AP Calc AB - Study Guide

Average velocity of a particle given velocity

(1/(b-a))∫[a->b] v(t)dt

∫du/(u√u²-a²)

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

∫du/(a²+u²)

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

∫xⁿdx

(xⁿ⁺¹/n+1)+C

Mean Value Theorem (MVT)

-Definition: If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) there exists a number c in (a,b) such that f'(c) = (f(b) - (fa))/(b-a) -Verbally says: Average RoC = Instantaneous RoC -Graphically says: Tangent line is parallel to the secant line.

Tangent Line Approximation

-If f"(x) is positive and f'(x) (slopes of f(x)) is increasing, then the tangent line to f gives an underestimate. -If f"(x) is negative and f'(x) (slopes of f(x)) is decreasing, then the tangent line to f gives an overestimate.

Area Enclosed by Intersecting Curves

-Intersection points give the limits of integration (the interval) -Follow same formula for area between curves.

Rolle's Theorem

-Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). -f(a) = f(b), then there exists at least one number where f'(c) = 0. -If these conditions hold true, then there is at least one number between a and b so that the tangent line is horizontal.

∫sin(x)dx

-cos(x) + C

∫csc²(x)dx

-cot(x) + C

∫csc(u)du

-ln|csc(u) + cot(u)| + C

sin²θ+cos²θ

1

How to find absolute extrema

1. Derivative/CRITICAL NUMBERS -> Set equal to 0 or where f'(c) = DNE. 2. List critical numbers and "closed interval" values in order on a table. 3. Plug in #2 to the original equation. 4. Identify the absolute extrema.

Left-Hand and Right-Hand Riemann Sums

1. Draw a rough graph of the function over the interval. 2. Use ∆x= (b-a)/n to determine each subinterval length on the interval [a,b] where n is the number of partitions. 3.Compute x-coordinates of rectangles at either left-end or right-end. 4. Compute the areas of each rectangle. 5. Find summation of the approximated areas of the rectangles.

How to Analyze the Graph of a Function

1. Find domain and range. 2. Find x and y-intercepts, vertical and horizontal asymptotes, symmetry of graph (if applicable). 3. Find critical points and intervals where function is increasing and/or decreasing. 4. Determine local/relative max and min points. 5. Determine concavity and points of inflection. 6. Sketch the curve.

U-Substitution in Indefinite Integrals

1. Find u and du 2. Plug into integral 3. Integrate (+ C!) 4. Plug in original equation 5. Simplify if needed.

U-Substitution in Definite Integrals

1. Find u and du 2. Turn x values into u values 3. Integrate 4. Plug in u values and substract (F(a) - F(b))

How to solve a related rates problem

1. Make a sketch and label the quantities. 2. Read the problems and identify all quantities as "KNOW", "GIVEN", and "FIND" with the appropriate information. 3. Write an equation involving the variables whose rates of change are either given or are to be determined. 4. Using the Chain Rule, IMPLICITLY DIFFERENTIATE both sides of the equation with respect to time, t. 5. AFTER STEP 4, substitute into the equation all known values for the variables and their rates of change. Then, solve for the required rates of change.

How to find particular solutions (sep. of variables)

1. Separate 2. Integrate. 3. Solve for C (plug condition into integration) 4. Solve for y

How to find general solutions (sep. of variables)

1. Separate the variables by multiplying or dividing by f(y) and then multiplying by dx. 2. Integrate both sides, placing the "+ C" on the x side. 3. Solve for y.

Steps for Optimization

1. Write an equation for what you want to maximize/minimize. 2. Use constraints to find relationships between the variables. 3. Rewrite equations in terms of one variable. 4. Identify critical values and use the candidate's test to find extrema. 5. Interpret the answer in context.

The Fundamental Theorem of Calculus (Part 1)

1.dF/dx = d/dx ∫[a->x] f(t)dt = f(x) 2. ∫[a->b] f(x)dx = F(b) - F(a)

Area Between Curves

A = ∫[a->b] [f(x) - g(x)]dx

Rule used for implicit differentiation

Chain Rule

The Fundamental Theorem of Calculus (Part 2)

If F(x) = ∫[0->g(x)] f(t)dt, then F'(x) = [f(g(x))][g'(x)]

Mean Value Theorem for Integrals

If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b,] such that: ∫[a->b] f(x)dx = f(c)(b-a)

Additive Interval Property

If f is integrable on the 3 closed intervals determined by a, b, and c, then the definite integral, from a TO b of f(x) with respect to x, is equal to: [the definite integral, from a TO c of f(x) with respect to x,] + [the definite integral, from b TO C of f(x) with respect to x]

Average Value of a Function on an Interval (How to find f(c))

If f is integrable on the closed interval [a,b], then the average (mean) value of f on the interval is: f(c) = [1/(b-a)] ∫[a->b]f(x)dx

Exponential Function Rule

If f(x) = aⁿ, then f'(x) = aⁿ+ln(a)

Constant Rule (Definition)

If f(x) = c, where c is any real number, then f'(x) = 0.

9th Rule: Cosine

If f(x) = cos(x), then f'(x) = -sin(x)

12th Rule: Cotangent

If f(x) = cot(x), then f'(x) = -csc²(x)

14th Rule: Cosecant

If f(x) = csc(x), then f'(x) = -cot(x)csc(x)

Derivative of eⁿ (Definition)

If f(x) = eⁿ, then f'(x) = eⁿ

Derivative of ln(x) (Definition)

If f(x) = ln(x), then f'(x) = 1/x

Logarithmic Function Rule

If f(x) = log(a), then f'(x) = 1/xln(a)

13th Rule: Secant

If f(x) = sec(x), then f'(x) = tan(x)sec(x)

10th Rule: Sine

If f(x) = sinx(x), then f'(x) = cosx

11th Rule: Tangent

If f(x) = tan(x), then f'(x) = sec²(x)

Power Rule (Definition)

If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹, where n∈R.

Extreme Value Theorem (EVT)

If f(x) is continuous on the closed interval [a,b], then f(x) has at least one absolute minimum and at least one absolute maximum on the closed interval.

Constant Multiple Rule (Definition)

If y = cf(x), where c is any real number, then dy/dx = cf'(x).

Second Derivative Test

If: -f"(x) is positive, then f'(x) is increasing (i.e. negative to positive) and f(x) is concave up. -f"(x) is negative, then f'(x) is decreasing (i.e. positive to negative) and f(x) is concave down. -f"(x) is neither positive nor negative, then f'(x) may have a possible relative min or max and f(x) may have a possible point of inflection.

First Derivative Test

If: -f'(x) changes from negative to positive at c, then f(c) is a relative minimum of f. -f'(x) changes from positive to negative at c, then f(c) is a relative maximum. -f'(x) does not change signs at c, then f'(c) is neither a relative minimum nor a relative maximum at c.

L'Hopital's Rule

States when f(x)/g(x) has an indeterminate form of the type 0/0 or ∞/∞ at x=a, then we can replace f(x)/g(x) by the quotient of the derivatives f'(x)/g'(x)

How to find derivative of an inverse function

Switch x and y and solver for y, find the derivative of the new function.

The Sum and Difference Rules (Definition)

The derivative of a sum or difference is the sum or difference of the derivatives.

Average Value of a Function

The value of f(c) given in the Mean Value Theorem for integrals.

Trapezoidal Sum Rule

Tₙ=(1/2)∆x [f(x₀)+2f(x₁)+2f(x₂)+...+2f(xₙ−₁)+f(xₙ)

Disk Method about the y-axis

V = π∫[a->b] x²dy

Disk Method about the x-axis

V = π∫[a->b] y²dx

Volumes by cross sections

V = ∫[a->b] A(x)dx (for example, could be any variable)

Washer Method

V = ∫π(R(x)²-r(x)²) dx

Point of Inflection

When the concavity of f changes. Where f"(x) = 0 or f"(x) = DNE.

∫aⁿdn

aⁿ/ln(a) + C

cot²θ + 1

csc²θ

19th Rule: Arccos

d/dx cos⁻¹(x) = -(1/(√1-u²))(du/dx), |u|<1

21st Rule: Arccot

d/dx cot⁻¹(x) = - (1/(1+u²))(du/dx), u is ARN

23rd Rule: Arccsc

d/dx csc⁻¹(x) = -[1/(|u|(√u²-1))](du/dx), |u|>1

22nd Rule: Arcsec

d/dx sec⁻¹(x) = [1/(|u|(√u²-1))](du/dx), |u|>1

18th Rule: Arcsin

d/dx sin⁻¹(x) = (1/(√1-u²))(du/dx), |u|<1

20th Rule: Arctan

d/dx tan⁻¹(x) = (1/(1+u²))(du/dx), u is ARN

Sum Rule (Notation)

d/dx[f(x) + g(x)] = f'(x) + g'(x)

Difference Rule (Notation)

d/dx[f(x) - g(x)] = f'(x) - g'(x)

Product Rule (Notation)

d/dx[f(x) × g(x)] = gf' + fg'

Quotient Rule (Notation)

d/dx[f(x)/(gx)] = (gf' - fg')/g²

Chain Rule

dy/dx = [f'(g(x))][g'(x)]

∫e^x dx

e^x + C

Alternate Form of a Derivative

f'(a) = lim (x goes to a) [(f(x) - f(a))/(x-a)]

Definition of a Derivative

f'(x) = lim (h goes to zero) [(f(x+h) - f(x))/h]

∫[a->b] kdx

k(b-a)

∫[a->b] kf(x)dx

k∫[a->b] f(x)dx

∫sec(u)du

ln|sec(u) + tan(u)| + C

∫cot(u)du

ln|sin(u)| +C

∫du/u

ln|u| + C

∫(1/x)dx

ln|x| + C

Final position of a particle is

s(a) + ∫[a->b] v(t)dt, where s(a) is the initial position.

∫sec(x)tan(x)dx

sec(x) + C

tan²θ+1

sec²θ

∫cos(x)dx

sin(x) + C

∫du/√(a²-u²)

sin⁻¹(u/a) + C

∫sec²(x)dx

tan(x) + C

General Solution to the Exponential Model: dy/dx = ky

y = Ce^(kx)

∫csc(x)cot(x)dx

−csc(x) + C

Definite Integral Property: x=a

∫[a->a] f(x)dx = 0

∫[a->b] [f(x) + g(x)]dx

∫[a->b] f(x)dx + ∫[a->b] g(x)dx

Displacement of a particle over the time interval [a,b]

∫[a->b] v(t)dt

Total distance traveled by the particle over the time interval [a,b]

∫[a->b] |v(t)|dt

Definite Integral Property: a>b

∫[b->a] f(x)dx = -∫[a->b] f(x)dx