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MATH 2250: Increasing, Decreasing Functions and First Derivative Test, Lecture notes of Calculus

The concepts of increasing and decreasing functions, the increasing/decreasing test, and the first derivative test. It includes examples, exercises, and instructions on how to determine intervals where functions are increasing or decreasing, and how to find relative and absolute extrema.

Typology: Lecture notes

2021/2022

Uploaded on 09/27/2022

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MATH 2250 Section 4.3 - Increasing/Decreasing Functions and First Derivative Test
Definitions
A function fis an increasing function if
the y-values on the graph increase as you
go from left to right.
A function fis an decreasing function if
the y-values on the graph decrease as you
go from left to right.
Example. f(x) = exExample. f(x) = ex
Most functions switch back and forth from increasing to decreasing.
Example. The graph of fis below. Assume that it includes all of the relevant information
about f, and that the domain of fis (−∞,). Determine intervals on which fis increasing
and decreasing.
Increasing/Decreasing Test Suppose fis differentiable on an open interval (a, b).
If f0(x)>0 for all xin (a, b), then fis increasing on (a, b)
If f0(x)<0 for all xin (a, b), then fis decreasing on (a, b).
If f0(x) = 0 for all xin (a, b), then fis constant on (a, b).
pf3
pf4
pf5

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MATH 2250 Section 4.3 - Increasing/Decreasing Functions and First Derivative Test Definitions A function f is an increasing function if the y-values on the graph increase as you go from left to right.

A function f is an decreasing function if the y-values on the graph decrease as you go from left to right.

Example. f (x) = ex^ Example. f (x) = e−x

Most functions switch back and forth from increasing to decreasing.

Example. The graph of f is below. Assume that it includes all of the relevant information about f , and that the domain of f is (−∞, ∞). Determine intervals on which f is increasing and decreasing.

Increasing/Decreasing Test Suppose f is differentiable on an open interval (a, b).

  • If f ′(x) > 0 for all x in (a, b), then f is increasing on (a, b)
  • If f ′(x) < 0 for all x in (a, b), then f is decreasing on (a, b).
  • If f ′(x) = 0 for all x in (a, b), then f is constant on (a, b).

Key observation: If f ′^ changes sign at c, then either f ′(c) = 0 or f ′(c) dne, so c is a critical number of f.

  1. Let f (x) be a continuous function whose derivative is f ′(x) = x(x + 3)^2 (x − 5). Deter- mine intervals where f (x) is increasing and decreasing.
  2. Find intervals where f (x) = 2x^3 + 3x^2 − 12 x + 6 is increasing and decreasing.

MATH 2250 Section 4.3 - Increasing/Decreasing Functions and First Derivative Test

  1. Find intervals where f (x) =

ln(x) x^3

is increasing and decreasing.

  1. Determine relative and absolute extrema of f (x) = x^1 /^5 (x − 10).
  1. Consider functions f satisfying all of the following conditions:
    • f (x) is differentiable on the interval 0 < x < 8
    • The critical points of f (x) in the interval 0 < x < 8 are x = 2, 4 , and 6. (f (x) has no other critical points in this interval.)
    • The table below shows some values of f (x) and of its derivative f ′(x). x 1 3 5 7 f (x) 3 6 11 0 f ′(x) -1?? -

For each of the statements below, decide whether the statement is true for all functions f satisfying all of the conditions described above, for SOME of these functions f , or for NONE of these functions f.

(a) f (x) has a local minimum at x = 2 (b) f ′(3) > 0 (c) f (x) has a local maximum at x = 4 (d) There is exactly one value of a with 3 < a < 7 such that f (x) has a local maximum at x = a.

  1. A portion of the graph of y = g(x) is shown below.

On the axes below, sketch the graph of y = g′(x). Be sure you pay close attention to each of the following:

  • where g′^ is defined
  • the value of g′(x) near each of x = − 5 , − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , 5
  • the sign of g′
  • where g′^ is increasing/decreasing/constant