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Problem 3.8.17a
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
sin y = 5x⁴−5; (1, π)
Problem 3.8.17b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
sin y = 5x⁴−5; (1, π)
Problem 3.8.27
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
sin x+sin y=y
Problem 3.8.28
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
y = xe^y
Problem 3.8.64b
Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.
Problem 3.8.65b
Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.
Problem 3.8.88
A challenging derivative Find dy/dx, where √3x⁷+y² = sin²y+100xy.
Problem 3.8.89
A challenging second derivative Find d²y/dx², where √y+xy=1.
Problem 3.8.10
Find the slope of the curve x²+y³=2 at each point where y=1 (see figure). <IMAGE>
Problem 3.8.19a
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
cos y = x; (0, π/2)
Problem 3.8.19b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
cos y = x; (0, π/2)
Problem 3.8.20a
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
tan xy = x+y; (0,0)
Problem 3.8.20b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
tan xy = x+y; (0,0)
Problem 3.8.23a
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
³√x+³√y⁴ = 2;(1,1)
Problem 3.8.23b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
³√x+³√y⁴ = 2;(1,1)
Problem 3.8.60a
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
a. Find equations of all lines tangent to the curve at the given value of x.
x+y³−y=1; x=1
Problem 3.8.60b
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
b. Graph the tangent lines on the given graph.
x+y³−y=1; x=1
Problem 3.8.62a
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
a. Find equations of all lines tangent to the curve at the given value of x.
4x³ =y²(4−x); x=2 (cissoid of Diocles)
Problem 3.8.62b
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
b. Graph the tangent lines on the given graph.
4x³ =y²(4−x); x=2 (cissoid of Diocles)
Problem 3.8.63a
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
a. Use implicit differentiation to find dy/dx.
Problem 3.8.63b
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.
Problem 3.8.63c
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
c. Solve the equation y(x²+4)=8 for y to find an explicit expression for y and then calculate dy/dx.
Problem 3.8.63d
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
d. Verify that the results of parts (a) and (c) are consistent.
Problem 3.8.74
73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. <IMAGE>
Exercise 46
Problem 3.8.76
73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. <IMAGE>
Exercise 48
Problem 3.8.80a
79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>
a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)
x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)
Problem 3.8.80b
79–82. {Use of Tech} Visualizing tangent and normal lines
b. Graph the tangent and normal lines on the given graph.
x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)
Problem 3.8.82a
79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>
a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)
(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)
Problem 3.8.82b
79–82. {Use of Tech} Visualizing tangent and normal lines
b. Graph the tangent and normal lines on the given graph.
(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)
Problem 3.8.58a
58–59. Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
xy^5/2+x^3/2y=12; (4, 1)