Calc 1 Final Exam Study Guide
Trigonometric Functions: Definition: Amplitude
For any periodic function of time, the Amplitude is half the distance between the maximum and minimum values of (if it exists).
Trigonometric Functions: Definition: Period
For any periodic function of time, the period is the smallest time needed for the function to execute one complete cycle.
Trigonometric Functions: Definition: Arctan
For any y, arctan y=x means tan x=y with -pie/2 less than x which is less than pie/2
Functions: Definition: Family of Functions
Formulas such as f(x)= b + mx, in which the constants m and b can take on various values, give a family of functions. All the functions in a family share certain properties. The constants m and b are called parameters
Limit: Theorem 1.4: CONTINUITY OF COMPOSITE FUNCTIONS
If f and g are continuous, and if the composite function f(g(x)) is defined on an interval, then f(g(x)) is continuous on that interval.
Shortcuts to Differentiation: Powers and Polynomials: Theorem 3.2: DERIVATIVE OF SUM AND DIFFERENCE
If f and g are differentiable, then d/dx[f(x) + g(x)]=f'(x) + g'(x) and d/dx[f(x)-g(x)]=f'(x)-g'(x)
Shortcuts to Differentiation: Powers and Polynomials: Theorem 3.1: DERIVATIVE OF A CONSTANT MULTIPLE
If f is differentiable and c is a constant, then: d/dx= [cf(x)]=cf'(x)
New Functions from Old: Definition: Even Function
For any function f, f is an even function if f-(x)= -f(x)
New Functions from Old: Definition: Odd Function
For any function f, f is an odd function if f(-x)= f(x)
New Functions from Old: Notes: Shifts and Stretches
- Multiplying a function by a constant, c, stretches the graph vertically (if c>1) or shrinks the graph vertically (if 0<c<1). A negative sign (if c<0) reflects the graph about the x-axis, in addition to stretching or shrinking. -Replacing y by (y-k) moves the graph up by k (down if negative). -Replacing x by (x-h) moves the graph to the right by h (left if h is negative).
What are for ways to express a function?
-Formula -Words -Table -Graph
What are the four mathematical concepts of calculus
-Functions -Limits -Derivatives -Integrals
Derivative: Differentiability: Notes: When a function fails to have a derivative
-The function is not continuous at the point -The graph has a sharp corner at the point -The graph has a vertical tangent line.
Exponential Functions: Notes: numerical value of e
2.71828
Functions: Definition: Decreasing function
A function f is decreasing if the values of f(x) decrease as x increases. The graph of an increasing function falls as we move from left to right
Limit: Definition: Limit (Cauchy Formal Def)
A function f is defined on an interval around c, perhaps at the point x=c. We define the limit of the function f(x) as x approaches c, written the limit as x approaches c f(x), to be a number L (if one exists) such that f(x) is as close to L as we want whenever x is sufficiently close to c (but does not equal c.) If L exists, we write the limit as x approaches c f(x) = L
Functions: Definition: Increasing function
A function f is increasing if the values of f(x) increases as x increases. The graph of an increasing function climbs as we move from left to right.
Functions: Definition: Monotonic function
A function f is monotonic if it increases for all x or decreases for all x.
Functions: Defintion: Function
A function is rule that takes certain numbers as inputs and assigns to each a definite output number.
Functions: Formula: Linear
A linear function has the form y=mx+b m is the slope, or rate of change of y with respect to x b is the vertical intercept, or the value of y when x is zero.
Powers, Polynomials, and Rational Functions: Definition: Power function
A power function has the form f(x)=kx^p, where k and p are constant.
Random Note #1 Area formula
A=pier^2
Exponential Functions: Notes: Concavity of Exponential functions
All exponential functions are concave up.
Trigonometric Functions: Notes: Amplitude and Period
Amplitude = the absolute value of A Period = 2pie/the absolute value of B f(t)= Asin(Bt) and g(t)=Acos(Bt) The graph of a sinusoidal function is shifted horizontally by a distance absolute value of h when t is replaced by t-h or t+h. Functions of the form f(t)=Asin(Bt) + C and g(t)= Acos(Bt) + C have the graphs which are shifted vertically by C and oscillate about this value.
Trigonometric Functions: Notes: Angle of 1 radian
An angle of 1 radian is defined to be the angle at the center of a unit circle which cuts off an arc length 1, measured counter clockwise.
Exponential Functions: Notes: Exponential Decay
Any exponential decay function can be written, for some 0<a<1 and -k<0, as Q=Qsub0A^t or Q=Qsub0e^-kt Qsub0 being the initial quantity we say that Q is decaying at a continuous rate of k
Exponential Functions: Notes: Exponential Growth
Any exponential growth function can be written, for some a>1 and k>0, in the form P=psub0A^t or P=psub0e^kt Psub0 being the initial quantity we say that P is growing at a continuous rate of k
Trigonometric Functions: Definition: Arc length
Arc length = s= rtheta
Limit: Theorem 1.2: PROPERTIES OF LIMITS
Assuming all the limits on the right-hand side exist: 1. If b is a constant, the the limit as x approaches c (b(f(x)) = b (the limit as x approaches c f(x)). 2.The limit as x approaches c ((f(x)+g(x))= the limit as x approaches c of f(x) + the limit as x approaches c of g(x) 3.The limit as x approaches c (f(x)g(x))= the limit as x approaches c f(x) times the limit as x approaches c g(x). 4.The limit as x approaches c f(x)/g(x) = the limit as x approaches c f(x)/ the limit as x approaches c g(x) 5. For any constant k, the limit as x approaches c (k) = k 6. The limit as x approaches c (x) = c
Derivative: The Derivative at a point: Definition: Average rate of change
Average rate of change of f over the interval from a to a +h = (f(a+h)-f(a))/h
Derivative:How do we measure speed?: Difference quotient
Average velocity formula change in distance over change in time
Trigonometric Functions: Definition: Arcsin
For -1 less than or equal to y which is less than or equal to 1, arc sin y=x means sinx=y with -pie/2 less than or equal to x which is less than or equal to pie/2
Shortcuts to Differentiation: Powers and Polynomials: Rule: The Power Rule
For any constant real number n, d/dx(x^n)=nx^n-1
Trigonometric Functions: Definition: Cosine
Cos t = x
Functions: Discrete vs. Continuous Quantities
Discrete quantities are counted (temperature) whereas Continuous quantities are measured (time). Discrete quantities must be integers, continuous quantities can be any number
Trigonometric Functions: Notes: Domination
EVERY EXPONENTIAL GROWTH FUNCTION DOMINATES EVERY POWER FUNCTION
RANDOM NOTES: GRAVITATIONAL FORCE
F=k/r^2 or F=kr^-2
Derivative: The Second Derivative: Notes: What does it tell us?
If f''(x)>0 on an interval, then f' is increasing, so the graph of f is concave up there. If f''(x)<0 on an interval, then f' is decreasing, so the graph of f is concave up there. If the graph of f is concave up and f'' exists on an interval, then f" is greater than or equal to there. If the graph of f is concave up and f"exists on an interval, then f" is less than or equal to 0 there.
Derivative: The Derivative Function: Notes: What does the derivative tell us graphically?
If f'>0 on an interval, then f is increasing over that interval. If f'<0 on an interval, then f is decreasing over that interval.
Derivative: The Derivative Function: Notes: Derivative of an Exponential Function
If f(x) = x^n then f'(x) = nx^n-1
Derivative: Differentiability: Theorem: A DIFFERENTIABLE FUNCTION IS CONTINUOUS
If f(x) is differentiable at a point x=a, then f(x) is continuous at x=a.
Derivative: The Derivative Function: Notes: Derivative of a constant function
If f(x)= k, then f'(x)=0
Derivative: The Derivative Function: Notes: Derivative of a Linear Function
If f(x)=b + mx, then f'(x)= Slope = m
Derivative:How do we measure speed?: Definition: Average Velocity
If s(t) is the position of an object at time t, then the average velocity of the object over the interval a less than or equal to t which is less than or equal to b is, average velocity= change in position/change in time= (s(b)- s(a))/b-a In words, the average velocity of an object over an interval is the net change in position during the interval divided by the change in time. Average velocity= s(a+h)-s(a)/h
Trigonometric Functions: Definition: Tangent Function
If t is any number where cos t does not equal zero, we define the tangent function as follows: tan t= sin t/ cos t
Trigonometric Functions: Definition: Vertical Asymptote
If the graph of y=f(x) approaches the vertical line x=K as x approaches K from one side or the other, that is, if y approaches infinity or y approaches negative infinity when x approaches K, then the line x=K is called a vertical asymptote.
New Functions from Old: NOTES: Inverse Function Symmetry
If the x- and y- axes have the same scales, the graph of f^-1 is the reflection of the graph of f about the line y=x
Shortcuts to Differentiation:The Product and Quotient Rules: Theorem 3.4: THE QUOTIENT RULE
If u=f(x) and v=g(x) are differentiable, then (f/g)'= (f'g-g'f)/g can also be written d/dx(u/v)=(((du/dx)(v))-((u)(dv/dx))/v^2 In words: The derivative of a quotient is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the denominator squared.
Shortcuts to Differentiation:The Product and Quotient Rules: Theorem 3.3: THE PRODUCT RULE
If u=f(x) and v=g(x) are differentiable, then (fg)'=f'g+fg' can also be written d(uv)/dx= (du/dx(v))+(u)(dv/dx) In words, the derivative of a product is the derivative of the first times the derivative of the second plus the first times the derivative of the second.
Derivative: The Second Derivative: Notes: Interpretations of a function
If y=s(t) is the position of an object at time t, then: -Velocity: v(t)=dy/dx= s'(t) -Acceleration: a(t)=(d^2y)/(dt^2)= s"(t)=v'(t)
Derivative: The Second Derivative: Definition: Instantaneous Acceleration
Instantaneous Acceleration= v'(t)= the limit as h approaches 0 of (v(t+h)-v(t))/h
Exponential Functions: Notes: Rate of growth
Large values of a mean fast growth; values of a near 0 mean fast decay.
Derivative:How do we measure speed?: Definition: Instantaneous Velocity
Let s(t) be the position at time t. Then the instantaneous velocity at t=a is defined as the limit as h approaches 0 of s(a+h)-s(a)/h In words, the instantaneous velocity of an object at time t=a is given by the limit of the average velocity over an interval, as the interval shrinks around a. The instantaneous velocity is the slope of a curve at a point.
Exponential Functions: Notes: Concavity of a line
Neither concave down or concave up
Trigonometric Functions: Notes: Continuity
No breaks, holes, or jumps
New Functions from Old: PROPERTIES OF NATURAL LOGARITHMS
Note the ln x is not defined when x is 0 or negative. 1. ln (AB)= ln A + ln B 2. ln (A/B)= ln A - ln B 3. ln (A^p)= pln(A) 4. ln(e^x)=x 5. e^(lnx)=x Additionally, ln 1=0 because e^0=1
New Functions from Old: PROPERTIES OF LOGARITHMS
Note the log x is not defined when x is 0 or negative. 1. log (AB) = log A + log B 2. log (A/B)= log A - log B 3. log (A^p)= plog(A) 4. log(10^x)=x 5. 10^(logs)=x Additionally, log 1=0 because 10^0=1
Trigonometric Functions: Notes: Power Function Degrees
Quadratic (n=2) Cubic (n=3) Quartic (n=4) Quintic (n=5)
Shortcuts to Differentiation: The Chain Rule: Notes: Intuition Behind the Chain Rule
Rate of change of composite function= rate of change of outside function x rate of change of inside function
Trigonometric Functions: Notes: Phase shift
Shifting sin or cos functions to the left (+) or to the right (-)
Trigonometric Functions: Definition: Sine
Sin t= y
Limit: Note: Limits at infinity
Sometimes we want to know what happens to to f(x) as x gets large, that is, the end behavior of f. If f(x) gets as close to a number L as we please when x gets sufficiently large, then we write the limit as x approaches infinity f(x)= L. Similarly, if f(x) approaches L when x is negative and has a sufficiently large absolute value, then we write the limit as x approaches negative infinity f(x) = L.
Intro to Continuity: Theorem 1.1: INTERMEDIATE VALUE THEOREM
Suppose f is continuous on a closed interval [a,b]. If K is any number between f(a) and f(b), then there is at least one number c in [a,b] such than f(c)=K
Limit: Theorem 1.3:CONTINUITY OF SUMS, PRODUCTS, AND QUOTIENTS OF FUNCTIONS
Suppose that f and g are continuous on an interval and that b is a constant. Then, on that same interval, 1. bf(x) is continuous 2. f(x) + g(x) is continuous 3. f(x)g(x) is continuous 4. f(x)/g(x) is continuous provided g(x) does not equal 0 on the interval.
Derivative:How do we measure speed?: Notes: Average Velocity & how it relates to slope
The average velocity over any time interval a is than or equal to t which is less than or equal to b is the slope of the line joining the points on the graph of s(t) corresponding to t=a and t=b.
Derivative: The Derivative at a point: Definition: Derivative of f at a
The derivative of f at a, written f'(a), is defined as Rate of change of f at a= f'(a)= limit at h approaches 0 of (f(a+h)-f(a))/h. If the limit exists, then f is said to be differentiable. The derivative at point A can be interpreted as: -The slope of the curve at A. -The slope of the tangent line to the curve at A.
Functions: Definition: Domain
The domain is the set of all input numbers
Exponential Functions: Definition: Doubling-time
The doubling time of an exponentially increasing quantity is the time required for the quantity to double.
Limit: Definition: Continuity
The function f is continuous at x =c if f is defined at x = c and if the limit as x approaches c f(x)= f(c) In words, f (x) is as close as we want to f (c) provided x is close enough to c. The function is continuous on an interval [a,b] if it is continuous at every point in the interval.
Derivative: Differentiability: Definition: Differentiable
The function f is differentiable at x if the limit as x approaches zero of f(x+h)-f(x)/h exists. Thus, the graph of f has a non vertical tangent line at x. The value of the limit and the slope of the tangent line are the derivative of f at x.
Exponential Functions: Definition: Concave down
The graph of a function is concave down if it bends downwards from left to right.
Exponential Functions: Definition: Concave up
The graph of a function is concave up if it bends upwards as we move left to right.
New Functions from Old: Definition: Asymptotes of Logs and Natural Logs
The graph of y= logs has a vertical asymptote at x=o, whereas y=10x has a horizontal asymptote at y=0 One big difference b/w the graph of a log vs a natural logs is that logs grow extremely quickly, whereas natural logs grow extremely slowly. The graphs of logs and natural logs are reflections of one another about the line y=x.
Exponential Functions: Definition: Half-life
The half-life of an exponentially decaying quantity is the time required for the quantity to be reduced by a factor of one half.
Logarithmic Functions: Definition: Logarithm
The logarithm to base 10 of x, written logsub10x, is the power of 10 we need to get x. In other words, logsub10x=c means 10^c=x we often write logx in place go logsub10x
New Functions from Old: Definition: Natural Logarithm
The natural logarithm of x, written ln x, is the power of e needed to get x. In other words ln x = c means e^c=x
Functions: Definition: Closed Interval
The set of numbers t such that a is less than or each to t and t is less than or equal to b is called a closed interval and written [a,b]
Functions: Definition: Open Interval
The set of numbers t such that a is less than t and t is less than b is called an open interval and written (a,b)
Functions: Definition: Function
The set of resulting output numbers
Exponential Functions: Notes: Recognizing an exponential function
To recognize that a table of t and P values comes from an exponential function, look for ratios of P values that are constant for equally spaced t values.
Functions: Notes: Linear Functions
To recognize that a table of x and y values comes from a linear function, y=b + mx, look for differences in y-values that are constant for equally spaced x values.
Trigonometric Functions: Definition: Horizontal Asymptote
Trigonometric Functions: Definition:If the graph of y=f(x) approaches a horizontal line y= L as x approaches infinity or x approaches negative infinity, then the line y= L is called a horizontal asymptote. This occurs when f(x) approaches L as x approaches infinity or f(x) approaches L as x approaches negative infinity.
RANDOM NOTES: VOLUME OF A SPHERE
V=g(r)=4/3pier^3
Function: Definition: Inverse Proportionality
We also say that one quantity is inversely proportional to another if one is proportional to the reciprocal of the other. i.e. 1/t
New Functions from Old: Notes: Composite function
We are thinking of a composite function as a function of a function, which is written as y=f(g(x))
Trigonometric Functions: Notes: Radians and Angles
We assume that the angles are always in radians unless specified otherwise.
Limit: Definition: Limit (Restated formal)
We define the limit as x approaches f(x) to be the number L (if one exists) such that for every epsilon greater than 0 (as small as we want), there is a delta greater than 0 (sufficiently small) such that if the absolute value of x - c is less than delta and x does not equal c, then the absolute value of f(x) - L is less than epsilon.
Exponential Functions: Definition: General Exponential Functions
We say P is an exponential function of t with base a if P=Psub0A^t Where psub0 is the initial quantity (when t=0) and a is a factor by which P changes when t increases by 1. If a > 1, we have exponential growth If a < 1 we have exponential decay
Function: Definition: Proportionality
We say y is (directly) proportional to x if there is a nonzero constant k such that y=kx The k is called the constant of proportionality
Limits: Notation: Limit
We write the limit as x approaches c f(x) = L if the values of f(x) approach L as x approaches c.
Limit: Note: When limits do not exist
Whenever there is no number L such that the limit as x approaches c f(x) = L, we say that the limit as x approaches c f(x) does not exist.
Shortcuts to Differentiation: The Exponential Function: Rule: a^x
d/dx(a^x)=ln(a)a^x
Shortcuts to Differentiation: The Exponential Function: Rule: e^x
d/dx(e^x)=e^x
Functions: The output is called the...
dependent variable
Shortcuts to Differentiation: The Chain Rule: Rule: The Chain Rule
dy/dx=(dy/dz)(dz/dx)
Derivative: Interpretations of Derivative
f'(x)= dy/dx
New Functions from Old: Definition: Inverse Function
f^-1(y)= x mean y=f(x) A function has an inverse if and only if its graph intersects any horizontal line at most once.
Functions: The input is called the...
independent variable
Functions: Formula: Slope
m= rise/run = y2-y1/x2-x1
Trigonometric Functions: Equation: Unit Circle
x^2 + y^2 = 1