4. Applications of Derivatives

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

d. Verify that the results of parts (a) and (c) are consistent.

Vertical tangent lines

b. Does the curve have any horizontal tangent lines? Explain.

58–59. Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

xy^5/2+x^3/2y=12; (4, 1)

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x³ + y³ = 18xy

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

cos y = x; (0, π/2)

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x²y + xy² = 6

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)

Consider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .

27–40. Implicit differentiation Use implicit differentiation to find dy/dx.

√x⁴+y² = 5x+2y³

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)

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.

Find dr/dθ in Exercises 15–18.

r – 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).

Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>

y = cx²; x²+2y² = k, where c and k are constants

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)

Second Derivatives

In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.

2√y = x – y