BackNormal 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?
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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?
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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)
Normal lines parallel to a line Find the normal lines to the curve xy + 2x – y = 0 that are parallel to the line 2x + y = 0.