Chapter 4

Question 1

First Derivative Test

Answer 1

Take the derivative of f(x)
Find the zeros from f1(x), these zeros display critical points on the graph
Plug 0 values in for f1(x), if the values are negative, the graph is concave down on that section. If they are positive, it is concave up.

Question 2

Second Derivative Test

Answer 2

Take the second derivative of f(x), or the derivative of f1(x)
plug the zero values into the fII(x) function. if the value is negative, it represents a local maximum. If it is positive, it represents a local minimum. If it is 0, refer to fI(x) for its value

Question 3

Mean Value Theorem

Answer 3

If you are looking at a graph, with 2 points comprising an integral, there exists a point whose tangent slope is equal to the secant slope of the two points.

f1(c) = (f(point2) - f(point1)/point2 - point1

Find the derivative of f(x), set it equal to the slope from plugging values in and solve

Question 4

Newtons Method

Answer 4

Use Calculator:
y1 = f(x)
y2 = fI(x)
y3 = y1/y2
y4 = x - y3

There will be an equation and an x value given, y4 = your new x value, complete until you reach the desired steps

Question 5

Shortest Distance of Point on a graph from origin

Answer 5

Line = 3x + 4, D = sqrt(x-0^2)(y-0)^2
D = x^2 + y^2

1. Substitute y with slope ==> D = x^2 +(3x+4)^2
2. Take derivative of equation above: 2x + 2(3x+4)(3)
   20x-24 = -6/5
3. Plug value back into original equation

Question 6

Linear Approximation

Answer 6

L(x) = f(a) + fI(a)(x-a)

1. f(x) is gien as f(x) = sqrt(x+3)
2. Find Derivative of Equation = 1/2sqrt(x+3)
3. Linearize by plugging in a value = 1 + 1/4(x-1)
4. Plug desired values in for X

Question 7

L’Hospitale’s Rule

Answer 7

When given a limit as a fraction, when you plug in X and the top is either 0/0 or infinity/infinity, take the derivative of both the top and bottom and plug x back in. repeat until you don’t receive an indeterminate form

Question 8

Indeterminate forms of a function

Answer 8

0/0
(0)(infinity)
(0^0)
1^infinity
infinity^0

any kind of ininity as the numerator and denominator:
infinity/infinity
-infinity/infinity
infinity/-infinity
-infinity/-infinity