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We use basic algebra analysis, to help solve limits graphically and algebraically.

Limits graphically help us visualize and check Limits solved Algebraically.

We use solving limits, algebraically and graphically, to find rates

We use algebraic techniques to

solve for the indeterminate form,

which helps solve limits

We use our skills at solving limits

to deal with limits that involve infinity

We use solving limits and limits involving infinity in graphing rational functions

We use limits to help us solve for derivatives

We use the definition of derivatives to solve derivatives

Physics applications can help solve rates involving derivatives

Power terms can help solve derivatives and rates

Product rule, quotient rule, and chain rule all help solve for derivatives and rates

Differentiation is more work with derivatives

Implicit differentiation can help solve for rates

Derivatives are used to solve for evps and concavity

Optimization rules help solve for business, and derivatives help solve for optimization

Integrals undo derivatives

Integration rules help solve for Reimann sums

Integration helps graph integrals, and graphing them can help visualize and check a Reimann sum

Average value help solve for integrals

The FTC solves for definite integrals

U-substitution helps solve for integrals

Basic trig helps us solve any equation with trig functions

Trig derivatives help solve parametric equations, and physics applications

L'hopitals rule uses derivatives to sol limit

We use derivatives to find EVP's and PI's

Trig integrals undo trig derivatives

Expo and log uses integrals and derivatives

Expanding logs helps when solving for derivatives and integrals

Integrals undo derivatives

Slope fields help visualize derivatives and integrals

Integrals and derivatives help solve for growth and decay problems

The snow problem is a special growth and decay problem

Limits: Indeterminate form

The indeterminate form is when a lim=0/0

To avoid this you must evaluate the lim differently. Some techniques are:

Factor special cubic functions: example:

Factor trinomials:

Synthetic division

Rationalize the numerator

simplify complex fraction

Fast add/ Keep it change it flip it:

Limits: Rates

The slope of the tangent at a point is the rate the y-values change with

respect to its x-values,The problem with the slope formula is two points are required. Tangents on a graph only have one known point. A second point on the tangent is not known. The slope of the tangent is then estimated by picking a second

point that is on the graph.The only way for point Q to be a reasonable choice is if it is

as close to point P as possible.

Limits involving infinity

A secant is a line that connects two points on a function.

The line PQ is a secant. The slope of that secant line PQ can be used to

estimate of the slope of the tangent at point P.

The slopes of the tangents and secants are used to estimate the rates of change of a function around a certain x-value.

A zero is the x-intercept

A VA, vertical asymptote, are x-values used in a function will not have a Y value

HA is horizontal asymptotes, these are the end behavior of the function

Average speed is the slop of the secant and uses a time interval.

Instantaneous rate is the slope of the Tangent, and uses and exact point in time.

Limits can = infinity:

The graph is moving in same

direction on both sides of the VA.

This is an INFINITE LIMIT

at x = 5.

To calculate an estimated value of instantaneous rate will need to use the slope formula where the 2nd time is a guess

To calculate the Exact value you will use a limit where the time interval approaches zero.

Sketch a graph of the function:

You can use a table of values to solve for a instantaneous rate, by estimated the exact time. You take 2 points surrounding the time that you want to find the rate of, and plug them into the average rate formula.

Examples

A watermelon, initially at rest, is dropped from the top of a tall building. The watermelon falls a distance in feet

modeled by the equation ​​(​​) = ​​​​​​^2 during the first ​​ seconds of fall.

1) Find the average speed of the watermelon during the first 4 seconds of fall.

2) Find the velocity of the watermelon at ​​ = 4 seconds. Confirm your answer algebraically

1.

2.

Derivatives Differentiable and Continuous

Definition:

A function is differentiable at x=c if the tangent slope is a numerical value.

Theorems:

If f(x) is differentiable at x=a, then f(x) is CONTINUOUS at x=a.

If f(x) is NOT continuous at x=a, then it is NOT differentiable at x=a.

Property:

If f(x) is continuous at x=a, then it is not guaranteed f(x) is differentiable at x=a.

Gradual change(differentiable):

(both are continuous)

Abrupt change (not differentiable):

Derivatives Product and Quotient Rules

Derivative rules-

Product rule:

Quotient rule:

Examples

At what domain points does the function appear to be:

a) Differentiable?

b) Continuous but not differentiable?

c) Neither continuous nor differentiable?

a) All points in [−​​, ​​], b) None, c) None

Limits Algebraically

Examples:

a) All points in [−​​, ​​], b) None, c) None

Notation:

To evaluate a limit algebraically, substitute x=c into the function f(x).

Limit properties:

Derivatives Physics Applications

Position of an object is s = f(t)

Velocity of an object is change in distance over the change in time: s'(t)=v(t)

Average velocity is average rate of change

Acceleration of an object is change in velocity over change in time: s''(t)=v'(t)=a(t)

Using a table to solve for limits:

Example: real world

Plug in the numbers to find what the lim is as x approaches -2

Examples

The number of gallons of water in a tank ​​ minutes after the tank has started to drain is

How fast is the water running out at the end of 10 min? What is the average rate at which the water flows out

during the first 10 min?

At the end of 10 minutes: 800 gallons/minute. Average over first 10 minutes: 10,000 gallons/minute.

L=0

L=5

L=1

L=3

L= -3/4

Limits Graphing Rational Functions

Unit 1: Limits and continuity

Derivatives Chain Rule

Example:

Derivatives Rules for Power Terms

original equation: f(x) = y derivative: f ’(x) = y’ = dy/dx

This notation indicates to differentiate f(x) with respect to x

Rules of differentiation-:

Constant:

Power terms:

Sum of terms:

Unit 2: Derivatives

HIGHER ORDER DERIVATIVES: Second derivative: f ''(x) or y’’ Third derivative: f '''(x) or y’’’

Examples

dy/dx= x^2+x+1

dy/dx=-2x

dy/dx=2

iii

Limits Graphically

Derivatives Other Rates

Rates are connected to the derivatives. From the physics work, the derivative is velocity and the second derivative is acceleration, where time was used in either derivative to get a rate. x can represent radius, height, number of units, pressure (exerted on a volume of gas, say). And the rates evaluated would have different kinds of units

Examples:

a) Write the volume ​​ of a cube as a function of the side lengh ​​.

b) Find the (instantaneous) rate of change of the volume ​​ with respect to a side ​​.

c) Evaluate the rate of change of ​​ of ​​ = 1 and ​​ = 75.

d) If ​​ is measured in inches and ​​ is measured in cubic inches, what unit would be appropriate for ​​​​/​​​​?

Limits Continuity and Discontinuity

=6

=3

Interval notation is used for Domain, x-values

f(3)= Undefined, hole

0

3

- Infinity

DNE

= Infinity

Derivatives Definition

All 3 must be true

Continuity only occurs if:

A derivative is an equation used to calculate the slope of a tangent at a point on f(x) where x = a.

Discontinuity Occurs at any x-value that causes the function to be undefined

Formula for definition of derivatives:

Write the equation of a tangent line:

1.Take the derivative of the equation

2. Use equation of a line (y=mx+b) to write the wquation

Removable discontinuity at x = c: There is not a defined point at x=c on the path of the graph, but there is a limit

Non-removable discontinuity at x=c: - There is not a limit at x = c.

Differentiable Means the slope of a tangent exists; the slope is a numerical value.

Example:

Comparing the graphs of f(x) and f ’(x): Where f(x) is increasing, the derivative is above the x-axis.

Where f(x) is decreasing, the derivative is below the x-axis. Where f(x) is changing directions, the derivative crosses the x-axis.

a. yes, it =0 b. yes it is 0 c. yes d. Yes

Example

4) b 5) a 6) d 7) c

Basic Algebra Analysis

y=x

Name: Linear

Equation: y=mx+b

Picture:

Name: Cubic

Equation:

Picture:

Name: Quadratic/parabola

Equation:

Picture:

Name: 4th degree power function

Equation:

Picture:

Name: rational function

No general equation

Picture

Name: Odd radical function

No general equation

Name: absolute value function

No general equation

Picture:

Name: exponential

No general equation

Picture:

Name: Greatest integer function

No general equation

Picture:

Growth and decay

Formula y=Ce^kt

Law of cooling:

Formula- T(f)=Ce^-kt +ts

When C=T(0)-T(s) (Surrounding temp)

Y: quantity at time t

C: Initial amount

e: Base e

K: Proportional constant

(rate/%)

Example:

A certain strain of bacteria that is growing on your kitchen counter doubles every 5 minutes. Assuming that you start with only one bacterium, how many bacteria could be present at the end of 96 minutes?

Slope fields

Unit 6: Expo and Log

Condensing and expanding logs

Expo: Integrals

Examples:

Examples:

Expo: Derivatives

Example:

A Cessna plane takes off from an airport at sea level and its altitude (in feet) at time t (in minutes) is given by

h = 2000 ln (t + 1).

Find the rate of climb at time t = 3 min.

At t = 3, we have v = 2000/4 = 500 feet/min.

Derivatives Optimization Geometry

Optimization problems ask you to solve for a measurement or quantity that will Maximize an amount such as volume, area, or profit. Or, minimize an amount such as cost, distance, or

perimeter.

Examples:

What is the smallest possible area for a right triangle whose hypotenuse is 5 cm long, and what are its

dimensions?

Smallest perimeter = 16 in., dimensions are 4 in. by 4 in

You are planning to make an open rectangular box from an 8- by 15-in. piece of cardboard by cutting congruent

squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you

can make this way, and what is its volume?

Derivatives Concavity

Derivatives Optimization Business

Like a frown

Concavity is the shape of a function: up or down

Like a cup

  • C(x) cost to make x number of items
  • c(x) average cost- cost per one item
  • p(x) selling price of one item: need to meet costs, suitable price with respect to customer demand
  • R(x) revenue-total amount of money made after selling x number of items

  • P(x) profit-leftover revenue money after paying all costs

Points of inflection (PI) are were the function changes shape, form up to down or down to up

Point B is a PI

Concave up

Example:

Example

Use the First Derivative Test to determine the local extreme values of the function, and identify any absolute

extrema.

Concave down

Local minimum at (1/2, -5/4)

-5/4 is an absolute minimum

Calculus

Points A and C are evps

Derivatives EVPs

EVP are turning points on the graph, gradual or abrupt

Unit 3: Differentiation

An abrupt change is sharp turn/cusp, has a vertical tangent

A gradual change is like a mountain top or a valley and has a horizontal tangent

EVPs are either maxima or minima: they can be relative or Absolute (no other value is bigger)

The x-coordinate of an EVP is called a critical number

Examples

Find the extreme values and where they occur.

Derivatives More Related Rates

Derivatives Implicit Differentiation

Implicit form is when an equation does not separate the independent variable (typically x)from the dependent variable (typically y)

Implicit differentiation is a method used to differentiate equations in implicit form.

Some related rates problems may involve triangles, so you must use Pythagorean theorem to solve them

First you must differentiate your equation and then use Pythagorean theorem to solve for the missing variable.

A ladder 20 feet long leans against a building. If the bottom of the ladder slides away from the building horizontally at a rate of 4 ft/sec, how fast is the ladder sliding down the house when the top of the ladder is 8 feet from the ground.

Interpretation of the notation dy/dx  derivative of variable y with respect to variable x

Interpretation of the notation dz/dt  derivative of variable z with respect to variable t

Interpretation of the notation dx/dx  derivative of variable x with respect to variable x

Examples

Since the problem is about a right triangle relating three sides we use can use z2=x2+y2

2z dz/dt=2x dx/dt+2y dy/dt

z dz/dt=x dx/dt+y dy/dt

(20)(0)=x(4)+(8)dy/dt

Derivatives Related Rates

The rate of change of something in calculus implies a derivative is used. All derivatives in this section are done implicitly with respect to time (dt).

“x-term is increasing at a rate of 2 ft/sec” translates to dx/dt = 2 ft/s

“rate the y-term is changing” translates to dy/dt = ?? ft/s

Examples

Trig Integrals of Inverse Trig

Steps to the triangle method for derivatives:

(1) Rewrite as a non-inverse

(2) Draw and label a right 

(3) Work derivative and get dy/dx.

(4) Use the triangle; dy/dx has to be in terms of x only.

Instead of using the triangle method there are six derivative rules for the inverse trig

functions. The formulas (for the derivative rules) are not difficult to use, however you will

then need to memorize them.

Basic Trig

Trig Integrals of the 6 Trig Derivatives

Quadrental angles – angles that land on an axis: 0, 90, 180, 270, 360, etc. (can be negatives and radians)

Reference angles: 30 60 and 90 (all other angles besides quadrental angles, have one of these reference angles.

In Q2 the special angles are 120, 135, 150. In Q3 they are 210, 225, 240. In Q4 they are 300, 315, 330.

chain rule involves the derivative of

the inside. So if something other than a simple x is one of the integral endpoints, then its derivative

needs to be included since it is eventually put inside f(t)

Angles can be measured in radians or degrees, most calculus problems will use radians

How to convert:

Radians to degrees Degrees to radians

Basic trig cont.

Examples:

Trig ratios

Graphs of trig functions

Trig properties:

Special right

Triangles:

How to solve trig equations

(1) Get trig ratio alone on one side of equal sign. Try to make sine, cosine, or tangent.

(2) Figure which quadrants the sign of the ratio can fit. Think about the signs of x and y.

(3) From the ratio, use the special triangle involved to determine the reference angle.

(4) Use the reference angle to get the value of the solution-angle.

Examples

Unit 5: Trig

Trig EVPs and PIs

Trig Derivative Rules

Rules:

Functions like cosine and sine have many gradual-change turning points, no abrupt changes, and no vertical asymptotes (VAs).

secant and cosecant, have gradual-change turning points and VAs.

Tangent and cotangent, on the hand, have no turning points, but have VAs.

Examples:

Examples

Find the critical numbers and extreme values for each indicated interval; indicate each as a max/min.

Trig L’Hopital’s Rule

Trig Parametric Equations

This rule is only for when you solve a lim and get the indeterminate form

Parametric equations are a pair of equations, x(t) and y(t) where x(t) is a equation modeling horizontal movement and y(t) is an equation modeling vertical movement.

There are 3 variables involved:x,y, and t, t is called the parameter.

When calculating rate with parametric equations, you get dy/dt and dx/dt, but you still must find dy/dx. To do this:

Examples:

Examples:

Trig Implicit Differentiation and Rates Application

Implicit differentiation works the derivative of each term with respect to each variable. Each derivative of each term multiplies the corresponding rate for that variable. It works the same as earlier, but this time involves trigonometric functions

Integration Introduction to Anti-derivatives

Integration Fundamental Theorem of Calculus

This is a method for calculating the exact value of a definite integral

The result of integrating is an Integral, also called an Anti-derivative .

Example:

Formula:

Notation:

A boy launches his toy rocket 15 feet away with his remote control. The toy

rocket gains altitude at a rate of 2.5 ft per sec. Find the rate at which the angle of

elevation is changing when the rocket has gained an altitude of 15 ft.

where f(x) is continuous on the interval [a, b].

-No “+C” even though FTC begins with setting up an indefinite integral. “+C” is only for when the answer is an equation; not for definite integrals where the answer is a number value.

There are definite integrals: C=a number, you solve the equation and your answer is the number, this is called the particular solution

indefinite integrals: F(x) + C, your answer is an equation

This equation is called the general solution

Examples:

Evaluate each integral using the Fundamental Theorem.

Examples:

Let x be the distance between the boy and the toy rocket, h be the altitude of the rocket and be the angle of elevation of the rocket.

Given: dh/dt = 2.5 ft/sec

(rate of change of the toy rocket’s altitude)

x = 15 ft

Unknown: d θ/dt = ?

tan = h/x

Tan =h/15 Now take the derivative:

Integral rules-

Constant:

Power term:

=1

Unit 4: Integration

Integration U- Substitution

Integration Definite Riemann Sum

Composite functions are made up of an expression on the inside and another on the outside. To differentiate composite functions need to use the chain rule derivative. To undo this you use substitution.

For a function over the closed interval [a, b] that does not produce a region where geometry area formulas can be used, it is possible to approximate its definite integral by using a method known as Riemann's Sum.

The inside of the integral will be replaced with variable u. By the time the substitutions are made for different parts, what is left is a new integral in terms of variable u

Left endpoint:

  • Starts at x=a
  • The LEFT endpoint of each rectangle is on f(x)
  • the rectangles are too short
  • Lower Sum

Right endpoint

  • Finishes at x=b
  • the rectangles are too tall
  • The RIGHT endpoint of each rectangle is on f(x)
  • Upper Sum

1. Set u = inside ,From that get the differential derivative du.

2. Use both u and du to substitute parts of the original integral.

3. Integrate, ∫ ( ) = F(u) This is an integral in terms of u; it is not F(x) +C.

4. Convert F(u) to be in terms of x  The answer is the integral F(x) + C .

Go back to step 1, use u = inside and replace all u's in F(u) with the original inside.

The more rectangles drawn, the more accurate the area is.

Area rectangle = bh, base = change inx and height = f(xi) (y-value at each rectangle corner) n = # rectangles

Example:

Use the x-coordinates calculated above to get the heights of each rectangle. Then use the area and height to calculate the areas. Then take the sum of all the areas.

Example

Calculate the approximate area under each curve by using the

Riemann Sums ​​​​, ​​​​, ​​​​ for each given ​​ increments. For each problem clearly show the value of Δ​​ and write the area

formula in terms of the function, Area = (value of Δ​​) (the equation for ​​(​​))

Trig uses derivatives and integrals

Integration Definite Graphs

Integration Average Value

Average value of a function is the average y-value for a given interval. Said another way, it is the average y-value from f(a) to f(b).

f(x) must be a continuous function for all x in the (closed) interval [a, b] .

One way to solve for definite integrals is working directly with the shape of the curve between the graph and the x-axis.

Examples

Net area is the value of the definite integral. It is the area of the region that is between the curve and the x-axis. The region above the x-axis is positive area, and the region is under the x-axis is negative area. The definite integral is the sum of positive and negative areas.

Total area is the area of the region between a curve and the x-axis such that all negative area is regarded as positive area. Will use the absolute value of any negative area. Total area is a positive value.

There is no rule for integrating absolute-value functions. Need to make equations for each segment of an absolute-value function

Integral properties:

Example:

Given the graph. Write a definite integral expression that represents the area of shaded region. Be sure to use the

proper notation. Then, evaluate each definite integral using geometry area formulas.

Calculus Mind Map

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