Lesson13IncDecFunctions Flashcards
What does a positive derivative tell you?
It tells you at which points the graph is increasing (positive slope of tangent line),
At what points on the graph of
f(x) = x³ is the graph increasing?
f’(x) = 3x²
It is always increasing.
Increasing means if you take any 2 x’s, the higher one always gives a higher y-value.
How do you find the various intervals of a function where it is increasing and where it is decreasing?
First, find the critical points.
Then, check points in-between
On which intervals is the following function increasing/decreasing?
f(x) = x³ - 3/2x²
f’(x) = 3x² - 3x :: 3x(x-1)
critical points are 1 and 0
at >1 it increases
at<0 in increases
at ½ in decreases
What are the relative extrema of
f(x) = ½x - sinx on (0,2π)
f’(x) = ½ - cosx = 0
½ = cosx 60° and 300° or π/3, 5π/3
at x < π/3 it’s negative
at π/2, it’s positive
at 2π it’s negative
π/3 = relative minimum
5π/3 = relative maximum
Find the critical numbers for
f(x) = (x² - 4)⅔
f’(x) = ⅔(x² - 4)-⅓ * 2x
= 4x / 3(x² - 4)⅓
If x is ±2, the denominator is 0, so derivative is undefined
(this is because they are sharp points)
If x = 0, The numerator is 0
Find the length and width of a rectangle of 80 perimeter with maximum area.
2x + 2y = 80
y = 40 - x
x*(40 - x) = 80
40x - x² = 80
A’ = 40 - 2x
2x = 40
x = 20 :: y = 20
What is Rolle’s Theorem?
f(x) is continuous on [a,b] and is differentiable on (a,b)
and f(a) = f(b)
:: there is a derivative of a point between a and b which is 0.
Does f(x) = x4 - 2x², [-2,2]
apply to Rolle’s Theorem? If so where are its points with derivative 0?
It appliies because f(2)=8 and f(-2)=8
f’(x) = 4x² - 4x
f’(x) = 4x(x - 1)(x + 1)
f’(x) = 0 when x = 0, ±1
What is the Mean Value Theorem?
It is a variation of Rolle’s Theorem.
f(x) is continuous on {a,b] and differentiable on (a,b)
There is a point between a and b that has the same slope as the line that connects a and b.
Apply the Mean Value theorem to
f(x) = 5 - 4/x, [1,4]
The slope between the endpoints is f(4)-f(1)/4-1 = 1
f’(x) = 4x-2
4x-2 = 1 when x=±2
;; the slope = 1 when x = 2
(-2 is outside the interval)
Give a practical example of the Mean Value Theorem.
You drive 240 miles in 3 hours and get a speeding ticket at the end. At some point, you have to have driven 80 mph.
What does a second derivative tell you?
concavity
How are tangent lines related to concavity?
They stay fully outside the graph
If they are above the curve, it is concave downward. If below the curve, the curve is concave upward.
What is an example of a function that has both a section with concave down and a section with concave up?
f(x) = x³
