AP Calc Final

Integral: du/ u(sqrt(u^2-a^2)

(1/a)arcsec(abs(u)/a)) + C

Integral: du/ a^2+u^2

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

find the slope of the tangent line to the graph of the function at the given point.

1. use f(x) to find derivative 2. plug in x from given point to find the slope of the tangent line at that specific point

dx

"with respect to x"

d/dx(logaU)

(1/(lna)U)(u')

d/dx(a^u)

(lna)a^u(u')

d/dx(a^x)

(lna)a^x

Steps for Long Division of Polynomials

*Step 1:* Make sure the polynomial is written in descending order. If any terms are missing, use a zero to fill in the missing term (this will help with the spacing). In this case, the problem is ready as is. *Step 2:* Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol. *Step 3:* Multiply (or distribute) the answer obtained in the previous step by the polynomial in front of the division symbol. *Step 4:* Subtract and bring down the next term. *Step 5:* Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol. *Step 6:* Multiply (or distribute) the answer obtained in the previous step by the polynomial in front of the division symbol. *Step 7:* Subtract and bring down the next term. *Step 8:* Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol. *Step 9:* Multiply (or distribute) the answer obtained in the previous step by the polynomial in front of the division symbol. *Step 10:* Subtract and notice there are no more terms to bring down. *Step 11:* Write the final answer. The term remaining after the last subtract step is the remainder and must be written as a fraction in the final answer.

Integration

- the process by which an antiderivative is calculated

Differentials Cheat Sheet

-You will need to know other formulas for shapes -Just go slowly and rely on thought and algebra skills -Requires implicit differentiation to determine propagated error aka dx

sinxdx

-cosx+c

(cscx)^2dx

-cotx+c

d/dx [cotx]

-csc^2x

cscxcotxdx

-cscx

You can evaluate a definite integral in two ways:

1. use the limit definition, or 2. check to see whether the definite integral represents the area of a common geometric region

Absolute value functions in integration

-find where the function is equal to 0 because this is where the function changes -then split it into two with the area to the left being the function negated and the one to right is the regular function -finally add the answers together because of the additive interval property

∫f(x)dx

-is called the indefinite integral of f. -is the family of anti-derivatives, or the most general anti-derivative of f.

d/dx [cosx]

-sinx

d/dx[arccotu]

-u'/(1+u^2)

d/dx[arccscu]

-u'/(lul√(u^2-1))

d/dx[arccosu]

-u'/√(1-u^2)

Additive Interval Property:

-you can add different sections of an interval's definite integrals to get the entire one -constants can be pulled out and then multiplied at the end

The Existence of an Inverse

1) A function possesses an inverse if and only if it is one-to-one. 2) If f is strictly monotonic (either consistently increasing or decreasing/ no peaks or valleys) on its entire domain, then it is one-to-one and, hence, possesses an inverse.

What are the three ways to evaluate a limit?

1) Graphically 2) Algebraically 3) Numerically

Important Observations about inverse functions

1) If g is the inverse of f, then f is the inverse of g. 2) The domain of f^-1 is equal to the range of f, and the range of f^-1 is equal to the domain of f. 3) A function need not possess an inverse, but if it does, the inverse is unique.

What are the three types of discontinuity?

1) Removable 2) Jump 3) Infinite(asymptotic)

Guidelines for Finding the Inverse of a Function

1) Use Theorem 5.7 to determine whether the function given by y=f(x) has an inverse. 2) Interchange x and y. 3) Solve for y. The resulting equation is y=f^-1(x) 4) Define the domain of f^-1(x) to be the range of f. 5) Verify that f(f^-1(x))=x and f^-1(f(x))

Steps for Related Rates

1. Draw a sketch if applicable. The only dimensions you put on your sketch should be those that do not change. 2. Write down, in calculus notation, the rates you know and want. 3. Write an equation relating the quantities that are changing. 4. Differentiate it implicitly, with respect to time. 5. Substitute known quantities. 6. Solve for the required rate.

Guidelines for Finding Extrema on a Closed Interval

1. Find the critical numbers of f in ( a , b ) -Find the derivative and then find the "roots" of the derivative 2. Evaluate f at each critical number in ( a , b ) 3. Evaluate f at each endpoint [ a , b ] 4. The least of these values is a minimum. The greatest is the maximum. 5. Check to see if your numbers are the interval

Relative Extrema

1. If there is an open interval containing c on which f (c) is a maximum, then f (c) is called a relative maximum of f , or you can say that f has a relative maximum at the point ( c , f (c) ) 2. If there is an open interval containing c on which f (c) is a minimum, then f (c) is called a relative minimum of f , or you can say that f has a relative minimum at the point ( c , f (c) )

Guidelines for Finding Intervals on Which a Function is Increasing or Decreasing:

1. Locate the critical numbers (zeros of f ' and any points of discontinuity) of f to determine the test intervals. 2. Determine the sign of f ' (x) at one test value in the interval. 3. Then determine if f is increasing or decreasing.

Primary Methods for Integration of Exponential Functions

1. U-Substitution 2. Integration by Parts 3. Algebraic Simplification before approaching the problem using the first two methods

Steps for Curve Sketching

1.)Domain 2.)Intercepts- set f(x)=0 3.)Asymptotes- v.a. set denominator to 0, h.a. limit at infinity 4.)Increasing and Decreasing Intervals-first derivative test 5.)Relative Extrema- first derivative test 6.)Concavity and inflection points- second derivative test 7.)Graph

Steps for Optimization Problems

1.)Given and unknowns 2.)Picture/notation 3.)Relation (and we want what we're solving for in terms of just 1 variable, e.g., y= x-40)- find and replace is done in this part 4.)Find the max or min 5.)Find the unknown by plugging the max/min back into the original equation

d/dx(logaX)

1/((lna)x)

Definite vs. Indefinite Integral

A definite integral is a number. [ + , - , 0] *it is not necessary to include the "C" with definite integrals * An indefinite integral is a family of functions

Definition of Decreasing Function

A function f is decreasing on an interval if for any two numbers x1 and x2 in the interval, x1 < x2 implies f (x1) > f (x2). *Y1>Y2*

Even and Odd Functions w/ Substitution

A function f is even if f (x)= f (x). It is odd if f (-x)=-f(x) 1. If f is even, then ∫f(x) on interval [-a,a] =2∫f(x) on [0,a] 2. If f is odd, the ∫f(x) on interval [-a,a]=0

Definition of Increasing Function

A function f is increasing on an interval if for any two numbers x1 and x2 in the interval, x1 < x2 implies f (x1) < f (x2). *Y1<Y2*

Trigonometric Ratio

A ratio of the lengths of two sides in a right triangle to the angle.

(4.4 Begins) The First Fundamental Theorem of Calculus

If f is continuous on the interval [a,b] and F is any function that satisfies F '(x) = f(x) throughout this interval then ∫f(x)dx= F(b)-F(a)

Limit

If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c is L

The Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) for all x in an open interval containing,(except possibly c itself) and if lim x ->c h(x)=L=lim x->c g(x) then lim x->a f(x) = lim x->a h(x) = L then lim x->a g(x)= L

Define Limits at Infinity

As f(x) approaches a real number (L), for each real number on the x-axis (e), as x increases towards infinity, f(x) will remain within the horizontal lines between L+e and L-e

Define a horizontal asymptote

As x approaches infinity (neg or positive), f(x) will equals L (the horizontal asymptote).

Limits of Polynomial and Rational Functions

If p is a polynomial function and c is a real number then lim p(x) = p(c) x->c If r is a rational function given by r(x) = p(x)/q(x) and c is a real number such that q(c) not equal to 0, then lim r(x) = r(c) = p(c)/q(c) x->c

The Second Fundamental Theorem of Calculus

Assume that f(x) is continuous on an open interval I containing a. Then the area function: A(x)=∫f(t)dt is an antiderivative of f(x) on I; that is, A'(x) = f(x). Equivalently, (d/dx)∫f(t)dt=f(x) *x is the upper limit for all of this* Conditions for this rule: f(t) is a derivative of an integral Derivative matches upper limit of integration Lower limit of integration is a constant.

lne=

∫(1,e) (1/t)dt=1

What does C mean?

C is the constant real number usually an x value that we try to determine to limit of.

arccot x

D: (-∞,∞) R: (0,π)

arccsc x

D: (⁻∞ , -1] U [1 , + ∞) R: (-π/2, 0) U (0, π/2]

arcsec x

D: (⁻∞ , -1] U [1 , + ∞) R: [0 , π/2) U (π/2 , π]

How to determine discontinuity of a piecewise function

Determine the limit of where there is a possible discontinuity (where the the separate functions meets)

arcsin

Domain: [-1, 1] Range: [-π/2, π/2]

arccos

Domain: [-1, 1] Range: [0, π]

arctan

Domain: [-∞, ∞] Range: [-π/2, π/2]

Differential Equation

Equation that contains one or more derivatives; order is the order of the highest derivative involved

∫f(x)dx=

F(x)+C where F'(x)=f(x)

How to find symmetry

Find f(-x)

Guidelines for Finding Extrema on an Open Interval

Find the domain of the original function first 1. Find the critical numbers of f in ( a , b ) -Find the derivative and then find the "roots" of the derivative 2. Evaluate f at each critical number in ( a , b ) 3. The least of these values is a minimum. The greatest is the maximum.

Solving a Right Triangle

Finding all side lengths and angle measures of a triangle.

What happens if the one-sided portions of a limit do not match? That is the curve approaches different values depending on direction.

It does not exist as it causes a discontinuity.

Positive (negative) k value

Growth (decay)

Relative Extrema Occur Only at Critical Numbers

If f has a relative minimum or relative maximum at x = c , the c is a critical number of f

Graphical Test for the Existence of an Inverse Function

Horizontal Line Test to see if it is one to one (each y has only one x)

The Extreme Value Theorem

If f is continuous on a closed interval [ a , b ], the f has both a minimum and a maximum on the interval. The EVT is an existence theorem because it tells of the existence of a minimum and maximum values, but does not show how to find them.

Theorem 3.8: Points of Inflection

If ( c , f (c) ) is a point of inflection on the graph of f , then either f "(c) = 0 or f "(c) does not exist at x = c.

Are rational and radical functions continuous?

In their domain they are continuous, but on an open interval no.

∫

Integral Sign

f(x)

Integrand−the function to be integrated

Mean Value Theorem

Let F be a function that satisfies the following hypotheses: F is continuous on the closed interval [a,b] F is differentiable on the open interval (a,b) Then there is a number c in (a,b) such that: ∫f(x)dx=f(c)(b-a) or F(b)-F(a)=f(c)(b-a)

Theorem 3.6: The First Derivative Test

Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f (c) can be classified as follows: 1. If f ' (x) changes from negative to positive, then f has a relative minimum. 2. If f ' (x) changes from positive to negative, then f has a relative maximum. 3. If f ' (x) is positive (or negative) on both sides of c then f (c) is neither a relative minimum nor a relative maximum. Think of no change...

The Definite Integral as a Function of x

Let f be a continuous function on [a, b] and x varies between a and b. If x varies, the following is a function of x denoted by g(x): g(x)=∫f(t)dt It should be clear there is an inverse relationship between the derivative and the integral. Thus, the derivative of the integral function is simply the original function.

Theorem 3.9: Second Derivative Test

Let f be a function such that f "(c) = 0 and the second derivative of f exists on an open interval containing c. 1) If f "(c) > 0, the f has a relative minimum at (c , f (c)). 2) If f "(c) < 0, the f has a relative maximum at (c , f (c)). If f "(c) = 0, the test fails. That is, f may have a relative maximum, a relative maximum, or neither. In such cases, you can use the First Derivative Test.

Theorem 3.5: Test for Increasing and Decreasing Functions

Let f be a function that is continuous on the closed interval [a , b] and differentiable on the open interval (a , b). 1. If f ' (x) > 0 for all x in (a , b), then f is increasing on [a , b]. 2. If f ' (x) < 0 for all x in (a , b), then f is decreasing on [a , b]. 3. If f ' (x) = 0 for all x in (a , b), then f is constant on [a , b].

The Derivative of an Inverse Function

Let f be a function that is differentiable on an Interval I. If f possesses an inverse function g, then g is differentiable at any x for which f'(g(x))≠0 g'(x)=1/f'(g(x)), f'(g(x))≠0

Continuity and Differentiability of Inverse Functions

Let f be a function whose domain is an interval I. If f possesses an inverse, then the following statements are true. 1. If f is continuous on its domain, then f^-1(x) is continuous on its domain. 2. If f is increasing on its domain, then f^-1(x) in increasing on its domain. 3. If f is decreasing on its domain, then f^-1(x) is decreasing on its domain. 4. If f is differentiable at c and f'(c)≠0, then f^-1 is differentiable at f(c).

Theorem 3.7: Test for Concavity:

Let f be a function whose second derivative exists on an open interval I. 1.) If f "(x) > 0 for all x in I, then the graph of f is concave upward in I. 2.) If f "(x) < 0 for all x in I, then the graph of f is concave downward in I.

Part 3 of The Fundamental Theorem of Calculus

Let f be continuous on the interval [a,b]. Then the function g defined by: g(x)=∫(a,x) f(t) dt, a≤x≤b is continuous on [a,b] and differentiable on (a,b) and g (x)=f '(x).

Definition of a Critical Number

Let f be defined at c . If f ' (c) = 0 or if f is not differentiable at c , then c is a critical number of f ***The critical numbers of a function need not produce relative extrema!

Definition of Concavity:

Let f be differentiable on an open interval I. The graph of f is concave upward on I if f ' is increasing on the interval. The graph of f is concave downward on I if f ' is decreasing on the interval.

What does L mean?

Limit

Upper and Lower Sums

Lower sum s(n) = f(mi)∆x Upper Sum S(n) = f(Mi)∆x f(mi or Mi)= you basically plug ∆x's formula into the equation of the line wherever there is an x ∆x=(b-a)/n or the interval end points over n which represents the number of rectangles as they approach infinity since it represents approaching infinity you never plug a number into it even if a certain amount of rectangles is given to you in a diagram. only plug in if there is a number in the place of the max term aka n and its not asking for a limit but instead just to evaluate and find the sum

Infinite(asymptotic) Discontinuity?

Means the limit approaches infinite at some point. The curve has asymptotes.

Slope Field

Pictorial representation of both the general and particular solutions to a differential equation

How to find concavity

Set second derivative=0 and do a number line sign analysis

How is a limit evaluated NUMERICALLY?

Plug into values close to a from the right and left. Observe the trend. Usually the last resort of the three ways to evaluate a limit.

How to find intercepts

Set x to 0 to find y-intercept Set y to 0 to find x-intercept

Which functions are always continuous on open intervals except for in some piecewise functions

Polynomial and trig functions

Hill

Relative Maximum

Valley

Relative Minimum

How to find vertical asymptotes

Set denominator= 0 (None if continuous)

How to find intervals of increasing/decreasing

Set derivative=0 and do a number line sign analysis

Remember!

Show all work to receive full credit

Areas of Common Geometric Figures

Sketch, then evaluate using knowledge of common formulas

General Solution

Solution including +C

Steps for Implicit Differentiation

Step 1: Notate y as y(x) to remember that y is a function of x Step 2: Differentiate the function with respect to x. this means every y must be multiplied by dy/dx Step 3: Isolate dy/dx to one side Step 4: Replace f(x) with y and simplify

(4.5 Begins) Integration by Substitution

Steps: 1. Choose inside or one of trig functions to be "u" 2. Derive, du=____dx 3. Integrate u so ∫udx= (u^2)/2 *this depends on properties of original f(x) for sinx it would be sinu * 4. Plug in original "u" to integrated function 5. Multiply your finished answer by (1/coefficient of _____) found from step 2. 6. Don't forget to add +C

How to find local max/mins

Take smallest/largest values from previous part

Reflective Property of Inverse Functions

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

What is the limit of a rational function if the numerator has an exponent value greater than the denominator?

The limit of the function does not exist.

What is the limit of a rational function if the numerator and denominator have the same exponent value?

The limit of the function is the ratio of the leading coefficients

Net Change as the Integral of a Rate

The net change in s(t) over an interval [t1,t2] is given by the integral:

It is often easier to prove a function has an inverse then to find the inverse

This basically means, find the derivative and set it equal to zero prove that it does not change from increasing to decreasing, aka monotonic.

The Upper Limit of the Integral is a Function of x

Use the First Fundamental Theorem of Calculus to evaluate the integral: (g(x) is the upper limit and a is the lower limit) ∫f(t)dt = F(g(x))-F(a) Find the derivative of the result: F'(g(x))-F'(a) a is a constant so F'(g(x))×g'(x)-0 f(g(x))×g'(x)

Composite Functions and The Second Fundamental Theorem of Calculus

When the upper limit of the integral is a function of x rather than x itself, We can use the Second Fundamental Theorem of Calculus together with the Chain Rule to differentiate the integral. think of it like when we use just x you're multiplying by its derivative but thats one so we dont really see it but with this you do see the derivative so you just tack it on. i am an effing god peace out

Composite Functions and The Second Fundamental Theorem of Calculus (aka when the conditions aren't met)

When the upper limit of the integral is a function of x rather than x itself, We can use the Second Fundamental Theorem of Calculus together with the Chain Rule to differentiate the integral. think of it like when we use just x you're multiplying by its derivative but thats one so we dont really see it but with this you do see the derivative so you just tack it on. i am an effing god peace out

Initial Value Problem (Finding the position at t=6 given s(0) and v(t))

When we can obtain a particular solution based on initial condition given (s(0)+int(v(t) from 0 to 6) bc it is just initial condition plus change.)

Inverse Trigonometric Ratios

You can use sin⁻¹, cos⁻¹ and tan⁻¹ to find the measure of an angle in a right triangle given its sides.

What is the limit of a rational function whose denominator exponent is GREATER than the numerator, but the coefficient of the denominator is < 0?

Zero as x approaches negative infinity

What is the limit of a rational function whose denominator exponent is GREATER than the numerator, but the coefficient of the denominator is > 0?

Zero as x approaches positive infinity

(4.1 begins) Antiderivative

a function whose derivative is f. i.e. A function F is an anti-derivative of f if F'(x)=f(x)

(5.5 begins) Definition of Exponential Function to Base a

a^x=e^(lna)x

with ln expressions

always make to simplify using properties before trying calculus methods

Primary Equation

an equation/formula for the quantity to be optimized.

Secondary Equation (Optimization)

an equation/formula relating the independent variables of the primary equation. *Purpose:* to reduce the primary equation to one having a single independent variable before attempting to maximize or minimize it.

(3.9 Begins) The differential of x (denoted by dx) is

any nonzero real number

Integral: du/ sqrt(a^2-u^2)

arcsin(u/a)+C

cosxdx

sinx+c

Steps for u substitution in this case

choose u, derive to find ratio of coefficients find integral of original equation with u for example: ∫1/u du= lnlul+c plug what you chose as u into the integrated function

nΣ i=1 c

cn, c is a constant

Apply the Fundamental Theorem of Calculus to the function f(t)=1/t

d/dx(∫(1,x) (1/t)dt=1/x this means d/dx(lnx)=1/x

If a and b are the same

definite integral equals 0

Determining Average Value

determine f(c) using mvt and then use this to find c

end behavior

determined by limits at infinity (x value is + or - infinity) lim as x -> + or - infinity of f(x)

for other rational functions (polynomial/polynomial)

divide each term in the numerator and the denominator by the highest power of the denominator. all terms with an x in the denominator after you divide cancel out

The differential of y (denoted by dy) is given by

dy = f'(x) dx

________ is *inversely* proportional

dy/dx=K/x

∫e^xdx

e^x+c

Mean Value Theorem

f is CONTINUOUS on [a,b] and DIFFERENTIABLE on (a,b) then there exist a point c in (a,b) where f'(c) = f(b)-f(a)/b-a 1) it guarantees the existence of a tangent line that is parallel to the secant line through the points (a , f(a) ) and (b , f(b) ) → the endpoints of the interval 2) there must be a point on the open interval (a , b ) at which the instantaneous rate of change is equal to the average rate of change over the interval [a , b]

Rolle's Theorem

f is CONTINUOUS on [a,b] and DIFFERENTIABLE on (a,b). 1) it guarantees the existence of an extreme value in the interior of a closed interval 2) there must be at least one point between a and b at which the derivative is 0

A function g is the inverse of the function f if

f(g(x))=x for each x in the domain of g and g(f(x))=x for each x in the domain of f

Solution to f¹(x)=cos(x²) for which f(3)=5

f(x)=5 + ∫(3→x)cos(t²)dt

f(x)= lnx has the same derivative as

g(x)=ln|x|

A function F is called an anti-derivative of f on an interval I

if F'(x)=f(x) for all x in I

Limits of piecewise functions and V.A.'s

increase without bound

How to find horizontal asymptotes

limit as x approaches infinity

The natural logarithm function is defined as

lnx=∫(1,x) (1/t) dt, x>0

∫(u'/u)dx

ln|u|+c

if the exponent is a variable

logarithmic differentiation

(4.2 Begins) The sum of terms is written as

n ∑ ai= a1 a2 a3 . . . an i=1 where is the index of summation, is the ith term of the sum, and the upper and lower bounds of summation are and 1. *NOTE:* The upper and lower bounds must be constant with respect to the index of summation. However, the lower bound doesn't have to be 1. Any integer less than or equal to the upper bound is legitimate.

(4.3 Begins) Definite Integral Definition

n b lim ∑ f(ci)∆x= ∫ f(x)dx ll∆ll i=1 a f(ci)∆x=RIEMANN SUM ∫ f(x)dx= DEFINITE INTEGRAL If the limit exists, f is integrable on [a, b] a is the lower limit of integration b is the upper limit of integration

nΣ i=1 i^2

n(n+1)(2n+1)/6

nΣ i=1 i

n(n+1)/2

nΣ i=1 i^3

n^2(n+1)^2/4

position formula

s(t)=½gt²+v₀t+s₀

secxtanxdx

secx+c

The rate of change of N with respect to S is proportional to 500-s

separation of variables u substitution ignore k, pull out, and multiply at end bc is it a constant ∫dN/ds=∫(500-s)k ∫dN=∫(500-s)kds u=500-s du=-1ds N=⁻k∫udu N=(-k(500-s)^2)/2

The concavity of f is related to the slope of the derivative.

slope of the derivative.

dont forget to

split up integrals when the numerator has addition

Arctan(1)

tan(x)=1 x=π/4

(secx)^2dx

tanx+c

Curve Sketching- Testing Intervals

test every interval where one point equaled zero at some point including v.a. ultimately, if it is not in the domain you should not include it, but can just to verify the rest of the information you found.

if a becomes the upper limit and b is the lower limit

the answer is just the negative version of a as the lower limit and b as the upper limit

The range of f =

the domain of f^-1

If graph is sharp and peaked

the function is not differentiable at the relative max or relative min

If graph is smooth and rounded,

the graph as a horizontal tangent line at the relative max or relative min

A function can be represented as a set of ordered pairs. By interchanging the first and second coordinates we form

the inverse function.

Informal Definition of the Points of Inflection:

the point of the graph when the concavity changes from up to down (or down to up)

The domain of f =

the range of f^-1

Feasible Domain

the values of "x" that make sense in the problem. ex: can't have negative lengths

If F is an anti-derivative of f on an interval I

then all anti-derivatives of f on I will be of the form F(x)+C where C is a possible constant. The c is called the constant of integration.

d/dx[arctanu]

u'/(1+u^2)

d/dx[arcsecu]

u'/(lul√(u^2-1))

d/dx(lnu)=

u'/u f'(lnx)= f'(x)/x=1/x

d/dx[arcsinu]

u'/√(1-u^2)

A function with two H.A.'s

x>0 x=sq.r.t of (x^2) x<0 x=-(sq.r.t of (x^2))

Point Slope Formula

y - y1 = m(x - x1)

Exponential Change Formula

y = Ce^kt C-initial value k-proportional

When f(f^-1(x)) or f^-1(f(x))

y=x

Finding Upper and Lower Sums for a Region

you use the formulas but it is proof that the two of them are equal to each other as n approaches infinity therefore this is simply one general formula you can use Area = f(ci)∆x (this is called Riemann Sum)

The relative maximum or relative minimum of f are the

zeros of f '

(4.6 Begins) Trapezoidal Rule

∫f(x)dx= (b-a)/ 2n [f(x₀) + 2f(x₂) + 2f(x₃)..... f(x)]

Limits that fail to exist

-Unbounded Behavior- Asymptotes of any kind, could be going in different directions -Behavior that differ from the right and left -Oscillating behavior- like an ekg, has no significant behavior and therefore doesn't approach anything so no limit.

d/dx [cscx]

-cscx•cotx

Quotient Rule

(f/g)' = gf' - fg' / g^2

Product Rule

(f·g)'=f'·g+f·g'

Strategy for Finding Limits

1. Learn to recognize which limits can be evaluated by direct substitution. 2. If the limit of f(x) as approaches c cannot be evaluated by direct substitution, try to find a function g that agree with f for all x other than x = c. 3. Prove it analytically. 4. Use a graph or table to reinforce your conclusion.

A function f is continous at c when these three conditions are met.

1. The function f(c) is defined. 2. The limit of f(x) exists at f(c). 3. The limit of f(x) exist at f(c), and is equal to f(c).

Find an equation of the line that is tangent to the graph of f and parallel to the given line

1. determine the slope of the given line 2. set slope of tangent formula equal to slope of line 3. solve for x (the point at which the line is tangent to graph and parallel to the given line) usually we are solving for the slope, but since they are parallel we use the slope of the other line to determine the point at which the line is tangent 4. plug in x to original function to find y 5. plug into y=mx+b to find b and get a final equation of the line.

Limit of Trigonometric Functions

1. lim sin x = sin c 2. lim cos x = cos c x->c x->c 3. lim tan x = tan c 4. lim cot x = cot c x->c x->c 5. lim sec x = sec c 6. lim csc x = csc c x->c x->c

Continuity on an Open Interval

A function is continuous on an open interval (a,b) when the function is continuous at each point in the interval. A function that is continuous on the entire real number line (-∞,∞).

Derivative

A function which gives the slope of a curve; that is, the slope of the line tangent to a function. New function f'(x) that will calculate the slope of the tangent line for any chosen x-value

Removable Discontinuity

A hole in a graph. That is, a discontinuity that can be "repaired" by filling in a single point. In other words, a removable discontinuity is a point at which a graph is not connected but can made connected by filling in a single point.

What is a infinite limit?

A limit that increases or decreases without bound as the limit gets closer to the x-value. Basically, the y-values get really big in either positive/negative directions as you get close. The limit is considered to not exist as infinite is not a number.

What is the Chain Rule?

A way to differentiate functions within functions.

What is Implicit Differentiation?

A way to take the derivative of a term with respect to another variable without having to isolate either variable

What are the three rules for a limit to be continuous?

All 3 ensure there's no discontinuities. Condition 3 is the short-hand rule to immediately tell if something is continuous. The former two conditions are just an extension of it.

Limit of a Function Involving a Radical

Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is even lim sqrt(x)^n = sqrt(c)^n x->c

Continuous function?

Basically if there is no discontinuity along an interval. That is, the graph can be drawn in 1 continuous motion without lifting your pencil.

Intermediate Value Theorem

Basically, on a continuous interval. Each value between the two endpoints have corresponding pairs between the inputs and outputs.

Fractional Power limit law?

Basically, the limit must be positive if you are rooting by an even number.

Chain Rule

F' (x) = f'(g(x)) · g' (x)

How to simplify a function to an equation so they at all but one point

Factor Rationalize the numerator Simplify complex fraction

One-sided limits

For values evaluated from the left (negative superscript) and right (positive superscript) side respectively

Limit of a Composite Function

If f and g are functions such that lim g(x) = L and lim f(x) = f(L), x->c x->L then lim f(g(x)) = f(lim (g(x)) = f(L) x->c x->c

Non-removable Discontinuity

No limit -Unbounded Behavior- Asymptotes of any kind, could be going in different directions -Behavior that differ from the right and left -Oscillating behavior- like an ekg, has no significant behavior and therefore doesn't approach anything so no limit.

Do rationalizing and other methods always work?

No, and in this case just plug into the original equation and states that the limit dne

Not Differentiable

Non Continuous Functions, absolute values,VAs, and those with multiples y-values for an x

The limit of f(x) does not exist at f(c), but the other conditions are met.

Non-removable discontinuity because there is no limit

How is a limit evaluated graphically?

Observe the value ON THE CURVE (ignore the point) simultaneously approached as you approach a on the x-axis from the left and right.

Removable discontinuity?

Occurs when the limit exists, but it is not equal to f(a). This results in a hole on the curve.

What is a limit EVALUATED at infinite?

Occurs when the x-value increases/decreases without bound. The y-value is ± ∞ or a finite number. It determines the END BEHAVIOR of a function.

Functions with limits that can be found through Direct Substitution

Polynomials Radicals Trig Functions

When evaluating a limit analytically what should you always try first?

Simply try to solve by plugging a (what x approaches) into the function.

Find the derivative using limit definition

Step 1 Write f(x + h) and f(x). Step 2 Compute f(x + h) − f(x). Combine like terms. If h is a common factor of the terms, factor the expression by removing the common factor h. Step 3 Simply f(x + h) − f(x)/h -As h → 0 in the last step, we must cancel the zero factor h in the denominator in Step 3. Step 4 Compute lim h→0 f(x + h) − f(x)/h by letting h → 0 in the simplified expression.

For a rational limit EVALUATED AT infinite: what is true if the degree of numerator < degree of denominator?

The limit is equal to 0 since infinite is in the denominator.

For rational limits: when a big number such as ±∞ is approached in the denominator ______

The limit is equal to 0. Dividing by a big number produces a small number so it makes sense that it approaches 0.

For a rational limit EVALUATED at infinite: what is true if the degree of numerator = degree of denominator?

The limit is equal to the ratio of their leading coefficients.

For a rational limit EVALUATED at infinite: what is true if the degree of numerator > degree of denominator

The limit is ∞ or -∞

If f(c) is not defined, but the other conditions are met.

There is still a limit for when x->c so the discontinuity is removable.

The limit of f(x) exists at f(c), but is not equal to f(c), but the other conditions are met.

There is still a limit for when x->c so the discontinuity is removable.

Basic Limits

This just means you plug c in x's place to find limit in direct substitution Also if f(x)=2. y=2 so now matter what x value it will always be approaching 2 lim 2 = 2 lim x = c lim x^n = c^n x->c x->c x->c

How to determine a removable discontinuity analytically

When rearranging an equation to simplify so that it agrees at all but one point, that one point is the removable discontinuity because it can be removed and and the graph appears the same, minus the hole where the x value was.

A tangent line to the function f(x) at the point is

a line that just touches the graph of the function at the point in question and is "parallel" (in some way) to the graph at that point. Take a look at the graph below.

What can an absolute value function be rewritten as?

a piecewise lxl/x {-1 x<0, 1 x>0, x cannot = 0 then plug in for zero because that is where there is a possible discontinuity lim x->0^-(from the left)= -1 lim x-.0^+(from the right)=1 limit dne bc left and right behavior differ

speed

absolute value of velocity, direction doesnt matter

First thing to do when finding limits

always try direct substitution first

Asymptotes and Discontinuity

anytime you have asymptote it is a non-removable discontinuity. however, the converse of this statement is not true.

average velocity

change in distance/change in time=∆s/∆t

d/dx [sinx]

cosx

the constant rule

d/dx [c]=0

integral of v(t)

displacement/ change in position from lower limit to upper limit

Other notations used to denote the derivative of y = f (x)

f '(x) y' d [ f (x)] Dx[y] dy/dx

The derivative of f at x is given by

f '(x) = lim f (x + ∆x) - f (x)/∆x ∆x ➝ 0 provided the limit exists. For all x for which this limit exists f 'is a function of x.

velocity formula

first derivative of s(t) s'(t)=gt+v₀

find the constant a, or the constants a and b, such that the function is continuous on the entire real line.

first find the possible the discontinuity and plug it in as c to find the limit. then plug it into the next piece to solve for a by setting it equal to the limit obtain in the first step. this is all done bc we are told to assume that the function is continuous therefore the limits must be the same

Slope of a secant line

is a straight line joining two points on a function. (See below.) It is also equivalent to the average rate of change, or simply the slope between two points.

Scalar Multiple

lim [b f(x)] = bL lim 2(x^2)= 2(4) b=2, l=f(c) x->c x->2

Sum or Difference

lim [f(x) +/- g(x)] = L+/- K L=lim of f(x) K= lim of g(x) x->c

Power

lim [f(x)]^n = L^n lim (x-1)^2 = (1)^2 L=1 n=2 x->c x->2

Product

lim [f(x)g(x)] = LK L=lim of f(x) K= lim of g(x) x->c

Quotient

lim f(x)/g(x) = L/K, provided K is not equal to 0 x->c

Special Limits

lim sinx/x =1 lim 1-cosx/x= 0 x->0 x->0 rule only applies when x's have the same coefficients lim sin3x/2x= (1/2)sin3x/x= 3/3(1/2)sin3x/3x= (3/2)sin3x/3x= (3/2)(1)= 3/2 x->0

Can you rationalize complex fractions?

nah fam, need to multiply by denominators multiplied together

Does f(c) matter in determining a limit?

no, the left and right behavior it what determines a limit. If the behaviors match there is a limit, so although f(c)=1 the limit can be 3 or any other real number

To receive full credit

plug for each condition. when asked to discuss continuity, state which

d/dx [tanx]

sec^2x

acceleration formula

second derivative of s(t) s''(t)=g= -32

d/dx [secx]

secx•tanx

When asked to discuss the continuity of each function,

state the discontinuity and explain why. also, if possible, plug the equation in for the three conditions to express it analytically as well.

For rational limits: when a small number is being approached in the denominator______

the limit is equal to infinite. When a number is divided by a small number it produces a big number. So it makes sense that the limit approaches ±∞.

integral of speed (lv(t)l)

total distance travelled

How to draw the graphs of derivatives

y is your slope value and x is the same for the original function, the interval at which the function exists. Example: For the parent quadratic, the slope starts out as negative, but at the vertex the slope is 0 so the x value for the vertex will by the x value for derivative. Then the slope proceeds to get more positive so the derivative will look similar to the line y=x. you can also find the derivative and graph it when given which graph is the original