Calculus Volume 1 · 免费预览
隐函数求导法
当曲线由关于 x 和 y 的方程定义，而不是由解出 y 的公式给出时，对等式两边关于 x 求导，再解出 dy/dx，即可得到斜率，从而处理圆、椭圆以及相关变化率问题。
1. Functions hiding in equations
Functions hiding in equations
2. The general procedure
The general procedure
3. Higher derivatives implicitly
Higher derivatives implicitly
4. Curves with no closed form
Curves with no closed form
5. Why the chain rule appears
In implicit differentiation of x^2 + y^2 = 25, the derivative of y^2 with respect to x is 2y dy/dx. Why does the factor dy/dx appear?
6. Slope on the circle
Using implicit differentiation, what is the slope dy/dx of the circle x^2 + y^2 = 25 at the point (3, 4)?
7. Vertical tangents
For the circle x^2 + y^2 = 25, the derivative dy/dx = -x/y is undefined where y = 0. What geometric feature occurs at those points?
8. Product rule necessity
Differentiating the term xy in the equation x^2 + xy + y^2 = 7, which expression results?
9. Second derivative sign on the upper semicircle
The second derivative of the circle's upper semicircle works out to -25/y^3, which is negative since y is positive up there. What does the negative sign confirm geometrically?
10. Horizontal tangent on the lower circle
For the circle x^2 + y^2 = 1 with derivative dy/dx = -x/y, what is the tangent slope at the point (0, -1) on the bottom?
11. Implicit derivative of xy equals one
Differentiating xy = 1 implicitly gives a relation between dy/dx, x, and y. What is dy/dx? Answer like -y/x.
12. Slope on the upper unit semicircle
For x^2 + y^2 = 1, the derivative is dy/dx = -x/y. At the point (0, 1) on top, what is the slope? Answer with a number.
13. Product term count
In the equation x^2 + xy = 4, implicit differentiation produces one term from x^2, one from y in the product, and one from x dy/dx. How many total terms before solving? Answer with a number.
14. Ellipse slope formula
Implicit differentiation of x^2/9 + y^2/4 = 1 gives dy/dx equal to a constant times x/y. What is that constant, as a fraction? Answer like -4/9.