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Hall's Review 3.2
Final Review
| Question | Answer |
|---|---|
| Define a tangent line | The tangent line to the curve y = f(x) at the point P(c, f(c)) is that line through P with slope m= lim (as h approaches 0) (f(c+h)- f(c))/h proivided the limit exists and is not infiniety or negative infinity. |
| Define derivative | The derivative of a function f is another function f’ whose value at any number x is f'(x) = lim(as h approaches 0) [f(x+h)- f(x)]/h |
| Thm. differenciability implies what? | continuity; If f’(c) exist, the f is continuous at c |
| Thm: Constant Function Rule | If f(x) = k, where k is a constant, then for any x, f’(x) = 0 |
| Thm: Identity Function Rule | If f(x) = x then f’(x) = 1 |
| Thm: Power Rule | If f(x) = x^n, where n is a positive integer, the f’(x) = nx^n-1; that is Dx(x^n) = nx^n-1 |
| Thm: Constant Multiple Rule | If k is a constant and f is a differentiable function, then (kf)’(x) = k * f’(x) |
| Thm: Sum Rule | If f and g are differentiable functions, then (f+g)’(x) = f’(x) + g’(x) |
| Thm: Difference Rule | If f and g are differentiable functions, then (f-g)’(x) = f’(x) - g’(x) |
| Thm: Product Rule | If f and g are differentiable functions, then (f*g)’(x) = f(x)g’(x) + g(x)f’(x) |
| Thm: Quotient Rule | Let f and g be differentiable functions with g(x) does not equal zero. Then (f/g)'(x) = [g(x)f'(x)-g'(x)f(x)]/(g(x))^2 |
| Derivative of sin x | cos x |
| Derivative of cos x | -sin x |
| Derivative of tan x | (sec x)^2 |
| Derivative of cot x | -(csc x)^2 |
| Derivative of sec x | sec x tan x |
| Derivative of csc x | -csc x cot x |
| Thm. Chain Rule | Let y = f(u) and u = g(x). If g is differentiable at x and f is differentiable at u = g(x), then the composite function f(g(x)) is differentiable at x and f(g(x))' = f'(g(x))g'(x) |
Created by:
agea
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