Extrema
Optimization
Implicit Differentiation
Derivatives
Elasticity
100
List the steps of the first derivative test
1. Find derivative 2. Set equal to 0 or DNE (to find CV) 3. Test points to the left and right of all CV (to determine increasing or decreasing) 4. Input extrema into orignal function 5. Write a sentence
100
What is optimization?
Maximizing or minimizing an equation
100
Differentiate implicitly to find dy/dx and find the slope at the given point 3x^3 - y^2 = 8 (2,4)
dy/dx= 9x^2/2y m= 4.5
100
f(x) = 3x^2
f'(x) = 6x
100
When looking to maximize profit given an elasticity formula, what value must we set the formula to?
Elasticity = 1
200
The second derivative test finds what potential extrema?
Point of inflection
200
Graph the function f(x) = x^3 - 3x + 2
Relative min at (1,0) Relative max at (-1,4) Point of inflection (0,2)
200
3(x^3)(y^2)
dy/dx = -3y/2x m=4.5
200
d(x) = 3(e^x) + e
d'(x) = (e^x)
200
When demand is inelastic, what effect will a small increase in price have on revenue?
The revenue will increase
300
Fill in the table f'(x)______ then f(x)______ f'(x)______ then f(x)______ f''(x)______ then f(x)______ f''(x)______ then f(x)______
f'(x) + then f(x) increasing f'(x) - then f(x) decreasing f''(x) + then f(x) concave up f''(x) -then f(x) concave down
300
A carpenter is building a rectangular shed with a fixed perimeter of 42 ft. What are the dimensions of the largest shed he can build? What is the area?
max area = 110.25 ft x=10.5 y=10.5
300
x(y^3) = 24
dy/dx = -y/3x
300
y = (x^2)(e^3x)
y' = 3(x^2)(e^3x) + 2x(e^3x)
300
Given q = D(x) = (200 - (x^3))^(1/2) find the elasticity when x = 3
Elasticity = 81/346
400
Determine with calculus the horizontal asymptote of f(x)= ((x^2)-4)/((4x^2)-5x-6)
HA when y = 1/4
400
Riverside Appliances is marketing a new refrigerator. It determines that in order to sell x refrigerators, the price per refrigerator is p = 280 - 0.4x and the cost of producing x refrigerators is C(x) = 5000 + 0.6x^2 Find maximum profit.
Max profit = $14,600 x= 140 refrigerators price = $224/fridge
400
Suppose that the price p, in dollars, and number of sales, x, of a mechanical pencil are related by 5p + 4x + 2px = 60 If both p and x are functions of time, measured in days, find the rate at which x is changing when x=3, p=5, and dp/dt = 1.5
dx/dt = -1.18 sales/day
400
y = 3lnx
y' = 3/x
500
Use calculus to determine the VA of f(x)= ((x^2)-4)/((4x^2)-5x-6)
VA at x = -3/4
500
Find the maximum volume of an open box from a piece of cardboard with the dimensions of 8 in. by 8 in.
Max volume = 37.9259 x=4/3
500
g(x) = ln ((x^2) - 5)
g'(x) = 2x/((x^2)-5)
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