Multivariable Calculus · free preview

Implicit Differentiation

The implicit function theorem in two and three variables, the gradient formula for slopes of implicitly defined curves and surfaces, and the geometric reason it works.

1. Curves defined without solving

Curves defined without solving

2. The circle and the folium

The circle and the folium

3. The geometric engine is the gradient

The geometric engine is the gradient

4. When the hypothesis fails

When the hypothesis fails

5. A slope by implicit differentiation

The curve x2y+xy2=6x^2 y + xy^2 = 6 passes through (1,2)(1, 2). What is the slope of the curve there?

6. A tangent plane to a sphere

The surface x2+y2+z2=14x^2 + y^2 + z^2 = 14 passes through (1,2,3)(1, 2, 3). What is its tangent plane there?

7. Where implicit differentiation refuses to work

On the circle x2+y2=25x^2 + y^2 = 25, at which point does the formula dy/dx=−Fx/Fydy/dx = -F_x/F_y break down?

8. Tangent line of a circle

For the circle x2+y2=25x^2 + y^2 = 25 at the point (3,4)(3, 4), give the slope of the tangent line as a fraction like -3/4.

9. A second point on the folium-like curve

For the curve x2y+xy2=6x^2 y + xy^2 = 6 at the point (2,1)(2, 1), give the slope as a fraction like -3/4.

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