Calc Medic - Calculus Quick Notes
Unit 1: Exploring Rates of Change
Functions and Function Notation
Functions are crucial for modeling relationships between variables, a key skill assessed on the AP exam.
Understanding domain and range is essential for contextual problems.
Function notation is used extensively; be prepared to interpret and manipulate functions.
Graphs of Functions
Visual analysis of graphs to determine increasing, decreasing intervals, and key points is fundamental.
Connect graph behavior to real-world scenarios.
Concavity
Understand how the sign of a graph's slope indicates increasing or decreasing behavior.
Concavity relates directly to the rate of change; interpret its meaning in context.
Concave up: Rate of change is increasing, often related to acceleration.
Concave down: Rate of change is decreasing, often related to deceleration.
Rates of Change
Calculate and interpret the average rate of change over an interval; this is a common AP question.
Average Rate of Change = \frac{\Delta y}{\Delta x} = \frac{f(x2) - f(x1)}{x2 - x1}
Estimate and interpret instantaneous rates of change.
Compare rates of change across different intervals.
Change in Functions
Linear Functions
Recognize that linear functions have a constant rate of change.
Interpret slope as a rate of change in applied problems.
f(x) = mx + b, where m is the constant rate of change (slope).
Quadratic Functions
Know that for quadratic functions, the change in output values over equal intervals grows linearly.
Understand the relationship between the concavity of a parabola and increasing/decreasing average rates of change.
f(x) = ax^2 + bx + c, where the average rates of change vary linearly.
Unit 2: Polynomial and Rational Functions
Polynomial Functions and Rates of Change
Be familiar with the key characteristics of polynomial functions:
Degree
Leading coefficient
Relative and absolute extrema (maxima and minima)
Points of inflection (where concavity changes)
The degree of a polynomial can be determined using finite differences.
Zeros of Polynomial Functions
Understand how a root's multiplicity affects the graph at an x-intercept:
Even multiplicity: The graph touches the x-axis but does not cross.
Odd multiplicity: The graph crosses the x-axis.
A polynomial of degree n has exactly n complex zeros and can be written as a product of n linear factors.
P(x) = a(x - r1)(x - r2)…(x - rn), where ri are the zeros.
Find all zeros of a polynomial function when given in factored form.
Imaginary zeros occur when the polynomial's graph does not intersect the x-axis; these are important for complete analysis.
Even and Odd Polynomials
Even functions: Symmetrical about the y-axis; f(-x) = f(x).
Odd functions: Symmetrical about the origin; f(-x) = -f(x).
Algebraically prove whether polynomial functions are even, odd, or neither.
Polynomial Functions and End Behavior
Determine the end behavior of a polynomial based on its degree and leading coefficient:
Even degree, positive leading coefficient: Both ends go up.
Even degree, negative leading coefficient: Both ends go down.
Odd degree, positive leading coefficient: Left end goes down, right end goes up.
Odd degree, negative leading coefficient: Left end goes up, right end goes down.
The end behavior is determined by its leading term.
Use limit notation to describe the end behavior:
\lim
x \to \infty} P(x) = \pm \infty\lim
x \to -\infty} P(x) = \pm \infty
Rational Functions and End Behavior
Interpret the behavior of a rational function within a given context, focusing on horizontal asymptotes.
Determine end behavior by comparing the degrees of the numerator and denominator:
Degree of numerator < degree of denominator: Horizontal asymptote at y = 0.
Degree of numerator = degree of denominator: Horizontal asymptote at y = \frac{leading coefficient of numerator}{leading coefficient of denominator}.
Degree of numerator > degree of denominator: No horizontal asymptote (may have a slant asymptote).
The end behavior is determined by the quotient of leading terms.
Graphs of Rational Functions
Key features of a rational function for AP exam success:
Domain
Intercepts
Holes
Vertical asymptotes
Identify these features from both the graph and the equation in factored form.
Use one-sided limit notation to describe behavior near vertical asymptotes:
\lim
x \to a^-} R(x) = \pm \infty\lim
x \to a^+} R(x) = \pm \infty
Determine the y-value of a hole by examining function outputs close to the x-value of the hole.
Factored and Standard Forms of Polynomials
Advantages of factored form: Easy to identify zeros.
Advantages of standard form: Easy to evaluate for specific values of x.
Convert polynomials from factored to standard forms and vice versa fluently.
When (x - k) is a factor, x = k is a zero.
Find all zeros of a polynomial function by hand or using technology.
Equivalent Representations of Rational Functions
Extend concepts of factors, dividends, divisors, quotients, and remainders to functions.
Divide polynomials using an area model.
Rewrite a rational function to reveal different characteristics, including slant asymptotes.
The Binomial Theorem
Generalize patterns for the expansion of binomials and connect to Pascal's triangle.
Expand binomial expressions using the Binomial Theorem:
(a + b)^n = \sum
k=0}^{n} {n \choose k} a^{n-k} b^k where {n \choose k} = \frac{n!}{k!(n-k)!}
Unit 3: Constructing Functions
A Library of Parent Functions
Understand parent functions as the most basic in a function family.
Key features of six parent functions:
Identity: f(x) = x
Absolute value: f(x) = |x|
Square root: f(x) = \sqrt{x}
Quadratic: f(x) = x^2
Cubic: f(x) = x^3
Reciprocal: f(x) = \frac{1}{x}
Analyze and compare the key features of parent functions.
Transformations of Functions
Construct a new function by applying translations, dilations, and reflections to a parent function.
Describe transformations from an equation or graph.
Determine the domain and range of a transformed function.
Piecewise Functions
Interpret and evaluate functions that have different rules for certain domain intervals.
Graph piecewise-defined functions.
Write equations for piecewise-defined functions from a graph or context.
Selecting a Function Model
Identify the appropriate function type based on observations about how quantities are changing.
Describe assumptions and restrictions related to a particular function model.
Constructing a Function Model
Construct a function model based on mathematical or contextual constraints.
Construct a function model using transformations from a parent function.
Use rational functions to model inversely proportional quantities.
Apply a function model to answer questions about a data set or scenario.
Unit 4: Exponential Functions
Change in Arithmetic Sequences
Sequences are functions with a domain of positive integers.
Write an explicit rule for arithmetic sequences using the common difference:
an = a1 + (n - 1)d, where d is the common difference.
Apply understanding of arithmetic sequences to determine the common difference and find missing terms.
Change in Geometric Sequences
Write an explicit rule for geometric sequences using the common ratio:
an = a1 * r^(n-1), where r is the common ratio.
Apply understanding of geometric sequences and exponents/roots to determine the common ratio and find missing terms.
Compare arithmetic and geometric sequences.
Change in Linear and Exponential Functions
Create linear and exponential functions using constant rates of change and constant proportions:
Linear: f(x) = mx + b, constant rate of change m.
Exponential: f(x) = a * b^x, constant proportion b.
Interpret the parameters of linear and exponential functions in context and describe growth patterns.
Describe similarities and differences between linear and exponential functions.
Exponential Functions
Recognize exponential growth or decay scenarios by identifying a fixed percent change or common ratio.
Write equations of the form y = ab^x to model growth or decay scenarios.
Graphing and Manipulating Exponential Functions
Graph functions of the form y = ab^x and identify key characteristics, including end behavior, concavity, domain/range, asymptotes, and intercepts.
Determine the growth factor from the graph of an exponential function, including transformed functions.
Apply knowledge of transformations to exponential functions.
Explain why two exponential functions are equivalent using exponent properties and transformations.
Modeling with the Natural Base, “e”
Describe the effects of compounding interest and derive the compound interest formula.
A = P(1 + \frac{r}{n})^{nt}, where A is the final amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the time in years.
Continuously compounded interest: A = Pe^{rt}.
Use an exponential model to make predictions about the dependent variable.
Understand "e" as the base rate of growth for all continually growing processes.
Constructing Exponential Models
Construct exponential models from an initial value and ratio or from two input-output pairs.
Use an exponential model to make predictions about the dependent variable.
Understand how equivalent forms of an exponential function can reveal different properties about its growth rate.
Using Regression Models
Use data set characteristics to decide whether a linear, quadratic, or exponential model is most appropriate.
Create a regression model for a scenario using technology.
Use a residual plot to validate whether a given model was appropriate.
Unit 5: Logarithmic Functions
Compositions of Functions
When two functions are composed, the output of one becomes the input of the second.
Write equations for compositions of functions:
(f \circ g)(x) = f(g(x))
Decompose a complicated function into a composite of two or more functions.
Reason about the domain of a composition of functions.
Intro to Inverse Functions
Solve equations of the form f(x) = c to recognize the need for a function that "undoes" the original function.
Understand the relationship between inputs and outputs of a function and its inverse; use this to evaluate inverse functions.
Find an inverse function algebraically.
Verify that one function is the inverse of another by composition:
f(f^{-1}(x)) = x and f^{-1}(f(x)) = x
Graphs of Inverse Functions
Understand why a function must be one-to-one (invertible) for the inverse mapping to be a function.
Explore relationships between the graph of a function and its inverse, including their domains and ranges.
The graph of the inverse function is a reflection of the original function across the line y = x.
Inverses of Exponential Functions
Understand that a logarithm represents the exponent to which the base must be raised to attain the input value; use this to evaluate logarithmic expressions.
y = log_b(x) is equivalent to b^y = x
Use exponential and logarithmic forms to write equivalent statements about powers.
Understand the inverse relationship between how inputs and outputs change in exponential versus logarithmic functions.
Understand the inverse relationship between exponential and logarithmic functions of the same base, including the natural base, e.
Graphs of Logarithmic Functions
Describe key features (domain, range, asymptotes, concavity, and end behavior) of the graph of a parent logarithmic function, y = log_b x.
Sketch parent logarithmic functions and their transformations.
Connect key features on the graphs of exponential and logarithmic functions.
Logarithm Properties
Discover and use the sum, difference, and power properties of logarithms to rewrite logarithmic expressions.
logb(mn) = logb(m) + log_b(n)
logb(\frac{m}{n}) = logb(m) - log_b(n)
log*b(m^p) = p * log*b(m)
Explain why two logarithmic functions are equivalent using logarithm properties and transformations.
Solving Exponential and Logarithmic Equations
Solve exponential and logarithmic equations by rewriting them in equivalent forms using properties.
Identify and exclude extraneous solutions.
Write equations for inverse functions by applying inverse operations.
Modeling with Logarithmic Functions
Understand that a logarithmic model takes quantities that grow proportionally and assigns them output values that grow linearly.
Identify situations that could be modeled with a logarithmic function.
Construct logarithmic models using input-output pairs or transformations.
Use logarithmic function models to predict values of the dependent variable.
Semi-log Plots
Linearize quantities exhibiting exponential growth or decay using a log transformation.
Interpret the parameters of exponential regression models and their associated linear regression models after a log transformation.
Unit 6: Exploring Sine and Cosine Functions
Periodic Phenomena
Identify periodic relationships between variables and construct their graphs.
Describe key features of a periodic function:
Period
Amplitude
Midline.
Angles on the Coordinate Plane
Understand how to measure angles in standard position and their properties.
Understand that a radian is an angle measure with an arc length of one radius.
2\pi radians = 360 degrees.
Label the angles on the unit circle in radians using proportional reasoning.
Defining Sine, Cosine, and Tangent for Any Angle
Extend the definition of sine, cosine, and tangent ratios to angles greater than 90° using the coordinate plane.
Understand why in a unit circle, sine and cosine ratios correspond with the y-value and x-value, respectively:
sin(\theta) = y
cos(\theta) = x
Understand that in a unit circle, the tangent of an angle is the ratio of the y-coordinate to the x-coordinate:
tan(\theta) = \frac{y}{x} = \frac{sin(\theta)}{cos(\theta)}
Use symmetry to identify relationships between the sine, cosine, and tangent values of angles in all four quadrants.
Coordinates on the Unit Circle
Use special right triangles to determine the coordinates at key points on the unit circle.
30-60-90 triangle and 45-45-90 triangle.
Evaluate sine, cosine, and tangent for key angles on the unit circle.
Find coordinates of points on circles where r \neq 1.
Graphs of Sine and Cosine
Construct graphs of the sine and cosine functions using values from the unit circle.
Identify key characteristics for the parent functions y = sin x and y = cos x, including domain, amplitude, midline, period, and symmetry.
Transformations of Sine and Cosine
Determine how the amplitude, period, domain, range, and midline of sinusoidal functions are affected by transformations:
y = A sin(B(x - C)) + D
Amplitude: |A|
Period: \frac{2\pi}{|B|}
Phase shift: C
Vertical shift: D
Graph transformed sine and cosine functions given an equation.
Modeling with Trigonometric Functions
Interpret a sinusoidal function's period, amplitude, midline, and range in context.
Construct a trigonometric model based on data points and key features.
Unit 7: Working with Trigonometric Functions
The Tangent Function
Understand that the tangent of an angle is determined by the slope of the terminal ray and use this to understand the behavior of the tangent function.
Describe key features of the graph of the tangent function, including its domain, range, x-intercepts, and period.
Identify how the graph of the parent tangent function is affected by transformations.
Inverse Trig Functions
Understand that inverse trigonometric functions input ratios and output angles.
Explain why and how the domains of sine, cosine, and tangent must be restricted to create an inverse function.
Evaluate inverse trig expressions.
Trigonometric Equations and Inequalities
Extend the process of inverse operations to trigonometric equations and inequalities.
Understand that using the unit circle gives infinite solutions while inverse trig functions give only one solution; expand using symmetry.
The Secant, Cosecant, and Cotangent Functions
Define the secant, cosecant, and cotangent functions as the reciprocal of the cosine, sine, and tangent functions, respectively:
sec(\theta) = \frac{1}{cos(\theta)}
csc(\theta) = \frac{1}{sin(\theta)}
cot(\theta) = \frac{1}{tan(\theta)}
Understand how the zeros, vertical asymptotes, and range are related for a trigonometric function and its reciprocal function.
Trigonometric Relationships
Explore relationships between all six trigonometric functions, including the Pythagorean identities:
sin^2(\theta) + cos^2(\theta) = 1
1 + tan^2(\theta) = sec^2(\theta)
1 + cot^2(\theta) = csc^2(\theta)
Use identities to establish and verify other trigonometric relationships and solve trigonometric equations.
Angle Sum Identities
Find exact values for the sine and cosine of angles not on the unit circle by writing the angle as a sum or difference of known angles:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
$$cos(a + b) = cos(a)cos(