Post #29

Substitution:

1. Label the parts of your integral. (x^2)(2x) u= (x^2) du=(2x)

2. Integrate (u du). (1/2)(u^2) +c

3. Plug u back in. (1/2)((x^2)^2) +c

Your final answer should be (1/2)(x^4) +c or (x^4)/2 +c


For substitution to work, one must recognize derivative properties that applies the product rule, quotient rule, or chain rule. Say if a problem asks you to find the integral of (x^3)(x). You would have to bring out a 1/3 then multiply your x by 3, making it look like (1/3) integrate (x^3)(3x). Then you just follow yours steps of substitution, but make sure to leave the 1/3 out until the last step then you distribute it in.

Table Issues

Using the Table, estimate f'(2.1).

f(2.0)=1.39
f(2.2)=1.73
f(2.4)=2.10
f(2.6)=2.48
f(2.8)=2.88
f(3.0)=3.30

This problem is really easy if you remember what a derivative is, which is a slope. All you need to do for this problem is find the slope of two points nearest to 2.1. After picking the two points, 2.0 and 2.2, you must use the slope formula, which is f(b)-f(a) over b-a or (f(b)-f(a))/(b-a). Using the chart, you plug in f(b)=1.73 and f(a)=1.39, while b=2.2 and a=2.0.

(f(b)-f(a))/(b-a)
(f(2.2)-f(2.0)/(2.2-2.0)
(1.73 - 1.39)/(.2)
(.34)/(.2)

slope= 1.70


Steps in order to optimize anything:

1. Identify primary and secondary equations. Primary deals with the variable that is being maximized or minimized. The secondary equation is usually the other equation that ties in all the information given in the problem.

2. Solve the secondary equation for one variable and then plug that variable back into the primary. If the primary equation only have one variable you can skip this step.

3. Take the derivative of the primary equation after plugging in the variable, set it equal to zero, and then solve for the variable.

4. Plug that variable back into the secondary equation in order to solve for the last missing variable. Check endpoint if necessary to find the maximum or minimum answers.