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mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Find an equation of the tangent line at the point specified.y = x3 + cos x, x = 0
A car travels down a highway at 25 m/s. An observer stands 150 m from the highway.(a) How fast is the distance from the observer to the car increasing when the car passes in front of the observer? Explain your answer without making any calculations.(b) How fast is the distance increasing 20 s later?
Calculate the derivative with respect to x of the other variable appearing in the equation.x sin y − y cos x = 2
Use the following table of values for the number A(t) of automobiles (in millions) manufactured in the United States in year t.Given the data, which of (A)–(C) in Figure 6 could be the graph of the derivative A'? Explain. t A(t) 1970 1971 1972 1973 1974 1975 1976 6.55 8.58 8.83 9.67 7.32 6.72 8.50
Let ƒ(x) = √x. Does ƒ (5 + h) equal √5 + h or √5 + √h? Compute the difference quotient at a = 5 with h = 1.
Compute the derivative of ƒ ∘ g.ƒ(u) = 2u + 1, g(x) = sin x
Find an equation of the tangent line at the point specified.y = tan θ, θ = π/6
xIn the setting of Example 5, at a certain moment, the tractor’s speed is 3 m/s and the bale is rising at 2 m/s. How far is the tractor from the bale at this moment? EXAMPLE 5 Farmer John's tractor, traveling at 3 m/s, pulls a rope attached to a bale of hay through a pulley. With dimensions as
Calculate the derivative with respect to x of the other variable appearing in the equation.x + cos(3x − y) = xy
Compute the derivative.y = 3x5 − 7x2 + 4
Let ƒ(x) = 1/√x. Compute ƒ'(5) by showing that f(5+h)-f(5) h 1 √5 √5 + h(√5 +h + √5)
Compute the derivative of ƒ ∘ g.ƒ(u) = u + u−1, g(x) = tan x
Find an equation of the tangent line at the point specified.y = sin t/1 + cos t, t = π/3
Calculate the derivative with respect to x of the other variable appearing in the equation.2x2 − x − y = x4 + y4
Placido pulls a rope attached to a wagon through a pulley at a rate of q m/s. With dimensions as in Figure 13:(a) Find a formula for the speed of the wagon in terms of q and the variable x in the figure.(b) Find the speed of the wagon when x = 0.6 if q = 0.5 m/s.Placido pulls a rope attached to a
Compute the derivative.y = 4x−3/2
Find an equation of the tangent line to the graph of ƒ(x) = 1/√ x at x = 9.
Compute the derivative of ƒ ∘ g.ƒ(u) = u/u − 1 , g(x) = csc x
Find an equation of the tangent line at the point specified.y = sin x + 3 cos x, x = 0
Show that x + yx−1 = 1 and y = x − x2 define the same curve [except that (0, 0) is not a solution of the first equation] and that implicit differentiation yields y'= yx−1 − x and y'= 1 − 2x. Explain why these formulas produce the same values for the derivative.
Compute the derivative.y = t−7.3
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = 2x2 + 10x, a = 3
Use the result in Example 3 to find d6/dx6 x−1.EXAMPLE 3 Calculate the first four derivatives of y = x-1. Then find the pattern and determine a general formula for y(n).
Julian is jogging around a circular track of radius 50 m. In a coordinate system with its origin at the center of the track, Julian’s x-coordinate is changing at a rate of −1.25 m/s when his coordinates are (40, 30). Find dy/dt at this moment.
Find the derivatives of ƒ(g(x)) and g(ƒ(u)).ƒ(u) = cos u, g(x) = x2 + 1
Compute the derivative.y = 4x2 − x−2
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = 4 − x2, a = −1
Find the derivatives of ƒ(g(x)) and g(ƒ(u)).ƒ(u) = u3, g(x) = 1/x + 1
Find an equation of the tangent line at the point specified.y = csc x − cot x, x = π/4
Find an equation of the tangent line at the point specified.y = (cot t)(cos t), t = π/3
Find dy/dx at the given point.(x + 2)2 − 6(2y + 3)2 = 3, (1, −1)
Compute the derivative. y = x + 1 x² + 1 2
Assume that the pressure P (in kilopascals) and volume V (in cubic centimeters) of an expanding gas are related by PVb = C, where b and C are constants (this holds in an adiabatic expansion, without heat gain or loss).Find dP/dt if b = 1.2, P = 8 kPa, V = 100 cm2, and dV/dt = 20 cm3/min.
Use the Chain Rule to find the derivative.y = sin(x2)
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(t) = t − 2t2, a = 3
Find an equation of the tangent line at the point specified.y = x cos2 x, x = π/4
Find dy/dx at the given point.sin2(3y) = x + y, (2 − π/4, π/4)
Compute the derivative. y = 3t - 2 4t - 9
Assume that the pressure P (in kilopascals) and volume V (in cubic centimeters) of an expanding gas are related by PVb = C, where b and C are constants (this holds in an adiabatic expansion, without heat gain or loss).Find b if P = 25 kPa, dP/dt = 12 kPa/min, V = 100 cm2, and dV/dt = 20 cm3/min.
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = 8x3, a = 1
Use the Chain Rule to find the derivative.y = sin2 x
Use the Chain Rule to find the derivative.y =√t2 + 9
Show that for an object falling according to Galileo’s formula, the average velocity over any time interval [t1, t2] is equal to the average of the instantaneous velocities at t1 and t2.
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = x3 + x, a = 0
Find an equation of the tangent line at the given point.xy + x2y2 = 6, (2, 1)
Use the Chain Rule to find the derivative.y = (t2 + 3t + 1)−5/2
Find an equation of the tangent line at the point specified. y = sin 9- cos 0 Ө 0 = Ө 元-4 π
Compute the derivative.y = (3t2 + 20t−3)6
Two parallel paths 15 m apart run east–west through the woods. Brooke jogs east on one path at 10 km/h, while Jamail walks west on the other path at 6 km/h. If they pass each other at time t = 0, how far apart are they 3 s later, and how fast is the distance between them changing at that moment?
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(t) = 2t3 + 4t, a = 4
Use Theorem 1 to verify the formula. THEOREM 1 Derivative of Sine and Cosine The functions y= sin x and y = cos x are differentiable and d dx sin x = cos x and d dx - sin x COS X = -
Find an equation of the tangent line at the given point.x2/3 + y2/3 = 2, (1, 1)
Use the Chain Rule to find the derivative.y = (x4 − x3 − 1)2/3
A particle travels along a curve y = ƒ(x) as in Figure 16. Let L(t) be the particle’s distance from the origin.(a) Show that if the particle’s location at time t is P = (x, ƒ (x)).(b) Calculate L'(t) when x = 1 and x = 2 if ƒ(x) = √3x2 − 8x + 9 and dx/dt = 4. dL dt x+ f(x)f'(x) √x²
Compute the derivative.y = (2 + 9x2)3/2
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = x−1, a = 8
Find an equation of the tangent line at the given point.x2 + sin y = xy2 + 1, (1, 0)
Use Theorem 1 to verify the formula.d/dx sec x = sec x tan x THEOREM 1 Derivative of Sine and Cosine The functions y = sin x and y = cos x are differentiable and d dx sin x = cos x and d dx - sin x COS X = -
Use the Chain Rule to find the derivative.y = (√x + 1 − 1)3/2
Let θ be the angle in Figure 16, where P = (x, ƒ (x)). In the setting of the previous exercise, show thatDifferentiate tan θ = ƒ(x)/x and observe that cos θ = x/√x2 + ƒ(x)2. de dt || xf'(x) - f(x) dx x² + f(x)² dt
Compute the derivative.y = (x + 1)3(x + 4)4
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = x + x−1, a = 4
Find an equation of the tangent line at the given point.sin(x − y) = x cos (y + π/4), (π/4, π/4)
Use the Chain Rule to find the derivative. = (x + 1)* y =
Use Theorem 1 to verify the formula.d/dx csc x = − csc x cot x THEOREM 1 Derivative of Sine and Cosine The functions y = sin x and y = cos x are differentiable and d dx sin x = cos x and d dx - sin x COS X = -
Compute the derivative. y = Z √1-z Z
Ethan finds that with h hours of tutoring, he is able to answer correctly S (h) percent of the problems on a math exam. Which would you expect to be larger: S'(3) or S'(30)? Explain.
Refer to the baseball diamond (a square of side 90 ft) in Figure 17.A baseball player runs from home plate toward first base at 20 ft/s. How fast is the player’s distance from second base changing when the player is halfway to first base? 15 ft/s Second base Home plate 90 ft First base 20 ft/s
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = 1/x + 3, a = −2
Find an equation of the tangent line at the given point.2x1/2 + 4y−1/2 = xy, (1, 4)
Use the Chain Rule to find the derivative.y = cos3(12θ)
Show that both y = sin x and y = cos x satisfy y"= −y.
Compute the derivative. y = 1 + x 3
Suppose θ(t) measures the angle between a clock’s minute and hour hands. What is θ'(t) at 3 o’clock?
Player 1 runs to first base at a speed of 20 ft/s, while player 2 runs from second base to third base at a speed of 15 ft/s. Let s be the distance between the two players. How fast is s changing when player 1 is 30 ft from home plate and player 2 is 60 ft from second base?
Find an equation of the tangent line at the given point. X x+1 + y y+1 = 1, (1, 1)
Find an equation of the tangent to the graph of y = ƒ'(x) at x = 3, where ƒ(x) = x4.
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(t) = 2/1 − t, a = −1
Use the Chain Rule to find the derivative. 1 y = sec = X
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = √x + 4, a = 1
Calculate the higher derivative.ƒ (θ), ƒ(θ) = θ sin θ
Compute the derivative. y = x² + √x
The conical watering pail in Figure 18 has a grid of holes. Water flows out through the holes at a rate of kA m3/min, where k is a constant and A is the surface area of the part of the cone in contact with the water. This surface area is A = πr √h2 + r2 and the volume is V = 1/3 πr2h.
Find an equation of the tangent line at the point specified.y = 2(sin θ + cos θ), θ = π/3
A particle moves counterclockwise around the ellipse with equation 9x2 + 16y2 = 25 (Figure 14).(a) In which of the four quadrants is dx/dt > 0? Explain.(b) Find a relation between dx/dt and dy/dt.(c) At what rate is the x-coordinate changing when the particle passes the point (1, 1) if its
Compute the derivative.y = (x4 − 9x)6
The base x of the right triangle in Figure 15 increases at a rate of 5 cm/s, while the height remains constant at h = 20. How fast is the angle θ changing when x = 20? 0 X 20
Find an equation of the tangent line at the point specified.y = x2(1 − sin x), x = 3π/2
To determine drug dosages, doctors estimate a person's body surface area (BSA) (in meters squared) using the formula BSA =√hm/60, where h is the height in centimeters and m the mass in kilograms. Calculate the rate of change of BSA with respect to mass for a person of constant height h = 180.
Find an equation of the tangent line at the given point. sin(2x - y) = x² R y (0, π)
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(t) = √3t + 5, a = −1
Use the Chain Rule to find the derivative.y = tan(θ2 − 4θ)
The atmospheric CO2 level A(t) at Mauna Loa, Hawaii, at time t (in parts per million by volume) is recorded by the Scripps Institution of Oceanography. Reading across, the annual values for the 4-year intervals are(a) Estimate A'(t) in 1962, 1970, 1978, 1986, 1994, 2002, and 2010.(b) In which of
Calculate the higher derivative. d² dt² cos² t
Compute the derivative. y = 1 (1-x) √2-x
A bowl contains water that evaporates at a rate proportional to the surface area of water exposed to the air (Figure 19). Let A(h) be the cross-sectional area of the bowl at height h.(a) Explain why V(h + Δh) − V(h) ≈ A(h)Δh if Δh is small.(b) Use (a) to argue that dV/dh = A(h).(c) Show that
Find an equation of the tangent line at the given point.x + √x = y2 + y4, (1, 1)
The second derivative ƒ" is shown in Figure 7. Which of (A) or (B) is the graph of ƒ and which is ƒ'? f"(x) y A· A (B) (A)
Use the Chain Rule to find the derivative.y = tan(θ + cos θ)
In Exercises 29–46, use the limit definition to compute ƒ'(a) and find an equation of the tangent line.ƒ(x) = 1/√x, a = 4
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