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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Calculate the derivative with respect to x of the other variable appearing in the equation.tan(x + y) = tan x + tan y
Use the following table of values for the number A(t) of automobiles (in millions) manufactured in the United States in year t.What is the interpretation of A(t)? Estimate A(1971). Does A(1974)
Compute the derivative of ƒ ∘ g.ƒ(u) = sin u, g(x) = 2x + 1
A police car traveling south toward Sioux Falls, Iowa, at 160 km/h pursues a truck traveling east away from Sioux Falls at 140 km/h (Figure 12). At time t = 0, the police car is 20 km north and the
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?
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
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
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
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 −
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
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
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
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,
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)
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
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)
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
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
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
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
Find an equation of the tangent line at the point specified.y = 2(sin θ + cos θ), θ = π/3
Use the method of Example 4 to compute dy/dx Ι P at P = (2, 1) on the curve y2x3 + y3x4 − 10x + y = 5.
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
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.
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
Calculate the higher derivative. d² dt² cos² t
Compute the derivative. y = 1 (1-x) √2-x

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