Differentiation
Integration
Application
Theorems
Differentiation 2.0
100
The derivative of f(x)=5x+2
f'(x)=5
100
The integral of f(x)=cosx
f'(x)=sinx
100
In reference to the axis of revolution, what strips do you use when using disk method to find area?
Strips perpendicular to the axis of revolution
100
Define the Mean Value Theorem (MVT)
Let f(x) be a continuous function on the closed interval [a,b] and differentiable on the open interval (a,b).There exists some c in (alb) such that:
f'(c) = ( f(b) - f(a) )/(b-a)
100
The derivative of g(t)=(2x)(4x)
g'(t)=16x
200
The derivative of h(x)=8
h'(x)=0
200
Given: f'(x)dx=f(x)+c
What does c represent?
c is the constant of integration
200
What's the formula for shell method?
A(x)=2Pi (radius) (height)
200
Suppose it took 14 seconds for a thermometer to rise from -19oC to 100OC. Show that sometime between t=0 and t=14 (sec) the mercury is rising at the exact rate of 8.5oC/sec
Mean Value Theorem :( f(14 - f(0) ) / (14-0) = 8.5
200
The derivative of v(t)=10x/5
v'(t)=2
300
The derivative of g(x)=8x2-3x
g'(x)=16x-3
300
Why must we add c when solving an indefinite integral?
Recall that when you take the derivative of some function, all constants become 0, as the derivative of a constant is always 0. This is why, when solving an indefinite integral, it is important to add c, to represent the missing constant from the integration.
300
What are the three volume formulas and when do you use them?
1. Disk Method - perpendicular strips to the axis of revolution& Every strip must touch the axis of revolution
2. Washer Method - perpendicular strips to the axis of revolution & every strip must not touch the axis of revolution
3. Shell Method - strips parallel to the axis of revolution
300
The three requirements for Rolle"s Theorem
1. It is continuous on the closed interval [a,b]
2. It is differentiable on the open interval (a,b)
3. f(a) = f(b)
300
The derivative of x(t)=sinx2
x'(t)=2xcosx2
400
The derivative of f(t)=X1/2+3x3-2
f'(t)=(1/2)x-1/2+9x2
400
For 0 ≤ t≤ 12, a particle moves along the x-axis. The velocity of the particle at time t is given by v(t)= cos(π /6t). The particle is at position x =-2 at time t= 0. Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time t =0 to time t =6.
∫06 |v(t)|dt
400
Find the volume of the solid of revolution formed by rotating the region bounded by the x-axis and the graph of 
from x=0 to x=1, about the x-axis.
400
Prove Rolles Value Theorem for the function
f(x) = x2-5x+4 on the interval [1,4]
- Continuous
-Differentiable: f'(x) = 2x-5
- f(1) = 0
- f(4) = 0 , therefore f(1) = f(4)
400
The derivative of g(x)=3x2+3x-2/3x2-4x+9
g'(x)=1
500
The derivative of h(t)=2x4+4x3-6x2-10x+3
h'(t)=8x3+12x2-12x-10
500
In cases where u-substitution does not work when integrating, what can we do?
In cases where u-substitution does not work, we aim to find other methods to solve. Some of which are expansion, factoring, trigonometric substitutions, and integration by parts.
500
The region R is enclosed by the graphs f(x) = x4 - 2.3x3 + 4 and y=4. Find the volume of the solid generated when R is rotated about the horizontal line y=-2. Write,but do not solve the expression.
(pi) Integral [0,2.3] [(4 +2)2 -(f(x) + 2)2dx
500
Find the absolute maximum and minimum values of f(x) =x2/3 on the interval [-2,3]
f'(x) = (2/3)(x-1/3)
f'(x) is not equal to 0 but is undefined when x=0. Make sure to check the endpoints x =-2 and x=3.
f(0) = (0,0) = Absolute minimum
f(3) = 91/3 = (3,91/3) = Absolute Max
f(-2) = (-2)2/3 = 41/3
500
The derivative of f(t)=4x2-34x+91/2x2+33-22
f'(t)=2