(A.3-A.4) Exponents, Radicals and Factoring Polynomials
(1.1 - 1.3) Distance Formula, Graphs of Equations, Equations of Lines
(1.4 - 1.6) Functions, Limits and Continuity
(2.1 - 2.6) Derivatives, Rates of Change: Velocity and Marginals
(2.7 - 2.8) Implicit Differentiation and Related Rates
100
Simplify [10(x+y)^3]/[4(x+y)^(-2)]
[5(x+y)^5]/2
100
Find the distance and midpoint of (3, -2) and (-3, 1)
d = 3radical(5)
midpoint: (0, -1/2)
100
Find the limit as x -> -1 of (x + 3)/(6x + 1)
-2/5
100
Find the derivative of: (3x^2 + 7)(x^2 - 2x)
12x^3 - 18x^2 + 14x - 14
100
Use implicit differentiation to find dy/dx of
y^2 + x^2 - 6y - 2x - 5 = 0
(1 - x)/(y - 3)
200
Simplify the cuberoot(54x^5)
3x[cuberoot(2x^2)]
200
What is the equation of the line passing through (-6, 5) and (5, 6)
y = (1/11)x + (61/11)
200
Find the limit as x -> 2 of (x + 1)/(x - 2)
Limit does not exist because unbounded. Approaches -infinity and infinity
200
Differentiate radical(5x^9 + 9x). You can keep negative exponents
(1/2)(5x^9 + 9x)^(-1/2)(45x^8 + 9)
200
Use implicit differentiation to find dy/dx of
x^2 + 3xy + y^3 = 10
- (2x + 3y)/[3(x + y^2)]
300
Simplify by factoring: 3x(x + 1)^(3/2) - 6(x + 1)^(1/2)
3(x + 1)^(1/2)(x - 1)(x + 2)
300
Given f(x) = x^2 + 1 and g(x) = x - 9, evaluate f[g(3)]
37
300
Describe the interval where the function is continuous:
f(x) = 1/(x + 4)^2.
Where is it discontinuous?
Is the discontinuity removable?
continuous at: (-infinity, -4) U (-4, infinity)
discontinuous at x = -4 and it is not removable.
300
A rock is dropped from a tower on the Brooklyn Bridge, 276 feet above the east River. Let t represent the time in seconds.
Find the velocity function for the rock
Find the average velocity over the interval [0,2]
Find the instantaneous velocity at t = 3
position function: s(t) = -16t^2 + 276
velocity function: s'(t) = -32t
average velocity: -32 ft/sec
instantaneous velocity: -96 ft/sec
300
Use implicit differentiation to find the equation of the tangent line of:
y^2 = x - y at the point (2, 1)
dy/dx = 1/(2y + 1)
at (2, 1) slope is 1/3 by plugging in y = 1
equation of tangent line: y = (1/3)x + (1/3)
400
Use synthetic division to factor:
x^3 - 2x^2 - x + 2 = (x - 2) ( )
(x^2 - 4x - 2)
400
A company had sales of $1,330,000 in 2007 and $1,800,000 in 2011. The company's sales can be modeled by a linear equation. Predict the sales in 2015. Let x = 7 for 2007
y = 117,500x + 507,500
2015, let x = 15: $2,270,000
400
The demand and supply equations for a product are
p = 65 - 2.1x and p = 43 + 1.9x respectively. Find the equilibrium point for this market
x = 5.5 and p = 5500
(5.5, 5500)
400
Find the marginal cost for producing x units of
C = 475 + 5.25x^(2/3)
3.5/[cuberoot(x)]
400
The radius of the circle is increasing at a rate of 2 inches per minute. Find the rate of change of the area at r = 10 inches. Round to 1 decimal place.
List:
1) Equation
2) Given rate
3) Find
Equation: A = Pi(r)^2
Given rate: dr/dt = 2 in/min
Find: dA/dt when r = 10 inches
dA/dt = 2Pi(r)(dr/dt)
= 125.7 in^2/min
500
Find the domain (interval notation) of: radical(5 - 2x)
(-infinity, 5/2]
500
Find the general form of the equation of the circle with diameter at the endpoints (0, -4) and (6, 4)
x^2 - 6x + y^2 = 16
500
Find the limit as x -> -3 of (x^2 + 2x - 3)/(x^2 + 4x + 3)
2
500
Find the derivative of
[(3x+2)^2]/[(x^2 + 1)^2]
- [2(3x+1)(3x^2+2x-3)]/(x^2+1)^3
500
The demand function is p = 211 - 0.002x and the cost function is C = 30x + 1,500,000.
what is the profit function?
Find the marginal profit when 80,000 units are produced
Profit function: P = R - C; R = xp
P = 181x - 0.002x^2 - 1,500,000
Marginal Profit: P'(x) = 181 - 0.004x
P'(80,000) = -$139/unit