Chapter 3 Definitions and Theorems

Transcription

1 Chapter 3 Definitions an Theorems (from 3.1) Definition of Tangent Line with slope of m If f is efine on an open interval containing c an the limit Δy lim Δx 0 Δx = lim f (c + Δx) f (c) = m Δx 0 Δx exists, then we call the line passing through (c, f (c)) with slope m the tangent line to the graph of f at the point (c, f (c)). Definition of the Derivative of a Function The erivative of f at x is given by f '(x) = lim Δx 0 f (x + Δx) f (x) Δx provie the limit exists. Alternate Form of the Derivative The erivative of f at c is given by f '(c) = lim x c f (x) f (c) x c provie the limit exists. Differentiability Implies Continuity If f is ifferentiable at x = c, then f is continuous at x = c. (from 3.2) Definition of Average Velocity If s(t) gives the position at time t of an object moving in a straight line, then the average velocity of the object over the interval [t, t + Δt] is given by average velocity = Δs Δt = s(t + Δt) s(t) Δt

2 Definition of Instantaneous Velocity If s = s(t) is the position function for an object moving along a straight line, then the velocity of the object at time t is given by Δs v(t) = lim Δt 0 Δt = lim s(t + Δt) s(t) = s'(t) Δt 0 Δt Definition of Acceleration If s is the position function for an object moving along a straight line, then the acceleration of the object at time t is given by a(t) = v'(t) where v(t) is the velocity at time t. Interpretations of the Derivative If the function given by y = f (x) is ifferentiable at x, then its erivative y = f '(x) = lim x x 0 enotes both f (x + Δx) f (x) Δx 1. the slope of the graph of f at x an 2. the instantaneous rate of change in y with respect to x (from 3.3) Constant Rule The erivative of a constant is zero. x c [ ] = 0, c is a real number Power Rule If n is a rational number, then x xn = nx n 1 Constant Multiple Rule If f is a ifferentiable function an c is a real number, then [ cf (x)] = cf '(x) x

3 Sum an Difference Rules The erivative of the sum (or ifference) of two ifferentiable functions is the sum (or ifference) of their erivatives. [ f (x) + g(x) ] = f '(x) + g'(x) Sum Rule x f (x) g(x) x [ ] = f '(x) g'(x) Difference Rule (from 3.4) Prouct Rule The prouct of two ifferentiable functions f an g is itself ifferentiable. Moreover, the erivative of fg is the first function times the erivative of the secon, plus the secon function times the erivative of the first. x f (x)g(x) [ ] = f (x)g'(x) + g(x) f '(x) Prouct Rule expane to three functions If f,g, an h are ifferentiable functions of x, then x f (x)g(x)h(x) [ ] = f '(x)g(x)h(x) + f (x)g'(x)h(x) + f (x)g(x)h'(x) Quotient Rule The quotient f g of two ifferentiable functions f an g is itself ifferentiable at all values of x for which g(x) 0. Moreover, the erivative of f g is given by the enominator times the erivative of the numerator minus the numerator times the erivative of the enominator ivie by the square of the enominator. x f (x) g(x) g(x) f '(x) f (x)g'(x) =, g(x) 0 [ g(x) ] 2

4 Summary of Differentiation General Differentiation rules: Let u an v be ifferentiable functions of x. Constant multiple Rule: Sum Rule: Difference Rule: Prouct Rule: Quotient Rule: [ x cu] = cu ' x u + v x u v [ x uv ] = uv'+ vu' u vu' uv' x v = v 2 [ ] = u'+ v' [ ] = u' v' Derivatives of Algebraic Functions: Constant Rule: Power Rule: (from 3.5) Chain Rule x c [ ] = 0 x xn = nx n 1 [ x x] = 1 If y = f (u) is a ifferentiable function of u an u = g(x) is a ifferentiable function of x, then y = f (g(x)) is a ifferentiable function of x an y x = y u iu x or, equivalently, x f (g(x)) [ ] = f '(g(x))g'(x)

5 General Power Rule If y = [ u(x) ] n, where u is a ifferentiable function of x an n is a rational number, then y x = n[ u(x) u ]n 1 x or, equivalently, x un = nu n 1 u '. (from 3.6) Implicit Differentiation Given an equation involving x an y, an assuming y is a ifferentiable function of x, we can fin y as follows: x 1. Differentiate both sies of the equation with respect to x. 2. Collect all terms involving y on the left sie of the equation an move all x other terms to the right sie of the equation. 3. Factor y x 4. Solve for y x out of the left sie of the equation. that oes not contain y x. by iviing both sies of the equation by the left-han factor

6 (from 3.7) Proceure for Solving Relate Rate Problems 1. Assign symbols to all given quantities an quantities to be etermine. Make a sketch an label the quantities if feasible. 2. Write an equation involving the variables whose rates of change either are given or are to be etermine. 3. Using the Chain Rule, implicitly ifferentiate both sies of the equation with respect to time, t. 4 Substitute into the resulting equation all known values for the variables an their rates of change. Then solve for the require rate of change.