Limits
Derivatives
areas between curves
Applications of Derivatives
Theorems
100
lim x->0(cos(x)/x)
does not exist
100
d/dx e^x
e^x
100
Set up the definite integral that gives the area of the region.
y=9(x^3-x)
y=0
*0 on top, -1 on bottom* 18(x^3-x)
100
What is the derivative of velocity?
Acceleration
100
Definition of Continuity
Definition of Continuity
1. lim x→c f(x) exists.
2. f(c) exists.
3. lim x→c f(x) = f(c)
200
Lim x->3 ((5x^2-8x-13)/(x^2-5)
2
200
d/dx log2(x)
1/(xln(2))
200
area sin (x), -sin(x), [0,2pi]
8
200
If a function has a critical point of f′(x) = 0 and if f has a local minimum here, what is the second derivative?
positive
200
Mean Value Theorem
f'(c) = (f(b) - f(a))/ (b - a)
300
Lim x->2((x^2-4)/(x-2))
4
300
d/dx f(g(x))
f’(g(x))g’(x)
300
area between f(x)=x,g(x)=x^2 on [0,2]
1
300
The rate at which the number of individuals are infected with the coronavirus, from the start of the pandemic, is given by f(t)=0.5e^t+t^2 in hundreds. How much greater is the rate of increase at which people are infected on the 6th day than the 2nd?
28 times higher
300
Second Fundamental Theorem of Calculus
If f is continuous on an open interval containing a, then for every x in the interval the derivative of the the integral of f(x) dx on said interval is equal to f(x)
400
Lim x->inf(2x^4-x^2-8x)
infitite
400
d/dx x^n
nx^(n-1)
400
y=4x^2ln(x)
y=16ln(x)
1.8982
400
An atom's position is given by x(t)=t^3/3 -4t^2+12t. When (time intervals) does the atom have a positive velocity?
[0,2)U(6, infinity)
400
When is L’Hospital’s Rule used?
when a limit problem involves a fraction of two functions and it’s an indeterminate form.
500
Lim x->inf (cosx/x)
diverges
500
d/dx cos^-1(x)
-1/sqrt(1-x^2)
500
y=3x, y=3/5x, y=28-x^2
8958/125
500
A jet is flying at a distance of d= f(t), which is a function of time(t), as given by d= f(t)=8t^2-4t+23. What is the acceleration of the jet?
16
500
What theorems are used to find this question?
Fundamental Theorem of Calculus
Chain Rule
Techniques of antidifferentiation (switching the order of integration)