Calculus I Cumulative

Radian/Degree Conversion

# r = (x180/pi)

Average Speed Equation

((f(b)-f(a))/b-a

Reflections

-f(x) reflect across x-axis f(-x) reflect across y-axis

Continuous Growth

A = (Pe)^(r*t) P=principal r=rate t=time

Compounded Growth

A = P(1(+/-) (r/n))^(n*t) P= principal r = rate n = number of times compounded per cycle t= number of cycles

antiderivative

A function F(x)+C is an antiderivative of f on an interval I, if F1(x)=f(x) for all x in I.

Fixed Point

A function f has a fixed point whenever f(x) = x. And thus, f(x) intersects the line y=x.

Neither Inc/Dec Function

A function is neither decreasing nor increasing, when it is either doing both or else is a horizontal line

Horizontal Line Test

A function is one-to-one (has an inverse function) if its graph intersects each horizontal line at most once

Critical Point

An interior point of the domain of a function f, where f1 is zero or undefined; therefore, to find critical points, set f1 equal to zero and find where the denominator of f1 equals zero. DON'T FORGET THAT THE ENDPOINTS ARE ALSO CRITICAL POINTS.

Even/Odd Trig Functions

Cos(x)/Sec(x) are even functions sin(x)/cos(x)/tan(x)/csc(x)/cot(x) are odd functions

Acceleration Problems

Given: 9.8m/sec^2 v(t)=9.8t+C1 s(t)=4.9t^2+C1x+C2

Fixed Point Problem

Given: f(x) continuous on [0,1]; f(0)=1 and f(1)=0 Solution: Graph illustrates that the graph of f(x) and y=x must intersect. By intermediate value theorem we can conclude that there must be some point c such that the two lines intersect: f(x)=x, f(x)-x=0, f(0)-0 = 1 and f(1)-1 = -1, thus there must be a zero in between. f(0)<f(c)<f(1)

Intermediate Value Theorem

If f is a continuous function on the interval (a,b), and c is any value in this interval then f(a)<f(c)<f(b). (ex: f(a) = + and f(b) = -, then f(c) = 0)

Corollary 3

If f is continuous on [a,b] and differentiable on (a,b); f1(x)>0 then f(x) is increasing; f1(x)<0 then f(x) is decreasing

Mean Value Theorem for Definite Integrals (MVTI)

If f is continuous on [a,b], then at some point c in [a,b], f(c) = (1/(b-a))(a-bint(f(x)dx)

Sandwich Theorem

If f(x) and g(x) are continuous at x=c, and lim(f(x))=lim(g(x))=L as x approaches c, and f(x)<h(x)<g(x), then lim(h(x)) as x approaches c = L

Extreme Value Theorem

If f(x) is a continuous function on the closed interval [a,b], there must be both an absolute maximum and an absolute minimum.

Mean Value Theorem

If f(x) is continuous on [a,b] and differentiable on (a,b), then there is at least one point "c" such that f1(c)=(f(b)-f(a))/(b-a). note: This guarantees that there is a point where the tangent has a slope equal to the average value of the derivative, or the slope of the secant on the closed interval. Proof: (a,f(a)) and (b,f(b)); g(x)=f(a)+((f(b)-f(a))/(b-a))*(x-a); vertical difference is h(x)=f(x)-g(x); h(x)=f(x)-f(a)-((f(b)-f(a))/(b-a))*(x-a); h1(x)=f1(x)-((f(b)-f(a))/(b-a)); f1(c)=((f(b)-f(a))/(b-a))

Sign Change of f1(x) and Local Extrema

If f1(x) changes sign at some point "c," then there must be a local max or min at that point. This works, unless it is a piecewise defined function, in which case a filled hole may create a unique local extrema.

Corollary 2

If f1(x)=0; then f(x)=C for some constant C. Proof: f1(X)=0; ((f(x2)-f(x1))/(x2-x1)) = 0; f(x2)-f(x1)=0; f(x1)=f(x2)

Local Max and Min

If there is a local maximum or minimum at x=c, then f1(c)=0 or f1(c)=und.

Change of Base Formula

Log(a)^U = (ln(U))/ln(a) NOTE: Used to prove the derivative of logarithms for base ten

vertical line test

No vertical line can intersect the graph of a function more than once

Normal Line Tangent Line

Normal: N=f(a)+(-1/f1(a))(x-a) Tangent: T=f(a)+f1(a)(x-a)

Linear Approximations

Simply find the derivative, multiple both sides by dx and then plug in the value of dx to find the change. And use the tangent equations to determine the slight changes caused by approximations from 2 to 2.1 for example.

Basic Trig Relationships

Sin(x)=O/H Cos(x)=A/H Tan(x)=O/A Csc(x)=H/O Sec(x)=H/A Cot(x)=A/O

Derivatives of Functions and Their Inverses

The derivative of the function is equal to the reciprocal of the derivative of the inverse function. The derivative of the inverse function is equal to the reciprocal of the derivative of the function.

Natural Domain

The largest set of x-values for which the formula gives real y-values

Real-Valued

The name of the range that is the set of all real numbers

a^u derivative proof

This same method, in step 1, is used to prove the power rule This may also be used to solve the derivative of x^x.

Project 1

We essentially prove, in 2, that f-1(f(x))'s derivative is 1/f1(x)

Natural Logarithm Solutions to Quotient Derivatives

We may solve derivatives involving quotients faster by taking the natural log of both sides of the equation (y and x's), and then we know ln(u/v)=ln(u)-ln(v), so we can simplify and then find the derivatives from there.

Cox(x) Graph

cos(pi/6) = rad(3)/2 cos(pi/4) = rad(2)/2 cos(pi/3) = 1/2 cos(pi/2) = 0 cos(2pi/3) = -1/2 cos(3pi/4) = -rad(2)/2 cos(5pi/6) = -rad(3)/2 cos(pi) = -1

Proof that ln(bx)=ln(b)+ln(x)

d(ln(bx))/dx = 1/x = d(ln(x))/dx; by corollary we know they differ by a constant; ln(bx)=ln(x)+C; at x=1 ln(b)=ln(1)+C; C=ln(b); ln(bx)=ln(b)+ln(x)

Proof of ln(x^r) = rln(x)

d(ln(x^r))/dx = (1/(x^r))*(rx^(r-1)) = r*(1/x) = d(rln(x)/dx ln(x^r)=rln(x)+C set x=1 ln(x^r) = rln(x)

Odd Function Definition

f(-x) = -f(x) Graph of f is symmetric about the origin (ie when rotated 180 degrees about the origin the graph looks the same as the original)

Even Function Definition

f(-x) = f(x) Graph of f is symmetric about y-axis

Neither Odd/Even Definition

f(-x) not equal to f(x) or -f(x), thus the graph is not symmetric about the y-axis or the origin.

Definition of Continuity

f(c) exists lim(f(x)) as x approaches c exists lim(f(x)) as x approaches c = f(c) Note: Unless domain of function is all real numbers, endpoints become non-inclusive for where the limit is defined.

POI (Problem of Interest)

f(f(x^a)) = (x^a)^2

Trig Function Transformations Equation

f(x) = A*sin((2pi/B)(x-c)) + D A = amplitude B = period of function C = horizontal shift D = vertical shift

Exponential Function

f(x) = a^x, where a is some constant. Passes through, (0,1) and (1,a)

Identity Function

f(x) = x

Inverse Functions Proof

f(x) = y and g=f^-1(x) and g(y)=x (f^-1(f(x)) should equal x) g(f(x)) = g(y) = x note: inverse function and original function should be symmetric across line y = x.

Local Maximum Definition

f(x) has a local maximum at "c" on some interval (a,b), such that f(c)(>/=)f(x)

Local Minimum Definition

f(x) has a local minimum at "c" on some interval (a,b), such that f(c)(</=)f(x)

Definition of Increasing Function

f(x1) > f(x2) for x1>x2

Definition of Decreasing Function

f(x1)<f(x2) for x1>x2

second derivative test

f1(c)=0 and f11(c)<0 L. max f1(c)=0 and f11(c)>0 L. min.

Derivative Rule for Inverses

f1(f-1(x))*(f-1(x))1=1 f1(f-1(x))=1/(f-1(x))1 You know what I mean. Ultimately, The slope of the inverse function is the reciprocal of the function's slope at the same point.

Corollary 1

f1(x)=g1(x) then, the two functions must differ by a constant. To find the constant, set x=0 and then solve for C.

Half Life Formula

half-life = ln(2)/k k=rate

Infinite Limits Problem 1

lim ((sqr(x+1)+2) + (x-3)) as x approaches inf. Multiply top and bottom by numerator's conjugate and then solve

Natural Logarithm solution to limit

lim (1/x)^(1/ln(x)) as x approaches inf. ln (y) = (1/ln(x))*ln(1/x) ln (y) = -ln(x)/ln(x) ln(y) = -1 as x approaches inf.

Tangent Line Basic Equation

lim (as h app. 0) of ((f(x+h)-f(x))/h)

absolute value limit problem

lim (as x approaches inf) (2x^3+1)/(abs(x^3)-2x) = (+/-) 2

Infinite Limits Problem 2

lim lim ((sqr(x+9)+4) / (x-7)) as x approaches inf. Divide top and bottom by the variable, in the denominator, with the largest exponent. Note: This works, as you can see, even when we are dealing with square roots; If we approach negative inf. then the answer must be negated, as a square root is present.

Solving Quotient Limits

lim((sqr(x+1)+2)/(x-3)) as x approaches 3 Multiply top and bottom by numerator's conjugate and then solve

Limit Identities

lim(sin(x)/x) as x approaches 0 is 1 lim((1-cos(x))/x) as x approaches 0 is 0

Derivative of Natural Log

ln(cu) = (1/u)*(du/dx); where c=constant

orthogonal

perpendicular

Trig Function One-to-one Domain Restrictions

sin = (-pi/2 , pi/2) cos = (0, pi) tan= (-pi/2 , pi/2) csc = (-pi/2,0) U (0,pi/2) sec = (0,pi/2) U (pi/2,0) cot = (0,pi)

Sin(x) Graph

sin(pi/6) = 1/2 sin(pi/4) = rad(2)/2 sin(pi/3) = rad(3)/2 sin(pi/2) = 1 sin(2pi/3) = rad(3)/2 sin(3pi/4) = rad(2)/2 sin(5pi/6) = 1/2 sin(pi) = 0

Rule about Square Roots

sqr(x^2) = abs(x)

Rolle's theorem

suppose that f(x) is continuous on [a,b] and differentiable on (a,b), and f(a)=f(b); then there must be some point "c" such that f1(c)=(f(b)-f(a))/(b-a) = 0 or f1(c)=0

Proof of (e^a)*(e^b)=e^(a+b)

y1=(e^x1) and y2=(e^x2) ln(y1)+ln(y2)=x1+x2 e^(x1+x2)=y1y2; we know what y1 and y2 are. (e^x1)*(e^x2)=e^(x1+x2)