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Problem 3.7.79
Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.
Problem 3.8.64.a
Vertical tangent lines
a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).
Problem 3.8.5
5–8. Calculate dy/dx using implicit differentiation.
x = y²
Problem 3.8.7
5–8. Calculate dy/dx using implicit differentiation.
sin y+2 = x
Problem 3.8.26a
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
(x+y)^2/3=y; (4, 4)
Problem 3.8.31
Use implicit differentiation to find dy/dx.
sin xy = x+y
Problem 3.8.32
Use implicit differentiation to find dy/dx.
exy = 2y
Problem 3.8.33
Use implicit differentiation to find dy/dx.
cos y2 + x = ey
Problem 3.8.35
Use implicit differentiation to find dy/dx.
x3 = (x + y) / (x - y)
Problem 3.8.37
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
6x³+7y³ = 13xy
Problem 3.8.39
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
√x⁴+y² = 5x+2y³
Problem 3.8.42a
Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².
a. Find dr/dh for a cone with a lateral surface area of A=1500π.
Problem 3.8.42b
Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².
b. Evaluate this derivative when r=30 and h=40.
Problem 3.8.44a
Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.
a. Find db/da for a torus with a volume of 64π².
Problem 3.8.44b
Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.
b. Evaluate this derivative when a=6 and b=10.
Problem 3.8.46a
45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
x³+y³=2xy; (1, 1)
Problem 3.8.46b
45–50. Tangent lines Carry out the following steps. <IMAGE>
b. Determine an equation of the line tangent to the curve at the given point.
x³+y³=2xy; (1, 1)
Problem 3.8.48a
45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
x⁴-x²y+y⁴=1; (−1, 1)
Problem 3.8.48b
45–50. Tangent lines Carry out the following steps. <IMAGE>
b. Determine an equation of the line tangent to the curve at the given point.
x⁴-x²y+y⁴=1; (−1, 1)
Problem 3.8.50a
45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
(x²+y²)²=25/4 xy²; (1, 2)
Problem 3.8.50b
45–50. Tangent lines Carry out the following steps. <IMAGE>
b. Determine an equation of the line tangent to the curve at the given point.
(x²+y²)²=25/4 xy²; (1, 2)
Problem 3.8.52
51–56. Second derivatives Find d²y/dx².
2x²+y² = 4
Problem 3.8.54
51–56. Second derivatives Find d²y/dx².
x⁴+y⁴ = 64
Problem 3.8.56
51–56. Second derivatives Find d²y/dx².
sin x + x²y =10
Problem 3.8.26b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
(x+y)^2/3=y; (4, 4)
Problem 3.8.8
5–8. Calculate dy/dx using implicit differentiation.
e^y-e^x = C, where C is constant
Problem 3.8.12
Consider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .
Problem 3.8.13b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
x⁴+y⁴ = 2;(1,−1)
Problem 3.8.14a
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
x = e^y; (2, ln 2)
Problem 3.8.14b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
x = e^y; (2, ln 2)