Calculus I Chapter 3 Differentiation Formulas HughsHallett Derivative
![• Constant Function Theorem Suppose that f is continuous on [a, b] and](https://slidetodoc.com/presentation_image_h2/7bfe15c4b1f63beb7e6eb83d2cc226fc/image-5.jpg "• Constant Function Theorem Suppose that f is continuous on [a, b] and")
![• Constant Function Theorem Suppose that f is continuous on [a, b] and](https://slidetodoc.com/presentation_image_h2/7bfe15c4b1f63beb7e6eb83d2cc226fc/image-6.jpg "• Constant Function Theorem Suppose that f is continuous on [a, b] and")
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Calculus I • Chapter 3 -- Differentiation Formulas – Hughs-Hallett
Derivative Formulas, Part 1
Derivative Formulas, Part 2
Linear Tangent Line Approximation • Suppose f is differentiable at x = a. Then, for values of x near a, the tangent line approximation to f(x) is: • The expression is called the local linearization of f near x = a. We are thinking of a as fixed, so that f(a) and f’(a) are constant. The error E(x), is defined by: • an d
• Constant Function Theorem Suppose that f is continuous on [a, b] and differentiable on (a, b). If f’(x) = 0 on (a, b), then f is constant on [a, b]. • The Mean Value Theorem If f is continuous on [a, b] and differentiable on (a, b), then there exists a number c, with a < c < b, such that
• Constant Function Theorem Suppose that f is continuous on [a, b] and differentiable on (a, b). If f’(x) = 0 on (a, b), then f is constant on [a, b]. • Increasing Function Theorem Suppose that f is continuous on [a, b] and differentiable on (a, b). • If f’(x) > 0 on (a, b), then f is increasing on [a, b]. • If f’(x) 0 on (a, b), then f is nondecreasing on [a, b] • The Racetrack Principle Suppose that g and h are continuous on [a, b] and differentiable on (a, b), and that g’(x) h’(x) for a < x < b. • If g(a) = h(a), then g(x) h(x) for a x b. • If g(b) = h(b), then g(x) h(x) for a x b.
Differential and Integral Formulas
