.(c) In general, does a 1-cm/s increase in v lead to a greater change in h at low or
Use the following fact derived from Newton’s Laws: An object released at an angle θ with initial velocity v ft/s travels a horizontal distanceA player located 18.1 ft from the basket launches a successful jump shot from a height of 10 ft (level with the rim of the basket), at an angle θ = 34°
The side s of a square carpet is measured at 6 m. Estimate the maximum error in the area A of the carpet if s is accurate to within 2 cm.
In the notation of Exercise 49, assume that a measurement yields V = 4. Estimate the maximum allowable error in V if P must have an error of less than 0.2 atm.Data From Exercise 49The volume (in liters) and pressure P (in atmospheres) of a certain gas satisfy PV = 24. A measurement yields V = 4
The dosage D of diphenhydramine for a dog of body mass w kg is D = 4.7w2/3 mg. Estimate the maximum allowable error in w for a cocker spaniel of mass w = 10 kg if the percentage error in D must be less than 3%.
A golfer hits a golf ball at an angle of θ = 23° with initial velocity v = 120 ft/s.(a) Estimate Δs if the ball is hit at the same velocity but the angle is increased by 3°.(b) Estimate Δs if the ball is hit at the same angle but the velocity is increased by 3 ft/s.
Approximate f (2) if the linearization of f (x) at a = 2 is L(x) = 2x + 4?
Compute the linearization of ƒ(x) = 3x − 4 at a = 0 and a = 2. Prove more generally that a linear function coincides with its linearization at x = a for all a.
Estimate √16.2 using the linearization L(x) of ƒ(x) = √x at a = 16. Plot f and L on the same set of axes and determine whether the estimate is too large or too small.
Estimate 1/√15 using a suitable linearization of ƒ(x) = 1/√x. Plot f and L on the same set of axes and determine whether the estimate is too large or too small. Use a calculator to compute the percentage error.
Approximate using linearization and use a calculator to compute the percentage error.1/√17
Approximate using linearization and use a calculator to compute the percentage error.1/101
Approximate using linearization and use a calculator to compute the percentage error.1/(10.03)2
Approximate using linearization and use a calculator to compute the percentage error.(17)1/4
Approximate using linearization and use a calculator to compute the percentage error.(64.1)1/3
Approximate using linearization and use a calculator to compute the percentage error.(1.2)5/3
Approximate using linearization and use a calculator to compute the percentage error.tan(0.04)
Approximate using linearization and use a calculator to compute the percentage error.cos (3.1/4)
Approximate using linearization and use a calculator to compute the percentage error.(3.1)/2/sin(3.1/2)
Compute the linearization L(x) of ƒ(x) = x2 − x3/2 at a = 4. Then plot ƒ − L and find an interval I around a = 4 such that |ƒ(x) − L(x)| ≤ 0.1 for x ∈ I.
Show that the Linear Approximation to ƒ(x) = √x at x = 9 yields the estimate √9 + h − 3 ≈ 1/6 h. Set K = 0.01 and show that |ƒ"(x)| ≤ K for x ≥ 9. Then verify numerically that the error E satisfies Eq. (5) for h = 10−n, for 1 ≤ n ≤ 4. ΕΣ-Κ (Δ.)2 E <
The Linear Approximation to ƒ(x) = tan x at x = π/4 yields the estimate tan π/4 + h − 1 ≈ 2h. Set K = 6.2 and show, using a plot, that |ƒ"(x)| ≤ K for x ∈ [π/4, π/4 + 0.1]. Then verify numerically that the error E satisfies Eq. (5) for h = 10−n, for 1 ≤ n ≤ 4. 1 ΕΣ-Κ
Apply the method of Exercise 69 to P = (0.5, 1) on y5 + y − 2x = 1 to estimate the y-coordinate of the point on the curve where x = 0.55.
Apply the method of Exercise 69 to P = (−1, 2) on y4 + 7xy = 2 to estimate the solution of y4 − 7.7y = 2 near y = 2.
Show that for any real number k, (1 + Δx)k ≈ 1 + kΔx for small Δx. Estimate (1.02)0.7 and (1.02)−0.3.
Let Δƒ= ƒ(1 + h) − ƒ(1), where ƒ(x) = x−1. Show directly that E = |Δƒ − ƒ'(1)h| is equal to h2/(1 + h). Then prove that E ≤ 2h2 if −1/2 ≤ h ≤ 1/2. In this case, 1/2 ≤ 1 + h ≤ 3/2.
In Exercises 9–12, refer to Figure 13.Estimate ƒ(1) andƒ'(2). 3.0 2.5 2.0 1.5 1.0 0.5 y 0.5 1.0 f(x). 1.5 2.0 2.5 3.0 FIGURE 13 ·x

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