AP Calculus AB Concepts

Average rate of change (2)

(F(b)-F(a))/(b-a) slope of the secant line

E^x function properties

1. Unless e is negative, it will always have a positive function value (Ie. E^-100= (+)) 2. E^0=1 3. E= 2.7... 4. Strictly increasing unless e is neg 5. Domain= all real numbers, range= y>0 6. Asymptotic (x->-∞), increases w/o bound (x->∞) 7. Continuous 8. Inverse of lnx

Limit Definition (various representations) (1)

1. f'(a)= limit h->0 [(f(a+h)-f(a)]/h 2. f'(a)= limit h->0 [f(x)-f(a)]/(x-a)

Applying understandings of differentiation to problems involving motion (4)

Acceleration is the derivative of velocity is the derivative of position.

Solving accumulation problems steps (8)

Amount gained between [a,b]= integral (a->b) f(x)dx Amount total at b given a= initial amount at a+ integral (a->b) f(x)dx

Sketching slope fields (7)

Plug in a coordinate into the dy/dx equation and sketch at each coordinate. When trying to trace one, estimate which route it would take. Does not need to be perfect for AP credit points.

How to take derivatives steps (2)

Power rule: d/dx[x^n]=n•x^(n-1) Product rule: (a•b)= a'b+ab' Quotient rule: low d high minus high d low over low low

Generalizing understandings of motion problems to other situations involving rates of change (4)

Speed: if the acceleration and velocity have the same sign at a given point, it is speeding up. If the acceleration and velocity have different signs at a given point, it is slowing down.

Determining volume with cross-sections steps and formulas (8)

Square: side squared Rectangle: base (integral)•height Triangle: 1/2 base•height Equilateral triangle: √3/4 side squared Semicircle: π/8 side squared Integrate (a->b) formula dx

How to use chain rule steps and its definition (3)

chain rule: a formula used to find the derivative of a composite function 1. Derivative of the outer most property while maintaining the inner parts 2. Multiply by the derivative of inner parts, continue step 1 if necessary

When does a function have a vertical asymptote in regard to its limit (1)

f has a vertical asymptote at x=c iff limx->c f(x)= ∞ or -∞ (includes one sided limits). When solving a function, if it =b/0, factor top and bottom and any terms can be simplified in N and D, there is a vert asymptote at the value (ie, (x-1)/(x-1), VA @x=1)

How to prove continuity in a closed interval (1)

f is continuous in (a,b) if: 1. Limit x->a+ f(x)= f(a) 2. Limit x->b- f(x)= f(b) Essentially, check the endpoints on the side that connects to the rest of the function.

Accumulation functions formula and when to use (6)

An accumulation function is a function defined using a definite integral in order to find the accumulated change over the time interval. F(c)=integral (a,b) f(x)dx/accumulated amount Total amount: f(c)= initial amount when x=a + accumulated amount over (a,b)

What are Antiderivatives (6)

An antiderivative is a function whose derivative is the original function. Whereas, the integral is the sum of the entire quantity.

Determining the average value of a function using definite integrals (8)

Average value (y-value) where when multiplied by its x-value counterpart, it is equivalent to the integral Formula: f(c)=1/b-a• integral (a->b) f(x) dx Example: find the average velocity, function in the integral should be the velocity function

Definition of continuity of a function at a point and over a domain (1)

Continuity: A function is said to be continuous in a given interval if there is no break in the graph of the function in the entire interval range. (Can be traced without lifting pencil). Therefore, the limit must equal the function value in order to prove continuity. Note: Continuity allows you to figure out the limit at a certain point without a graph or table.

Derivative of e rules

D/dx e^u= u'e^u D/dx a^u= u'•a^u•ln(a)

Derivative of ln and log

D/dx lnu= u'/u D/dx log (base a) u= u'/ulna

Derivative of arcsin(x) and arccos(x)

D/dx of arcsin/cos(X)= 1/√1-x^2 It is negative for cosine

Derivative of arcsec(x) and arccsc(x)

D/dx sec/csc(x)= 1/abs(x)√1-x^2 It is negative for cosecant

Derivative of arccot(x) and arctan(x)

D/dx tan/cot(x)= 1/1+x^2 It is negative for cotangent

How to solve definite integrals (6)

Definite integrals: find the area between a function's curve and the x-axis on a specific interval. Outputs a real number because it involves an upper and lower limits. 1. Find the indefinite integral (rules of integration), techniques include: integration by trig and u substitution methods 2. Find F(b), plug upper limit into the F(x) 3. Find F(a), plug lower limit into F(x) 4. Take the difference F(b)-F(a)

Properties of integrals (6)

Definition: (both boundaries are the same)=0, (boundaries go from right to left)=switch sign Constant multiple: move the constant out of the integral Sum and Difference: you can split up integral's contents if it is a + or - Additivity: if the boundaries connect, you can add the integrals Integrals of symmetric functions: f is even (ie parabola) then either side of the axis of symmetry is identical (times 2) If f is odd, then either side of the axis of symmetry is opposite (=0)

Deriving and applying a model for exponential growth and decay formula and steps(7)

Exponential growth: f(x)=a(1+r)^x, a=initial amt, r=growth rate, x=number of time intervals Exponential decay: f(x)=a(1-r)^x or N(t)=Noe^-kt Rate of population change: rN(1-N/K), r=max population growth rate, N=population size, K=population carryig cap

First Fundamental Theorem of Calculus steps (6)

F is any function that satisfies F'(x)=f(x) if f is continuous on [a,b] Essentially, antiderivatives exist. D/dx integral (a->x) f(x)dx= f(x)

How to use the first derivative test (5)

First Derivative Test: states that if we are given a continuous and differentiable function f, and c is a critical number(f'(x)=0) of function f, then f(c) can be classified as follows: if f'(x) changes from negative to positive, then f(c) is a relative minimum. Postive to negative, then f(c) is a relative maximum. Positive=increasing, negative=decreasing. 1. Take the derivative 2. Set it to zero and find the zeros 3. Put the zeros on a number line 4. Find the value (+ or -) between each zero 5. Determine if it is a rel min or max

Extreme Value Theorem for derivatives and steps (5)

If f is continuous on a closed interval (a,b), then f attains both an absolute maximum value and an absolute minimum value at some numbers in [a,b]. This tends to require the candidate's test to solve.

Sketching graphs of functions and their derivatives (5)

If the function has a positive slope, then its derivative is graphed above the x-axis. If the function has a negative slope, then its derivative is graphed below the x-axis. If the function is concave up, then its derivative is increasing. If the function is concave down, then its derivative is decreasing.

How to find if a piecewise is continuous at a certain point steps and reasoning (2)

If the left-hand limit, right-hand limit, (and the function value) at x=c exist and are equal to each other then f is said to be continuous at x=c. 1. Find the function value 2. Find the limit as x->c from either side 3. Check if they equal each other 4. Write justification: limit x->c- f(x)=f(c)= limit x->c+ f(x)

Behaviors of implicit relations (5)

Implicit funcitons: type of funciton that is not defined in a usual manner (multiple variables on the same side) Critical points on implicit functions: differentiate both sides in respect to x, solve for dy/dx, set dy/dx=0, use first derivative test for each critical point Points of inflection on implicit functions: differentiate the derivative with respect to x, may need to incorporate dy/dx to solve, set it equal to 0 to find POI, use second derivative test for each POI

How to solve indefinite integrals (6)

Indefinite integrals: find the antiderivative of a function, gives a function accompanied by an arbitrary constant C (no upper and lower boundaries) Meant for question like "What function, when differentiated, gives us f(x)?" 1. Take the antiderivative 2. Add +C 3. If you have to take the derivative of the integral, and a function is one of the boundaries, first plug it into the function within the integral and then take the derivative of the limit

Approximating integrals using Riemann Sums steps (6)

Integral from [a,b] f(x)dx= lim n->∞ n(sigma)i=1 f(xi)∆x ∆x=b-a/n (number of shapes), xi=a+i∆x Left riemann sum: use left y value Right riemann sum: use right y value Midpoint riemann sum: find the midpoint between the two x values, take the midpoints y value Trapezoidal riemann sum: find the midpoint between the two x value's function/y values Width(height)+width2(height2)+etc

Using definite integrals to determine accumulated change over an interval (6)

Integral from [a,b] of f'(x) dx= f(b)-f(a) It is equivalent to the area under the curve (f'x) from a to b

Optimization steps

1) identify all quantities and draw a sketch 2) primary equation and secondary equation (constraints) 3) reduce primary equation to only one variable by manipulating secondary equation 4) plug secondary equation into primary equation 5) find the derivative of that one single variable 6) get the variable and plug back in for other variable 7) make sure you answered the question in right units, etc.

Indeterminate forms of Limits (1)

1. 0/0 2. ∞/∞ 3. 0(∞) 4. 0^0 5. ∞^0 6. ∞-∞ 7. 1^∞ Indeterminite forms require further steps/rearrangement/simplification to find the limit

Integration by trig substitution (6)

1. Characterize the radical as "u^2+a^2", "u^2-a^2", and "a^2-u^2" where a=constant, u=trig 2. Find the equivalent trig identity (ie. 1-cos^2x=sin^2x is equivalent to a^2-u^2) 3. Multiply trig identity by any constants necessary 4. Set u^2 equal to its equivalent and solve for u 5. Differentiate and solve for dx in terms of du 6. Plug 5. In and simplify and then solve

Solving related rates problems steps (4)

1. Decide what the two variables are 2. Find an equation relating them (ie: dx/dt=dx/dh•dh/dt) 3. Take d/dx of both sides(x being any variable that the problem asks for, t is common) 4. Plug in all known values at the instant in question 5. Solve for the unknown rate

Implicit differentiation steps (3)

1. Differentiate both sides of the equation with respect to "x" 2. When taking the derivative of any term that has a "y" in it multiply the term by y' (or dy/dx) 3. Solve for y'

L'Hospital's Rule steps (4)

1. Direct substitution of the limit 2. Prove it becomes an indeterminate form 3. Set the previous limit to a new limit where you take the derivative of the N and D separately 4. Direct substitution again 5. Simplify

How to solve optimization problems (5)

1. Draw and label a picture that represents the problem 2. Find the objective function/equation for the drawing 3. Identify the constraints (relate the multiple variables to each other) 4. Solve for all unknown variables other than the one the equation is asking to be optimized 5. Reduce the objective function to one variable 6. Differentiate the objective function 7. Find the critical numbers and possible endpoints by setting the derivative to 0 8. Answer the question using the value found

Limit Properties (1)

1. Limit x->a (c•f(x))=c• limit x->a (f(x)), constant coefficients can be taken out of the limit 2. Limit x->a (f(x)+g(x))= limit x->a (f(x))+limit x->a (g(x)), multiple functions in one limit can be split up but still use the four major functions 3. Limit x->a [f(x)]^h= (limit x->a f(x))^h, the function's exponents can be applied to the entire limit

Ln(x) function properties

1. Ln (xy)= lnx+lny 2. Ln (x/y)= lnx-lny 3. Ln (x^y)= y•ln(x) 4. Ln(x) is undefined when x≤0 5. Ln(0)=undefined 5. Lim x->0+ ln(x)=-∞ 6. Ln(1)=0 7. Lim x->∞, ln(x)=∞

How to find if a piecewise is differentiable at a certain point steps and reasoning (2)

1. Must prove continuity first 2. Left-hand derivative must equal the right-hand derivative at each connection

Finding the area between curves steps (8)

1. Plug into calculator to find top/right function, you can also plug in a value in the middle to compare 2. Find the endpoints/where they intersect based on what the question asks (x or y) 3. Make your integral: integral (a->b) top/right-bottom/left dx 4. Solve!

Solving separable differential equations to find general and particular solutions (7)

1. Separate both variables on either side of the equals sign 2. Integrate both sides, make sure +C is only on the x side 3. Plug in given coordinate to find the constant of integration 4. Solve for y/equation General solutions end at +C, no need to plug in coordinate, just isolate y completely

Squeeze Theorem Steps to solve (1)

1. Set the main trig function inbetween its limits (usually -1 to 1) 2. Multiply, divide, add, subtract any remainder parts 3. Take the limit that the question asks of each of the 3 parts 4. Prove that both end's limits equal each other

Local linearity and approximation steps (4)

1. Take the derivative at the x value provided in the coordinate 2. Plug that slope into point-slope form: y-y1=m(x-x1) 3. Simplify 4. Plug in the x value needed to be approximated 5. Solve for y

Lim x->0 1-cosx/x

=0

lim x->0 sinx/x

=1

Integration by U Substitution (6)

Like in chain rule of derivatives, using "u" finding "du", spotting it in the integral, replacing it and solving. After taking the integral, re-substitute "u" U Sub 1. Substitute: changing the integral from a function of x to a function of u. 2. Differentiate to find dx in terms of du 3. Plug it in and simplify where necessary 4. Change the upper and inner limits using 1. Equation 5. Integrate 6. Solve using second fundamental theorem of calc

Asymptotes and limits at infinity steps (1)

Limits to infinity: when a function f increases or decreases without bound as x approaches a target value c. Appropriate notation: lim x->c f(x)= ∞, lim x-> f(x)=-∞, if one side goes to ∞ and the other to -∞ then limit=DNE

How do limits help us handle change at an instant and steps to take a limit? (1)

Limits: given a function f, a fixed input x=a, and a real number L, we say nthat f has a limit L as x approaches a provided that we can make f(x) as close to L as we like by taking x sufficiently close to (but not equal to) a. Essentially the value the function approaches at it goes towards "a". 1. Direct substitution (plug in the variable provided) 2. If you get b/0, where b is a constant, it is an asymptote and doesn't need further simplification 3. If you get an indeterminate form (ie. 0/0), rearrange (simplify, factor, etc) 4. Direct substitute again and voila:)

Interpreting verbal descriptions of change as separable differentiable equations (7)

Look for Rate In and Rate out. Make sure to label them and their units on the paper and in the calculator. Also be sure to check for initial amount and boundaries for each limit (sometimes doesn't start at 0 for example)

Mean Value Theorem for derivatives (5)

Mean Value theorem: as long as the function is continuous and differentiable in the interval, then there must be a line tangent to the curve somewhere in the interval, which is parallel to the secant line that connect the endpoints. In other words, instantaneous rate of change at x=c equals the average rate of change over a->b.

Squeeze Theorem and Reasoning (1)

Squeeze Theorem: if g(x)≤f(x)≤h(x) for all x≠c in some interval about c, and lim x->c g(x)= lim x->c h(x)=L, then, lim x->c f(x)=L In other words, if we "sandwich" the function f(x) between two other functions g(x) and h(x) that both have the same limit as x->c, then f(x) is forced to have the same limit too Typically used with endpoint limits of cosine and sine graphs.

Tanx and arctan x properties (1)

Tangent: - range: -∞ to ∞ - domain: -π/2 to π/2 Arctangent: - range: -π/2 to π/2 - domain: -∞ to ∞

What are definite integrals (6)

The definite integral is an integral to be evaluated between a lower limit "a" and an upper limit "b". To find the answer, it tends to require the first derivative test.

Define the derivative of a function at a point and as a function (2)

The derivative of a function at a point: equal to the slope of the tangent line to the graph of the function at that point. The derivative gives the instantaneous rate of change of the function at that point. The derivative of a funciton: equal to the instantaneous rate of change at any point on the graph.

Differentiation of general and particular inverse functions steps (3)

The derivative of an inverse function: (f^-1)'(x)= 1/f'(f^-1(x)) Reciprical of the derivative of the function when x equals the inverse's y-value 1. Make two coordinate pairs (one for the function and one for the inverse) 2. For the x value they give you, put that as the x value of the inverse and the y value of the function 3. Look at the table or solve for at what x value the functions y value equals the provided number 4. Use that number to fill out the functions x value and the inverse's y value 5. Take the inverse of the functions slope at its x value

Discontinuity definition (1)

The function will be discontinuous at x=a if: 1. F(c) is not defined 2. Limits from both, right and left, sides are not equal 3. Limits exist and equal each other but not the function value

Intermediate Value Theorem steps and Reasoning (1)

The intermediate value theorem (IVT): states that if a function f(x) is continuous over an interval [a, b], then the function takes on every value between f(a) and f(b). 1. Find f(a) and f(b) 2. IVT guarantees that all values between f(a) and f(b) if the function is continuous and a<c<b, then all y values between f(a) and f(b) are guaranteed to be "hit" at least once

What is the connection between continuity and differentiation? (2)

The relationship between continuous functions and differentiability is-- all differentiable functions are continuous but not all continuous functions are differentiable. (Ie. Vertical tangent lines or cusps/corners)

Second Fundamental Theorem of Calculus steps (6)

The second part of the fundamental theorem of calculus tells us that to find the definite integral of a function ƒ from 𝘢 to 𝘣, we need to take an antiderivative of ƒ, call it 𝘍, and calculate 𝘍(𝘣)-𝘍(𝘢). Integral (a,b) f(x)dx= F(b)-F(a) Equivalent to the area under f(x) between a and b

Applying differentiation in context (2)

To explain the meaning of a derivative you need to mention: 1. Specific time with units 2. Noun with units 3. Whether its increasing or decreasing "At t=—units, the — of — is (inc/dec) at a rate of — units."

How to use the second derivative test (5)

To find relative extrema 1. Take the second derivative 2. Plug in critical points 3. Positive=minimum, negative=maximum To find points of concavity 1. Take the second derivative 2. Set it equal to 0 and find the zeros (Points of Inflection) 3. Place the zeros on a number line 4. Find if the values in between are positive or negative by plugging it into the second derivative 5. If it goes from postive to negative or negative to positive, there is a point of concavity where concavity changes

Determining higher-order derivatives of functions steps (3)

To find the higher-order derivative, take the derivative multiple times. Find the derivative. To find the n-th derivative, look at how many apostrophes are included in the question and repeat the process that many times. Simplify the final expression if possible.

Types of discontinuities (1)

Types of discontinuities 1. Removable discontinuity: f(a) is not defined or not equal to its limits (ie. Holes) 2. Jump discontinuity: left-hand and right-hand limit for a function exists, but they are not equal to each other (ie. Some types of piece wise functions) 3. Infinite discontinuity: function diverges at x=a to give a discontinuous nature. In other words, f(a) is not defined as it does not approach a finite value. Therefore, a limit does not exist (ie. Vert asymptotes)

How to use the candidates' test and when to use it (5)

Used for finding absolute extrema and requires endpoints. 1. Take the derivative 2. Find the zeros 3. Put the zeros and endpoints into a table 4. Plug into main function 5. Largest=absolute max, smallest=absolute minimum

Determining volume using washer method steps and formula (8)

When to use: one function around one axis Π•integral(a->b) (top-bottom)^2 dx For around y-axis: switch variables in function, solve for y, plug into calculator, determine end points, create integral (top-bottom) OR integral (bottom y boundary->top y boundary) right-left

Determining volume using disc method steps and formula (8)

When to use: two functions around one axis Π integral (a->b) ((top-axis)^2-(bottom-axis)^2)dx

Identifying relevant mathematical information in verbal representations of real-world problems involving rates of change (4)

Word problems tend to inlude some sort of rate (tell by the units included), starting amount, sometimes rate in and rate out, and time intervals/endpoints.

Derivative of trig functions

sinx = cosx cosx = -sinx tanx = sec^2x cotx = -csc^2x secx = secxtanx cscx = -cscxcotx