Unformatted text preview: 3.3 Increasing & Decreasing Functions and the 1st Derivative Test Calculus Home Page Problems for 3.3 Title: intro (1 of 10) Find: Intervals where function is increasing or decreasing.
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Problems for 3.3 Title: intuitive approach (2 of 10) Definitions:
A function f is increasing on an interval if, for any two numbers x 1 and x2 in the interval, x1 < x2 implies that f(x 1) < f(x2). A function f is decreasing on an interval if, for any two numbers x 1 and x2 in the interval, x1 < x2 implies that f(x 1) > f(x2). How can we tell if a function is increasing or decreasing, if we do not see it's graph? Is there a way to test for increasing or decreasing?
Hint: Consider the slope of the tangent line.
Return to graphs. Calculus Home Page Problems for 3.3 Title: definitions (3 of 10) Conclude: Yes. We can tell if a function is increasing or decreasing, if we consider the slope of the tangent line. In particular we need to look at the sign of the slope. Is it positive or negative? How can we examine the sign of slope of the tangent line? Calculus Home Page Problems for 3.3 Title: How do we tell? (4 of 10) Test for Increasing & Decreasing Functions
Let f be a function that is continuous on [a,b], add differentiable on (a,b), then: 1. If f '(x) > 0 for all x on (a,b), then f is increasing on [a,b] 2. If f '(x) < 0 for all x on (a,b), then f is decreasing on [a,b] 3. If f '(x) = 0 for all x on (a,b), then f is constant on [a,b] Calculus Home Page Problems for 3.3 Title: Test for Incr or Decr (5 of 10) p. 186 # 4 Find the intervals where y is increasing and intervals where y is decreasing. y = (x 1) 2 Calculus Home Page Problems for 3.3 Title: example 1 (6 of 10) p. 186 # 6 Find the intervals where y is increasing and intervals where y is decreasing. y = x4 2x2 Calculus Home Page Problems for 3.3 Title: example 2 (7 of 10) Wow! We can use this approach to determine max and mins! The First Derivative Test for Relative Extrema
Let c be a Critical Number of the function f that is continuous on the open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as: 1. a relative min, if f ' (x) changes from negative to positive at c \ / . 2. a relative maximum, if f ' (x) changes from positive to negative at c / \ 3. neither a max nor a min if f ' (x) is positive on both sides of c / / or negative on both sides of c \ \ Calculus Home Page Problems for 3.3 Title: 1st Deriv Test, Extrema (8 of 10) 3.3, # 18. G: f(x) = x2 + 8x + 10 F: a) CNs, b) inter inc, decr, c) rel.extrema Calculus Home Page Problems for 3.3 Title: example 3 (9 of 10) Calculus Home Page Problems for 3.3 Title: Oct 247:38 PM (10 of 10) ...
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