Calculus

Continuity

1. The limit exists for that x (usually by showing one sided limits agree)
2. The function exists for that x (the function is defined)
3. The value of the function and the limit are equal

Intermediate Value Theorem

If f is continuous on [a,b] and k is any y-value between f(a) and f(b), then there exists at least one x-value c between a and b such that f(c)=k.

IVT
Guarantee?
Required?

Continuity on a closed interval
Every y-value b/w the y-values of the endpoints

When is a function NOT differentiable?

1. At a sharp corner or cusp
2. Vertical tangent line
3. Discontinuities

Differentiable Implies Continuity

If f is differentiable at x=c then f is continuous at x=c

4 basic derivative rules

1. The Constant Rule
2. The Power Rule
3. The Constant Multiple Rule
4. The Sum Rule

Constant Rule

If f(x)=k is a constant, then f'(x)=0

The Power Rule

If f(x)=x^n then f'(x)=nx^n-1

The Constant Multiple Rule

If y=k.f(x), k is constant, then y'= k.f'(x)

The Sum Rule

If y=f(x)+g(x) then y'= f'(x)+ g'(x)

Derivative of Sin

d/dx(sinx)=cosx

Derivative of Cos

d/dx(cosx)=-sinx

Position, Velocity, Acceleration

1.) Position measures where our object is at a given time, t
2.) Velocity measures of change in velocity over time (slope of position)
3.) Acceleration measure of change in velocity over time (slope of velocity)

The Second Derivative

You can take the derivative of the derivative to get the second derivative

The 2nd Derivative tells us:

1. The slope of the derivative graph
2. The concavity of the original function
3. How fast the derivative is changing
4. Acceleration of the position function

When to use the product rule?

when we have multiplication that cannot be resolved

When to use the quotient rule?

when we have a fraction that cannot be rewritten of reduced

The Other 4 Trig Functions

1. y= tanx --> y'= sec^2x
2. y= cotx --> y'=-csc^2x
3. y= secx --> y'= secxtanx
4. y= cscx --> y'= -cscxcotx

The Chain Rule

y= f(u) is differentiable function of u t u= g(x) is differentiable function of x, then y= f(g(x)) is a differentiable function of x and dy/dx (f((gx)))= f'(g(x))g'(x)

3 "Classic Chain Rule" Senarios

1.) Parentheses to a Power (General Power Rule)
2.) Trig function with a "funky angle"
3.) Trig function to a power

Implicit Differentiation

1. Differentiate both sides of the equation with respect to x
2. Collect all other terms involving dy/dx on the left side of the equation and move all terms to the right of the equation
3. Factor dy/dx out of the left side of the equation. (GCF)
4. Solve fo dy/dx

Related Rates

1.Determine what function/formula you need
2.Do Implicit Differentiation usually with respect to TIME
3.The problem will typically give you every number you need EXCEPT ONE. You may need to use the original equation to find missing measurements.
4.Solve for the one you want.

Strictly Monotonic

If its derivative never changes sign (it can be zero)
Pass the Horizontal Line Test
Have inverses that also function
Sometimes restrict the domain of a function to force it to have an inverse function

Reflective Property of Inverse Function

The graph of f contains the point (a,b) if and only if the graph of f-1 contains the point (b,a)

Definition of Inverse Function

A function g is the inverse function of the function f if f(g(x))=x of each x in the domain of g and g(f(x))=x for each x in the domain of f g is denoted by f-1 ("f inverse")

Natural Exponential Function

1.The domain of f(x)=e^x is (-inf.,inf.) and the range is (0,inf.)
2.The function f(x)=e^x is continuous, increasing and one to one it entire (passes the horizontal line test)
3.The graph of f(x)=e^x is concave upward on its entire domain
4.lim x--> -inf. e^x=0 and lim x--> inf. e^x= inf.

Definition of Extrema

f be defined on an interval I containing c.
1.f(c) is the minimum of f on I if f(c) < f(x) for all x in I
2.f(c) is the maximum of f on I if f(c) > f(x) for all x in I

Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval
Required: continuity
Guarantee: minimum/maximum

Critical Number

If f'(c)=0 or if f is not differentiable at c, the c is a critical number of f

The Candidates Test

To find extrema of a continous function f on a closed interval [a,b], use the following steps
1.Find the critical numbers of f in (a,b)
2.Evaluate f @ each critical # in (a,b)
3.Evaluate f @ each endpoint of [a,b]
4.The least of these values is the minimum. The greatest is the max.

Mean Value Theorem

Requires: Continuity on a closed interval
Differentiability on open interval
Guarantee: The slope of the tangent line will equal the secant line @ least once on the interval

Rolle’s Theorem

\-If a function (x) is continuous on \[a,b\], differentiable on (a,b) and is such that f(a)-f(b), the f’(x)=0 for some x=c on (a,b)

\-Requires continuity on \[a,b\], Differentiability on (a,b)

\-Guarantees: Horizontal tangent line on (a,b)

The First Derivative Test

\-Use to LOCAL EXTREMA


1. Find y’, find all x such that y’=0 or is undefined (critical values)
2. Divide the number line at the critical values
3. Test y’ for convenient values in each interval

y’>0 graph is increasing on the interval

y’

The Second Derivative

\-The concavity of the original function

\- The slope of the derivative (where y’ is increasing or decreasing)

\-The rate of change of the derivative

\- Acceleration in a PVA problem

Application the 2nd derivative test can be used to determine if the point is a local max of min

f”> 0 local min

f”< 0 local max

f”= 0 TEST FAILS

Inflection Points

Occurs when the 2nd derivative changes signs, usually (but not necessarily) passing through zero

Interpreting the 2nd derivative as concavity is common

Concave Up: tangent line is an underestimate

secant line is an overestimate

Concave Down: tangent line is an overestimate

secant line is an underestimate

As Acceleration

\-Acceleration is the rate of change of velocity

\-An object is SPEEDING UP when v(t) and a(t) have the same sign (they’re working together)

\- An object is SLOWING DOWN when v(t) and a(t) have opposite signs (they’re working against each other)

The Definite Integral

b

S F’(x) dx

a

Definite Integral of a Function

\-Gives the area between the function curve and the x-axis

\-Area below counts as negative

\-Integrating right to left reverses the signs

Definite Integral of a Derivative/Rat

\-The integral represents ACCUMULATES CHANGE in the original functions (not the rate) between the limits

\-New = Old + Change

b

F(b)= F(a) + S F’(x) dx

a

Accumulation

\-Given a starting amount

\-Given a rate increase AND a rate of decrease. Different forms (a table and an equation)- and have different start times

\-The net rate of change is INCREASE minus the DECREASE

\-Probably have to integrate both functions

\-Beware of piecewise functions

Riemann Sums

\- use rectangles to estimate a definite integral over an interval

\-rectangles are anchored from one of 3 points to each sub-interval

Left Endpoint (right most y point not used)

Right Endpoint (left most y point not used)

Midpoint (only uses even-numbered y’s)

Trapezoidal Sum

A=1/2b(h1+h2)

Over/Underestimates

Left RS: Increasing -Overestimate

Decreasing- Underestimate

Right RS: Increasing -Underestimate

Decreasing- Overestimate

Midpoint RS: Concave Up- Underestimate

Concave Down- Overestimate

Trapezoidal Sum: Concave Up- Overestimate

Concave Down- Underestimate

Net Displacement

b Final Position - Original Position

S v(t) dt

a

Distance Traveled

b

S l v(t) l dt

a

Direction Changes and Directions

\-v postitive-- moving up or right or fowards

\-v negative -- moving down, left, or backwards

\-Direction change-- v changes signs

Anti-derivatives

\-is the opposite of the derivative

\-F is an anti-derivative of f on an interval I if F’(x)= f(x) for all x in I

Integrating Fractions

Fractions (with variable denominators) must fit (or be made to fit) 1 of the derivative rules that is naturally a fraction. These include the derivative rules for ln(x) and the inverse trig functions.

Integration of Transcendental Functions

Transcendental functions whose values cannot be computed by hand except by using infinite series

Polynomial Long Division

\-Use when we have a fraction with a < degree on top. The division will break the fraction up

Completing the Square

\-Constant numerator and quadratic, we can sometimes integrate by completing the square

\- When it works, it turns into a variation on one of the inverse trig derivative form, allowing us to integrate into an inverse trig function

The Fundamental Theorem of Calculus

1\.The Fundamental Theorem of Cal. - Part 1 states:

x u(x)

d/dx(S f(t)dt)=f(x) and more generally d/dx(S f(t)dt)= (f(u(x))

a a

(u’(x), due to the chain rule (The integral is a function)

b

2\.  Part 2 states: S F(x)dx= F(b)-F(a), where F(x) is any anti-

a

derivative of f(x)

3\. Since F(x) is an anti-derivative of f(x), them we can also say that f(x)=F’(x), and therefore the FTC part 2 can rewritten as S F’(x)dx= F(b)-F(a)

4\. Finally, we can solve the rewritten equation above for

b

F(b): F(b)= F(a)+ S F’(x)dx. This new equation is of particular

a

interest if we are interested in the value of a function F(b), and we know the rate of change of a function F’(x), and some value of the function F(a).

S kf(u)du

k S f(u)du

S \[f(u)+/- g(u)\] du

S f(u)du +/- S g(u)du

S du

u + c

S u^n du

(u^n+1/ n+1) +c

n cannot equal -1

S du/u

ln lul +c

S e^u du

e^u + c

S a^u du

(1 / lna) a^u + c

S sinudu

\-cosu + c

S cosudu

sinu + c

S tanudu

\-ln lcosul + c

S cotudu

ln lsinul + c

S secudu

ln lsecu + tanul +c

S cscudu

\-ln lcscu+cotul + c

S sec^2udu

tanu + c

S csc^2udu

\-cotu + c

S secutanudu

secu + c

S cscucotudu

\-cscu + c

S du/ √a^2 - u^2

arcsin (u/a)+ c

S du/ a^2+ u^2

(1/a) arctan (u/a) +c

S du/ u √u^2- a^2

(1/a) arcsec ( lul/a ) + c

d/dx \[cu\]

cu’

d/dx \[u +/- v\]

u’ +/- v’

d/dx \[uv\]

uv’+ vu’

d/dx \[u/v\]

vu’-uv’/v^2

d/dx \[c\]

0

d/dx \[u^n\]

(nu^n-1)(u’)

d/dx \[x\]

1

d/dx \[lul\]

(u / lul )(u’)

u cannot equal 0

d/dx \[lnu\]

u’/u

d/dx \[e^x\]

(e^u)(u’)

d/dx \[logau\]

u’/(lna)u

d/dx \[a^u\]

(lna) (a^u)(u’)

d/dx \[sinu\]

(cosu)(u’)

d/dx \[cosu\]

\-(sinu)(u’)

d/dx \[tanu\]

(sec^2u)u’

d/dx \[cotu\]

\-(csc^2u)u’

d/dx \[secu\]

(secutanu)u’

d/dx \[cscu\]

\-(cscucotu)u’

d/dx \[arcsinu\]

u’/√1-u^2

d/dx \[arccosu\]

\-u^2/√1-^2

d/dx \[arctanu\]

u’/1+u^2

d/dx \[arccotu\]

\-u’/1+u^2

d/dx \[arcsecu\]

u’/lul√u^2-1

d/dx \[arccscu\]

\-u’/lul√u^2-1

The Area Problem

Slope Fields and Differential Equaions

Every Integrate Function has an INFINITE number of anti-derivatives, for each possible value of C.

The process of finding +c is called SOLVING A DIFFERENTIAL EQUATION. We need one point on the anti-derivatives graph.

We have the slope. We want the original function

There are entire courses devoted to differential equations