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mathematics
calculus 4th

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

Calculate d/dx cos u for the following choices of u(x):(a) u(x) = 9 − x2 (b) u(x) = x−1 (c) u(x) = tan x
Assume that a is a constant and that y is implicitly a function of x. Compute the derivative with respect to x of each of a2, x2, and y2.
Differentiate the expression with respect to x, assuming that y is implicitly a function of x.x 3√y
Compute the derivative.ƒ(x) = x sin x
Calculate y" and y"'.y = 20t4/5 − 6t2/3
In Exercises 3–8, compute ƒ(a) in two ways, using Eq. (1) and Eq. (2).ƒ(x) = 1/x , a = 2 f'(a) = lim h→0 f(a+h)-f(a) h
Assume that the radius r of a sphere is expanding at a rate of 30 cm/min. The volume of a sphere is V = 43 πr3 and its surface area is 4πr2. Determine the given rate.Surface area with respect to time at t = 2 min, assuming that r = 10 at t = 0
Calculate d/dx ƒ(x2 + 1) for the following choices of ƒ(u):(a) ƒ(u) = sin u (b) ƒ(u) = 3u3/2 (c) ƒ(u) = u2 − u
Differentiate the expression with respect to x, assuming that y is implicitly a function of x.tan(xy)
Compute the derivative.ƒ(x) = 9 sec x + 12 cot x
Calculate y" and y"'.y = x−9/5
A conical tank (as in Example 2) has height 3 m and radius 2 m at the base. Water flows in at a rate of 2 m3/min. How fast is the water level rising when the level is 1 m and when the level is 2 m? EXAMPLE 2 Filling a Conical Container Water flows into a conical container at a rate of 5 in.³/s.
In Exercises 9–12, refer to Figure 13.Find the slope of the secant line through (2, ƒ (2)) and (2.5, ƒ(2.5)). 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
Compute dƒ / dx if dƒ / du = 2 and du/dx = 6.
Differentiate the expression with respect to x, assuming that y is implicitly a function of x.y/y + 1
Compute the derivative.H(t) = sin t sec2 t
Calculate y" and y"'.y = z − 4/z
A conical tank (as in Example 2) has height 8 m and radius 4 m at the base. Water flows in at a rate of 3 m3/min. Determine dh/dt as a function of h, and provide a graph of this relationship. EXAMPLE 2 Filling a Conical Container Water flows into a conical container at a rate of 5 in.³/s. Assume
Compute d/dx І x = 2 if ƒ(u) = u2, u(2) = −5, and u'(2) = −5.
According to one model that takes into account air resistance, the acceleration a(t) (in m/s2) of a skydiver of mass m in free-fall satisfieswhere v(t) is velocity (negative since the object is falling) and k is a constant. Suppose that m = 75 kg and k = 0.24 kg/m.(a) What is the skydiver’s
Find all values of n such that y = xn satisfiesx2y"− 12y = 0
Find all values of n such that y = xn satisfies x²y" - 2xy = 4y
Figure 9 shows the graph of the position s of an object as a function of time t. For each interval [0, 10], [10, 20], and so on, indicate whether the acceleration is negative, zero, or positive. S 10 20 30 FIGURE 9 40 50 t
Figure 8 shows the graph of the position s of an object as a function of time t. Determine the intervals on which the acceleration is positive. S 10 20 30 Time 40 -t
Let ƒ(x) = x + 2/x − 1.Use a computer algebra system to compute the ƒ(k)(x) for 1 ≤ k ≤ 4. Can you find a general formula for ƒ(k)(x)?
Find the 100th derivative of p(x) = (x + x³ + x²) ¹0 (1 + x²) ¹¹ (x³ + x³ + x²)
Calculate the derivative indicated.ƒ(4)(1), ƒ(x) = x4
Calculate the derivative indicated. d'y dt² =1 y = 4t-³ +31²
Calculate the derivative indicated. de fl dt4 It=1 f(t) = 6tº - 215
Calculate the derivative indicated. dt4 t=16 =t-3/4 X =
Figure 6 shows ƒ, ƒ, and ƒ". Determine which is which. y inter for 1/2 3 X
Calculate the derivative indicated. f""(-3), f(x) = 12 X x3
Calculate the derivative indicated. f"(1), f(t) = t t +1
Calculate the derivative indicated. h"(1), h(w) = 1 √w + 1
Calculate the derivative indicated. √s s+1 g"(1), g(s) = -
Calculate y(k)(0) for 0 ≤ k ≤ 5, where y = x4 + ax3 + bx2 + cx + d (with a, b, c, d the constants).
Which of the following satisfy ƒ(k)(x) = 0 for all k ≥ 6? (a) f(x) = 7x4 +4+x=¹ (c) f(x) = √x (e) f(x) = x/5 (b) f(x) = x³ - 2 (d) f(x) = 1 x6 - (f) f(x) = 2x² + 3x5
(a) Calculate the first five derivatives of ƒ(x) = √x.(b) Show that ƒ(n)(x) is a multiple of x−n+1/2.(c) Show that ƒ(n)(x) alternates in sign as (−1)n−1 for n ≥ 1.(d) Find a formula for ƒ(n)(x) for n ≥ 2. Verify that the coefficient is ± 1 · 3 · 5 · · · 2n − 3/2n.
Find a general formula for f(n)(x).ƒ(x) = x−2
Find a general formula for ƒ(n)(x).ƒ(x) = (x + 2)−1
Find a general formula for ƒ(n)(x).ƒ(x) = x−1/2
Find a general formula for ƒ(n)(x).ƒ(x) = x−3/2
Find a general formula for ƒ(n)(x). f(x) = x+1 x2
Find a general formula for ƒ(n)(x). f(x) = x-1 √x
(a) Find the acceleration at time t = 5 min of a helicopter whose height is s(t) = 300t − 4t3 m.(b) Plot the acceleration s" for 0 ≤ t ≤ 6. How does this graph show that the helicopter is slowing down during this time interval?
In Exercises 1–8, find the rate of change.Volume of a cube with respect to its side s when s = 5
In Exercises 1–8, find the rate of change.Cube root 3√x with respect to x when x = 1, 8, 27
In Exercises 1–8, find the rate of change.The reciprocal 1/x with respect to x when x = 1, 2, 3
In Exercises 1–8, find the rate of change.The diameter of a circle with respect to radius
In Exercises 1–8, find the rate of change.Surface area A of a sphere with respect to radius r (A = 4πr2)
In Exercises 1–8, find the rate of change.Volume V of a cylinder with respect to radius if the height is equal to the radius
In Exercises 1–8, find the rate of change.Speed of sound v (in m/s) with respect to air temperature T (in kelvins), where v = 20 √T
Find the average velocity over each interval.(a) [0, 0.5] (b) [0.5, 1] (c) [1, 1.5] (d) [1, 2]
At what time is velocity at a maximum?
Match the descriptions (i)–(iii) with the intervals (a)–(c) in Figure 11.(i) Velocity increasing(ii) Velocity decreasing(iii) Velocity negative(a) [0, 0.5](b) [2.5, 3](c) [1.5, 2] 150 100 Distance (km) 50 0.5 1.0 1.5 2.0 2.5 3.0 1 (h)
At what t does the SDQ approximation give the smallest (i.e., closest to 0) rate of change of temperature? What is the rate of change?
Match each situation with the graph that best represents it.(a) Dusty’s batting average increased over the first 5 years of his career, but from year to year the amount of increase went down. Dusty’s batting average is s and the time since the beginning of his career is t.(b) In performing a
Match each situation with the graph that best represents it.(a) Rocky slowed down his car as it approached the moose in the road. The distance from the car to the moose is s and the time since he spotted the moose is t.(b) The rocket’s speed increased after liftoff until the fuel was used up. The
Figure 12 shows the height y of a mass oscillating at the end of a spring, through one cycle of the oscillation. Sketch the graph of velocity as a function of time. y FIGURE 12
The earth exerts a gravitational force of F(r) = (2.99 × 1016)/r2 newtons on an object with a mass of 75 kg located r meters from the center of the earth. Find the rate of change of force with respect to distance r at the surface of the earth.
Calculate the rate of change of escape velocity vesc = (2.82 × 107)r−1/2 m/s with respect to distance r from the center of the earth.
The power delivered by a battery to an apparatus of resistance R (in ohms) is P = 2.25R/(R + 0.5)2 watts (W). Find the rate of change of power with respect to resistance for R = 3 Ω and R = 5 Ω.
A particle moving along a line has position s(t) = t4 − 18t2 m at time t seconds. At which times does the particle pass through the origin? At which times is the particle instantaneously motionless (i.e., it has zero velocity)?
Plot the position of the particle in Exercise 27. What is the farthest distance to the left of the origin attained by the particle?Data From Exercise 27A particle moving along a line has position s(t) = t4 − 18t2 m at time t seconds. At which times does the particle pass through the origin? At
A projectile is launched in the air from the ground with an initial velocity v0 = 60 m/s. What is the maximum height that the projectile reaches?
Find the velocity of an air conditioner accidentally dropped from a height of 300 m at the moment it hits the ground.
A ball tossed in the air vertically from ground level returns to earth 4 s later. Find the initial velocity and maximum height of the ball.
Olivia is gazing out a window from the 10th floor of a building when a bucket (dropped by a window washer) passes by. She notes that it hits the ground 1.5 s later. Determine the floor from which the bucket was dropped if each floor is 5 m high and the window is in the middle of the 10th floor.
An object falls under the influence of gravity near the earth’s surface. Which of the following statements is true? Explain.(a) Distance traveled increases by equal amounts in equal time intervals.(b) Velocity increases by equal amounts in equal time intervals.(c) The derivative of velocity
By Faraday’s Law, if a conducting wire of length ℓ meters moves at velocity v m/s perpendicular to a magnetic field of strength B (in teslas), a voltage of size V = −Bv is induced in the wire. Assume that B = 2 and ℓ = 0.5.(a) Calculate dV/dv.(b) Find the rate of change of V with respect
The voltage V, current I, and resistance R in a circuit are related by Ohm’s Law: V = IR, where the units are volts, amperes, and ohms. Assume that voltage is constant with V = 12 volts (V). Calculate (specifying units):(a) The average rate of change of I with respect to R for the interval from R
The tangent lines to the graph of ƒ(x) = x2 grow steeper as x increases. At what rate do the slopes of the tangent lines increase?
According to Kleiber’s Law, the metabolic rate P (in kilocalories per day) and body mass m (in kilograms) of an animal are related by a three-quarter-power law P = 73.3m3/4. Estimate the increase in metabolic rate when body mass increases from 60 to 61 kg.
The dollar cost of producing x bagels is given by the function C(x) = 300 + 0.25x − 0.5(x/1000)3. Determine the cost of producing 2000 bagels and estimate the cost of the 2001st bagel. Compare your estimate with the actual cost of the 2001st bagel.
Suppose the dollar cost of producing x video cameras is C(x) = 500x − 0.003x2 + 10−8x3.(a) Estimate the marginal cost at production level x = 5000 and compare it with the actual cost C(5001) − C(5000).(b) Compare the marginal cost at x = 5000 with the average cost per camera, defined as
Let M(t) be the mass (in kilograms) of a plant as a function of time (in years). Recent studies by Niklas and Enquist have suggested that a remarkably wide range of plants (from algae and grass to palm trees) obey a three-quarter-power growth law—that is,(a) If a tree has a growth rate of 6 kg/yr
According to Stevens’s Law in psychology, the perceived magnitude of a stimulus is proportional (approximately) to a power of the actual intensity I of the stimulus. Experiments show that the perceived brightness B of a light satisfies B = kI2/3, where I is the light intensity, whereas the
The average cost per unit at production level x is defined as Cavg(x) = C(x)/x, where C(x) is the cost of producing x units. Average cost is a measure of the efficiency of the production process.The cost in dollars of producing alarm clocks is given by C(x) = 50x3 − 750x2 + 3740x + 3750, where
The average cost per unit at production level x is defined as Cavg(x) = C(x)/x, where C(x) is the cost of producing x units. Average cost is a measure of the efficiency of the production process.Show that Cavg(x) is equal to the slope of the line through the origin and the point (x,C(x)) on the
Figure 13 displays the voltage V across a capacitor as a function of time while the capacitor is being charged. Estimate the rate of change of voltage at t = 20 s. Indicate the values in your calculation and include proper units. Does voltage change more quickly or more slowly as time goes on?
The velocity (in centimeters per second) of blood molecules flowing through a capillary of radius 0.008 cm is v = 6.4 × 10−8 − 0.001r2, where r is the distance from the molecule to the center of the capillary. Find the rate of change of velocity with respect to r when r = 0.004 cm.
Use Figure 14 to estimate dT/dh at h = 30 and 70, where T is atmospheric temperature (in degrees Celsius) and h is altitude (in kilometers). Where is dT/dh equal to zero? T (°C) -100- Troposphere 10 Stratosphere 50 Mesosphere 100 Thermosphere 150 h (km)
In Exercises 7–14, use the Power Rule to compute the derivative. d dx X x=-2
In Exercises 7–14, use the Power Rule to compute the derivative. d dt -3 t=4
In Exercises 7–14, use the Power Rule to compute the derivative. d dt -2/5 t=1
In Exercises 7–14, use the Power Rule to compute the derivative. d dx -0.35
In Exercises 7–14, use the Power Rule to compute the derivative. d dx x14/3
In Exercises 7–14, use the Power Rule to compute the derivative. d dt √17
In Exercises 7–14, use the Power Rule to compute the derivative. 1P P
Find an equation of the tangent line to y = 1/√x at x = 9.
In Exercises 21–32, calculate the derivative.ƒ(x) = 2x3 − 3x2 + 5
In Exercises 21–32, calculate the derivative.ƒ(x) = 2x3 − 3x2 + 2x
In Exercises 21–32, calculate the derivative.ƒ(x) = 4x5/3 − 3x−2 − 12
In Exercises 21–32, calculate the derivative.ƒ(x) = x5/4 + 4x−3/2 + 11x
In Exercises 21–32, calculate the derivative.g(z) = 7z−5/14 + z−5 + 9
In Exercises 21–32, calculate the derivative. h(t) = 6 √t + 1 √t
In Exercises 21–32, calculate the derivative.ƒ(s) = 4√s + 3√s
In Exercises 21–32, calculate the derivative.W(y) = 6y4 + 7y2/3
In Exercises 21–32, calculate the derivative.g(x) = π2
In Exercises 21–32, calculate the derivative.ƒ(x) = xπ

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