The increasing decreasing theorem

Crtical point

If f(x) is a function and f(c) is a point in its domain, we call f(c) a critical point of f(x) if f'(c)=0 or f'(c) is undefined.

A relative maximum and minimum (local maximum and minimum)

AKA relative extrema

f(x) has a maximum at f(c) if f(c) is larger than any other point near the point of f(c).

f(x) has a minimum of f(c) if f(c) is smallest value near the point near f(c).

To graph y=f(x)

1. find f'(x) and f''(x) 
2. Make a sign chart for f'(x) and f''(x)
3. Plot special points
   • Local max and min
   • Inflection points
   • Intercepts

Three basic steps to go through with all curve sketching

1. Find f'(x) and f"(x).
2. Make a sign chart for f' & f".
3. Plot special points (max, min, inflection points, intercepts.

How to determine the max and min of an optimization problem on a closed interval: 

1. Find the critical points (the points at which f'=0 or is UD when the function is defined. 
2. Evaluate the function on the critical points and end points by checkin there value in the original function.
3. The largest of the critical points is the max; the smallest of the critical points is the min.

Area of a rectangle

Area=(side A)(side B)

Formula for volume of box:

V=(base)^2(width)

Formula for area of a triangle:

A of triangle = (1/2)(base)(height)

Volume of a cylinder:

Volume of a cylinder = π(radius)^2(height)

Surface of cylinder:

Surface of cylinder = 2πrh+2π(r)^2

Cost

Expressed in terms of amount of product in the x axis vs. the amount of money needed to produce the product in the y axis.

Fixed cost + variable cost = cost

The cost function, C(q), gives the total cost of producing a quantity of some good q.

Fixed cost

Variable cost

Revenue

A producers income composed of the amount of product multiplied by the amount of cost.

R(x) = yx

R(q) gives the total revenue received by a firm from selling a quantity q of some good. 

Because the price of a product decreases when the market has a large supply, the graph of a revenue function typically levels out over time. 

Profit

Profit = revenue - cost

Typically notated as ∏(q)

∏(q) = R(q) - C(q)

The amount of money earned by the company minus the cost of earning the money. 

The break even point

Average cost

(The total cost for production) /(the # of products produced)

Average cost = C(x)/x

Economy of scale

Marginal cost/revenue

The additional costs/revenue when additional products are added to an already existing service. 

Marginal costs (MC) = C'(q) ≈ C(q+1) - C(q)

Similarly,

Marginal revenue (MR) = R'(q) ≈ R(q+1) - R(q)

To distinguish marginal revenue (MR) and marginal costs (MC) from total costs and total revenue, total costs in notated C and total revenue is notated R.

Volume of sphere

(4/3)π(r)^3

Steps for solving optimization problems

1. Understand the problem. What value/s is it looking for?
2. Write an equation for the quantity being optimized (typically in the form of a formula such as area, volume, etc.)
3. If there is another variable, write an equation for for the variable with the help of the constraint.
4. Solve for one variable and substitute into the other equation.
5. Find an interval that makes sense for the problem. 
6. Take the derivative to find the global max and min (don't forget to check end points as well. 
7. Plug CPs into original function to compare values and determine global max and min.

L'Hopital's rule: 

How to determine the frequency of a recording based on time:

(The difference of velocity at the beginning and the end of an observation period)(time between measurements)>interval you wish to calculate.

Time interval between two consecutive measurements:

∆t = (b - a)/n

The distance, and total distance, an object traveled can be estimated:

Distance traveled: f(t)*∆t

When a function is monotonic (only increasing or decreasing), the difference between the over and underestimate is:

∆t = |f(b)-f(a)| (∆t)

The fundamental theorem of calculus

If ƒ is a continuous function from [a,b] and f (t) = F ' (t) then:

the integral of f(t) from [a,b] times the change in t = F(b) - F (a)

The average value of ƒ from a to b:

Average value of ƒ from [a,b] = [1/(b-a)][∫_a^bƒ(t)(∆t)]

Critical point

The extreme value theorem

Area of a circle

area of a circle = π(r)^2

Circumference = 2πr

Steps for solving optimization problems

1. Understand the problem
   • What quantity are you looking for? What variable is being optimized?
   • What values vary and how are they related?
   • Label things.
   • Give a name to whatever seems important.
2. Sketch a diagram of the problem.
   • Note how two or more variables are related to each other.
3. Write a formula for the problem. 
   • If there are two or more variables, write two or more equations for the variables (solve for one and substitute). 
   • The first equation will be the formula that needs to be optimized. The second equation will relate the formulas; that way you can solve and substitute them into the original formula.
   • The second equation usually comes from some kind of constraint.
4. Determine intervals of optimization problem.
5. Find CPs and plug CPs and endpoints into original formula to obtain values.

Steps for solving related rates problems

1. Identify variables.
2. What derivative is requested in the problem?
3. What formula is needed?
4. Take derivative implicitly (usually with respect to time).
5. Identify values in the function that correlate to information in the problem.

All antiderivatives that are possible for a specific function.

Ex: antiderivative of 2x is x^2+C