Linearization and Extreme Values of Functions 3.10 Linearization and Differentials Linear or Tangent Line Approximations of function values Equation of tangent to y = f(x) at (a, f(a)): Tangent line approximation or Linear approximation of f at a The linearization of f at a: L(x) = 1
Example: Find the linearization of f(x) = ln x at a = 1. Use it to approximate ln(1.1) and ln(.9). Example: Find the linearization of f(x) = x + 2 at a = 2, and use it to approximate 3.96 and 4.03. 2
Let y = f(x) be a differentiable function. The differential dx is an independent variable (it can be any real number). The differential dy is a dependent variable defined by: Geometrically dy = or dy dx = Note: 1. y = 2. dy = 3
Example: Find dy if y = x 5 + 37x. Then find the value of dy if x = 1 and dx =.02 Example: Find dy if y = x x 2 + 6 Evaluate dy when x = 2 and dx =.04. 4
Example: Use differentials to approximate 3 7.8. Actual value: 5
True Estimated Absolute Error f = f(a + dx) f(a) df = f (a)dx Relative Error Percentage Error f f(a) f f(a) 100 df f(a) df f(a) 100 Example: Approximate the error in using 6 inches as the measurement of the side of a cube to calculate its volume if the measurement might be off by.04 inches. Relative error: Percentage error: 6
Example: The radius of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle s circumference and area? 7
4.1 Maximum and Minimum Values Definition A function f has an absolute maximum at x = c if f(c) f(x) for all x in the domain D of f(x). f(c) is the absolute maximum of f on D. Definition A function f has an absolute minimum at x = c if f(c) f(x) for all x in the domain D of f(x). f(c) is the absolute minimum of f on D. Together they are called extreme values. 8
Definition A function f has a local (relative) maximum at c if there is an open interval I containing c such that for all x in I we have f(x) f(c). Definition A function f has a local (relative) minimum at c if there is an open interval I containing c such that for all x in I we have f(x) f(c). Examples: 1. f(x) = sin x 2. f(x) = x 9
3. Find the relative and absolute extreme of f(x) = 3x 4 + 16x 3 + 18x 2, for 4 x 1 10
x 2 1 x 0 4. Let f(x) = 0 x = 0 Find the absolute maximum and minimum values of f(x) on each of the following intervals: [ 1, 1] ( 2, 1) [1, 2] 11
Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f has an absolute maximum f(c) and an absolute minimum f(d), at some numbers c and d in [a, b]. 12
Fermat s Theorem: If f has a local extremum at x = c, and if f (c) exists, then f (c) = NOTE The converse of Fermat s Theorem is not true. If f (c) = 0 then f(c) is NOT necessarily a local extreme. 13
Definition A critical point of a function f is a number c in the domain of f such that f (c) = 0 or f (c) does not exist. If f has an extreme value at c, then c is a critical point Find all critical points of: 1. f(x) = 3 x + 1 2. g(x) = 1 x 3 3x 14
The Closed Interval Method To find the absolute maximum and minimum values of a continuous function f on a closed interval [a, b]: 1. Check that f is continuous 2. Find c such that f (c) = 0 3. Compute f(a), f(b), and f(c) for all c found in part 2 4. The largest of the values computed in part 3 is the absolute maximum and the smallest is the absolute minimum. Example: Find the maximum and minimum values of f(x) = 3x 4 + 16x 3 + 18x 2 on [ 4, 1] 15
Example: Find the absolute maximum and minimum values of f(x) = ln x x on the intervals: a) [1, e] b) [ 1 e, 1] c) [1, e 2 ] 16
[ Example: Find the absolute extreme of f(x) = tan x on π 3, π ]. 4 17
4.2 The Mean Value Theorem Motivation: A ball is thrown into the air from the ground. At some point the velocity is Rolle s Theorem Let f be a function that satisfies the following properties: 1. f is continuous on [a, b] 2. f is differentiable on (a, b) 3. f(a) = f(b) Then there is a number c in (a, b) such that f (c) = 0 Graphically: 18
Example: Show that f(x) = x 3 + 3x 2 has exactly one real root. 19
Mean Value Theorem Motivation: Suppose you drove 240 miles in 4 hours. What was you average velocity? At some point your velocity was Mean Value Theorem Let f be a function that satisfies the following conditions: 1. f is continuous on [a, b] 2. f is differentiable on (a, b) Then there is a number c in the interval (a, b) such that f(b) f(a) b a = f (c) Graphically 20
Example: Find the value of c implied by the Mean Value Theorem for f(x) = x 3 x 2 2x on [ 1, 1]. 21
Example: If f(1) = 2 and f (x) 3 for all 1 x 5, how small can f(5) possibly be? Example: Does there exist a function f such that f(3) = 5, f(0) = 1, and f(x) 1 for all x? Theorem: If f (x) = 0 for all x in an interval (a, b), then Corollary: If f (x) = g (x) for all x in an interval (a, b), then Homework Section 3.10 # 2, 3, 5, 11(a), 13(a), 15, 17, 23, 25, 27, 33, 34, 38. Section 4.1 # 1, 5, 17 31 (odd), 35, 41, 43, 49, 53, 59, 61. Section 4.2 # 5, 7, 11, 13, 15, 19, 25, 27. 22