Math 121 Test 3

y=(e^2x-2)^2, (0,1)

(Chain rule) 1. Bring 2 on outside 2. rewrite the equation 3. then write derivative after that plug in 0 for x - Anything to the 0 power is 1 Solve take your answer and add an X add Y value

y=(lnx^2)^2

(chain rule) 1. bring exponent outside 2. write parenthesis 3. write derivative of what is inside parenthesis 4. remember when taking the derivative of (ln) take the derivative then write 1 over what was inside

Find the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 60 meters Length: Width:

*ALWAYS 1. Divide the perimeter by 4

Campground Question: 125,000 x= y=

*ALWAYS 500,250

Campground Question: 180,000 x= y=

*ALWAYS 600,300

Consider the following. P = −0.1s3 + 6s2 + 400 Find the amount s of advertising (in thousands of dollars) that maximizes the profit P (in thousands of dollars). s= (s,P)=

*ALWAYS: 40 20,2000

Find the number of units x that produces the minimum average cost per unit C in the given equation. C = 0.001x3 + 7x + 2

1. Divide equation by x 2. Move the x out of the denominator 3. Take derivative 4. Move the x back to the bottom 5. Solve for x

Find the price that will maximize profit for the demand and cost functions, where p is the price, x is the number of units, and C is the cost. Demand Function: p = 56 − 0.1(radical x) Cost Function: C=26x+500

MEMORIZE: 3x^1/2 20 1. Subtract the first term in each function 2. Once you get the answer to step 1 set it equal to the equation at the top 3. solve for x 4. Plug x into the demand function to get answer

Find the derivative of the function. f(x) = ex2 + 4

Take derivative Rewrite the original equation to the right of the derivative and that is your answer

S(e^(2x-5))dx

Top: everything Bottom: first number (2)

S(5x^3+20x)^3(3x^2+4)dx

Top: take integral of the first part, only add one to outside exponent Bottom: multiply first number (5) by 4 always

S(x^3/(8-x^4))dx

Top: write bottom of fraction in absolute value Bottom: write the exponent from the top but negative Add C

(3 - shahidi)The graph shows the profit P (in thousands of dollars) of a company in terms of its advertising cost x (in thousands of dollars). (a) Estimate the interval on which the profit is increasing. (Enter your answer using interval notation.) (b) Estimate the interval on which the profit is decreasing. (Enter your answer using interval notation.) (c) Estimate the amount of money the company should spend (in dollars) on advertising in order to yield a maximum profit. (d) Estimate the point of diminishing returns.

[0,40), (40,61), $40,000, (20,2000)

C=130+35x-160ln(x), x>=1

a. divide everything by x b. (first number/last number)+1 plug into e^(x) plug that back into what you get for part a

x^2y-e^y-4=0

always no matter what the numbers are (-2xy)/(x^2-e^y)

y=(lnx)/(x^6)

quotient rule - (BT'-TB')/B^2 derivative of lax =1/x

f(x)=ln(7x)

take derivative inside parentheses answer - write derivate - write a fraction with 1 over the # in parentheses

S(4x^4-9x^2+7)dx

take integral of equation + C

S(x^2-3)^2(2x)dx

take integral of first set

dC/dx=1/30x + 10, $7000

take integral of the first term second term is second number with x third term is dollar amount

S(8/(8x-7))dx

take natural log write bottom of fraction in absolute value add C