Derivatives
**What is the derivative of 4x5+8x?
20x4+8
**What is the integral of 27x2-16x3?
9x3-4x4
**Separate the variables of the differential equation dy/dx=(1+x)(2y)
dy/2y=(1+x)dx
**Which theorem is not very nice to be around?
The mean value theorem
**What are the requirements for a function to be continuous at a certain point?
It must be have a value at the point, it must have a limit at that point, and the value of the function at the point must equal the limit.
**What can you learn from the sign of the second derivative?
The concavity of the function.
**What is the integral of -1/(1+x2)?
tan-1(x)
**Write a differential equation that expresses the rate of change of a population, where the rate of change is proportional to the population itself.
dP/dt=kP
**What must a function be for the extreme value theorem to work?
Continuous
**What is f'(2) of the function f(x)=g(h(x)), when g(2)=0, g'(4)=6, h(2)=4, and h'(2)=2?
12
**Is x=1 a local min, max, or POI of the function y=6x3-9x2-3?
Local min
Evaluate ∫sin(x)+10csc2(x)dx
−cos(x)−10cot(x)+c
**Explain the difference between a general solution and a specific solution to a differential equation.
A general solution can represent a family of functions, and has a c in it. A specific solution is only one equation, and you need an initial condition to find it. Both of them satisfy the original differential equation.
The mean value theorem states that between two x values where a continuous and differentiable function has the same y value there must be a what?
A relative min or max.
Two people are at an elevator. At the same time one person starts to walk away from the elevator at a rate of 2 m/s and the other person starts going up in the elevator at a rate of 7 m/s. At what rate is the distance between the two people changing 15 seconds later?
7.2801 m/s
**What is the derivative of sec-1(x)?
1/([x]sqrt(x2-1))
What is the area of the solid generated by rotating the area between y=2x2 and y=x3 (between x=0 and x=2) about the x axis?
22.978
Solve this differential equation for the general solution dx/dy=2xy-2
y=(3x2+c)1/3
If f(x)=x+lnx, what is the value of c for which the instantaneous rate of change of f at x=c is equal to the average rate of change of f over the interval [1,4]?
2.164
If you have 45m2 of material to build a box with a square base and no top, which dimensions will give it the maximum volume? (you need to find a length and height)
l=3.8730, h=1.9365
Find y′ by implicit differentiation for 4x2y7−2x=x5+4y3
y′=(8xy7−5x4−2)/(12y2−28x2y6)
Evaluate ∫√x(x2−14x)dx
(2/7)x7/2-(1/2)x1/2+c
Solve the differential equation dy/dx=(xy3)/((1+x2)1/2) with the initial condition (0, -1)
y(x)=-1/(√(3−2√(1+x2)))
evaluate limx→−∞((x2)/(e1-x)
0
limy→0+(cos(2y))1/(y^2)
e-2