4. Applications of Derivatives

The devil’s curve (Gabriel Cramer, 1750) Find the slopes of the devil’s curve y⁴ – 4y² = x⁴ – 9x² at the four indicated points.

The folium of Descartes (See Figure 3.27)

c. Find the coordinates of the point A in Figure 3.29 where the folium has a vertical tangent line.

Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?

" style="" width="340">

Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?

<IMAGE>

In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?

53. xy³ + x²y = 6

In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?

54. x³ + y² = sin²y

Computer Explorations

Use a CAS to perform the following steps in Exercises 55–62.

b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.

2y² + (xy)¹/³ = x² + 2, P(1,1)

Computer Explorations

Use a CAS to perform the following steps in Exercises 55–62.

b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.

x√(1 + 2y) + y = x², P(1,0)

Theory and Examples

Intersecting normal line The line that is normal to the curve x² + 2xy – 3y² = 0 at (1,1) intersects the curve at what other point?

Differentiating Implicitly

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

x²y + xy² = 6

Differentiating Implicitly

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

x³ + y³ = 18xy

Differentiating Implicitly

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

x = sec y

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.

y² = x² + 2x

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.

3 + sin y = y – x³

In Exercises 29 and 30, find the slope of the curve at the given points.

(x² + y²)² = (x – y)² at (1,0) and (1,–1)