AP Calculus Formulas

Special Trig Limits

• lim x→ 0 (sin x)/x (or x/sinx) = 1

• lim x→ 0 (1 - cos x)/x (or x/1-cosx) = 0

The Squeeze Theorem

A fundamental theorem in calculus used to find limits, which states that if a function is squeezed between two other functions that have the same limit at a certain point, then the function itself must also converge to that limit at that point.

Types of Discontinuities

There are three main types of discontinuities in calculus:

• removable (or soft) discontinuities where a hole exists in the graph

• jump discontinuities where the function has two different limits approaching from either side

• infinite discontinuities where the function approaches infinity at a certain point.

Formal Definition of Continuity

• f(c) is defined (c is in the domain

• lim as x approaches c f(x) exists

• lim as x approaches c f(x) = f(c)

Finding the Domain

• Denominators must not equal zero for fractions

• numbers under even roots must be equal or greater than zero

• numbers inside logarithms and natural logs must be greater than zero

Intermediate Value Theorem (for continuous functions)

Must meet the required conditions:

• f(x) is continuous on an interval [a,b]

• f(a) ≠ f(b)

• k is between f(a) and f(b)

• Conclusion: IVT applis and there exists a value c between a and b such that f(c) = k

Average Rate of Change

• (f(a+h)-f(a))/((a+h)-a) or (f(x)-f(a))/(x-a)

• equation of the secant line

Instantaneous Rate of Change

• lim h → 0 (f(a+h)-f(a))/h or lim x → a (f(x)-f(a))/(x-a)

• equation of tangent line

Definition of Derivative

• f’(x) = lim h→0 (f(x+h)-f(x))/h

Equation of Tangent Line

• y-f(a) = f’(a)(x-a)

• point-slope form

Differentiability

The derivative will fail to exist where the function has a:

• discontinuity (hole or jump)

• corner or cusp

• vertical tangent (undefined slope)

Differentiability will always imply continuity but continuity might not always imply differentiability

Power Rule

• f(x) = xⁿ

• ^{f’(x) =} nx^n-1

Derivative Rules

• constant: d/dx c = 0

• Constant Multiple: d/dx cu = c dy/dx

• Sum/Difference: d/dx(u±v) = du/dx ± dv/dx

Derivatives of cos x and sin x

• d/dx cosx = -sinx

• d/dx sinx = cosx

Derivatives of Exponential Functions

• d/dx a^x = a^xlna

• d/dx e^x = e^x

Derivatives of Logarithmic Functions

• d/dx log_ax = 1/x * 1/lna

• d/dx lnx = 1/x

The Product Rule

• h(x) = f * g

• h’(x) = f’g * fg’

The Quotient Rule

• h(x) = f/g

• h’(x) = (f’g-fg’)/ g²

Trig Derivatives

• d/dx sinx = cosx

• d/dx cosx = -sinx

• d/dx tanx = sec²x

• d/dx cotx = -csc²x

• d/dx secx = secxtanx

• d/dx cscx = -cscxcotx

The Chain Rule

• d/dx f(g(x)) = f’(g(x)) * g’(x)

Implicit Differentiation

• d/dx y = dy/dx

• d/dx y² = 2y dy/dx

• d/dx e^5y = e^5y * 5dy/dx

Horizontal and Vertical Tangent Lines

• horizontal tangent lines exist when the slope dy/dx = 0

• Vertical tangent lines exist when the slope dy/dx = undefined

Derivative of an Inverse Function

• d/dx [f^-1(x)] = 1/(f’(f^-1(x)))

Inverse Trig Derivatives

• d/dx sin^-1x = 1/(√1-x²)

• d/dx cos^-1x = -1/(√1-x²)

• d/dx sec^-1x = 1/(lxl√x²-1)

• d/dx csc^-1x = -1/(lxl√x²-1)

• d/dx tan^-1x = 1/(x²+1)

• d/dx cot^-1x = -1/(x²+1)

Domain of an Inverse Trig Function

• arcsin x: domain: -1≤x≤1; range: -π/2≤y≤π/2

• arccos x: domain: -1≤x≤1; range: 0≤y≤π

• arctan x: domain: -infinity≤x≤infinity; range: -π/2<y<π/2

Higher-Order Derivatives

• repeat the power rule for each derivative

Position, Velocity, and Acceleration

• velocity = rate of change of position

• v(t)<0 = particle is moving left (x-axis) or down (y-axis)

• v(t)>0 = particle is moving right (x-axis) or up (y-axis)

• v(t)=0 = particle is at rest

• average velocity = (s(b))-s(a))/(b-a) on some interval [a, b]

• speed = lvelocityl

Speeding up or Slowing down

• if velocity and acceleration have the same sign, then the particle is speeding up

• if velocity and acceleration have different signs, then the particle is slowing down

Solving Related Rates

• Draw a picture

• Make a list of all known and unknown rates and quantities

• Relate the variables in an equation

• Differentiate with respect to time

• Substitute the known quantities and rates and solve

Approximating with Local Linearity

• Concave up with a tangent line = underestimate

• Concave down with a tangent line = overestimate

L’Hospital’s Rule

• suppose f(a) = 0 and g(a) = 0 and lim x→a f(x)/g(x) = 0/0 or infinity/infinity then L’hospital’s rule apply

• lim x→ a f(x)/g(x) = f’(x)/g’(x)

The Mean Value Theorem

• if a function f is continuous over the interval [a,b] and differentiable over the interval (a,b) then there exists a point c within that open interval where the instantaneous rate of change equals the average rate of change over the interval

• (f(b)-f(a))/(b-a) = f’(c)

Extreme Value Theorem

• if a function f is continuous over the interval [a,b], then f has at least one minimum value and at least one maximum value on [a,b]

• global extrema = absolute extrema

• local extrema = relative extrema

Critical Point

• a point that has a possibility of being an extrema (max or min)

• How do you find a critical point: f’(x) DNE and/or f’(x) = 0

Increasing and Decreasing Intervals

• if the slope of the function is positive, the function is increasing

• if the slope of the function is negative, the function is decreasing

The First Derivative Test

Assume c and d are critical numbers of a function f

• there is a minimum value at x=c because f’ changes signs from negative to positive

• there is a maximum value at x=d because f’ changes signs from positive to negative

• if h(c) does not exist, then x=c cannot be a critical point

Candidates for Absolute Extrema (on an interval)

• Critical Points

• End Points

Determining Concavity

• if f’ is increasing and f’’>0, then f is concave up

• if f’ is decreasing and f’’<0, then f is concave down

Point of Inflection

• there is a point of inflection of f at x=c if f(c) is defined and f’’ changes signs at x=c

• a point of inflection is where the graph changes concavity

• Two common mistakes: assuming that f’’=0 means there is a point of inflection and assuming that f’’≠0means there is a point of inflection

The Second Derivative Test

• Suppose f’(c)=0, then

  • if f’’(c)>0, then f has a relative minimum at x=c

  • if f’’(c)<0, then f has a relative maximum at x=c

• If there is only one critical point, and that CP is an extrema (max or min), then it is an absolute extremum (max or min)

Connecting f, f’, f’’

• an object is speeding up if f’ and f’’ have the same signs

• an object is slowing down if f’ and f’’ have different signs

Solving Optimization Problems

• Draw a picture of(if applicable) and identify known and unknown quantities

• Write an equation (model) that will be optimized

• Write your equation in terms of a single variable

• Determine the desired max or min value with calculus techniques

• Determine the domain (endpoints) of your equation to verify if the endpoints represent a max or min

Accumulation of Change

• the area under the curve gives us the accumulation of change

• unit for area under the curve: the dependent unit multiplied by the independent unit

  • the unit for y times the unit for x

Approximating Areas with Riemann Sums

• left riemann sum: uses the left side of the rectangles to estimate the area under the curve

• right riemann sum: uses the right side of the rectangles to estimate the area under the curve

• midpoint riemann sum: uses the midpoint of the rectangles to estimate the area under the curve

• trapezoidal sum: uses trapezoids to estimate instead of rectangles

Overestimate or Underestimate (Riemann Sums)

• Increasing functions: underestimate for left riemann sums and overestimate for right riemann sums

• Decreasing functions: underestimate for right riemann sums and overestimate for left riemann sums

• Trapezoid estimation: underestimate for concave down and overestimate for concave up

Summation Notation

• the area under the curve of f(x) on the interval [a,b] is represented by lim n → infinity ∑ⁿ_k=1((b-a)/n)*f(a+(b-a)/n*k)

• (b-a)/n = delta x

• f(a+(b-a)/n*k) = f(x_k)

• could also be written as ∫^b_a f(x) dx

Fundamental Theorem of Calculus

• d/dx ∫^x_af(t) dt = f(x)

• ∫^b_af(x) dx = F(b) - F(a)

• f(t) is the integrand

• derivatives and integrals are inverses of each other

• antiderivative

Variation of the FTC

• if a is a constant, f is a continuous function, and g and h are differentiable then:

  • d/dx ∫_a^g(x) f(t) dt = f(g(x))*g’(x)

  • d/dx ∫_h(x)^g(x) f(t) dt = f(g(x))*g’(x) - f(h(x))*h’(x)

Behaviour of Accumulation Functions

• F’(x) = f(x)

• F(x) is increasing when F’(x) > 0 or f(x)>0

• F(x) is decreasing when F’(x)<0 or f(x)<0

• F(x) has relative max when F’(x) changes from + to - or when f(x) changes from + to -

• F(x) has relative min when F’(x) changes from - to + or when f(x) changes from - to +

• F(x) is concave up when F’’(x) > 0 or f’(x)>0

• F(x) is concave down when F’’(x) <0 or f’(x)<0

• F(x) has a point of inflection when F’’(x) changes signs or f’(x) changes signs

Properties of Definite Integrals

• Equivalent limits: ∫^a_af(x) dx = 0

• Reversal of limits: ∫^a_bf(x) dx = -∫^b_af(x) dx

• Multiply by constant (k=constant): ∫^b_akf(x) dx = k∫^b_af(x) dx

• Adjacent intervals (a<c<b): ∫^c_af(x) dx + ∫^b_cf(x) dx = ∫^b_af(x) dx

• Addition: ∫^b_a[f(x) + g(x)] dx = ∫^b_af(x) dx + ∫^b_ag(x) dx

• Subtraction: ∫^b_a[f(x) - g(x)] dx = ∫^b_af(x) dx - ∫^b_ag(x) dx

Antiderivative

• f(x) = xⁿ then F(x) = (xⁿ⁺¹)/(n+1)

• ∫sinxdx = -cosx + c

• ∫cosxdx = sinx + c

Indefinite Integrals

• Exponential:

  • ∫e^x dx = e^x + c

  • ∫a^x dx = 1/lna a^x + c

• Logarithm:

  • ∫1/x dx = ln|x| + c

• Trig:

  • ∫ cosx dx = sinx + c

  • ∫ sinx dx = -cosx + c

  • ∫ sec^x dx = tanx + c

  • ∫ cscxcotx dx = -cscx + c

  • ∫ secxtanx dx = secx + c

  • ∫ csc² x dx = -cotx + c

• Inverse Trig:

  • d/dx sin^-1(x) = 1/(√1-x²)

  • d/dx sec^-1(x) = 1/[|x|*(√x²-1)]

  • d/dx tan^-1(x) = 1/(x²+1)

U-substitution

• substitute part of the integrand into u and convert everything into du instead of dx

Integrating with Long Division and Completing the Square

• A method used in calculus for integrating rational functions by dividing the numerator by the denominator and simplifying the resulting expression, often involving completing the square for quadratic denominators.

Integration by Parts

• typically used for the integration of the product of two functions

• ∫f(x)g’(x) = fg - ∫f’g

• Tabular integration: differentiate to 0 for chosen f(x). Integrate your chosen g’(x) the same number of times. Follow the sign convention, which is plus/minus repeating

Linear Partial Fractions

• A method to decompose a rational function into a sum of simpler fractions, typically used for integrating functions with polynomial denominators. This technique facilitates the integration of more complex rational expressions.

Improper Integrals

• if f(x) is continuous on [a, infinity), then ∫_a^infinity f(x) dx = lim_t→infinity ∫_a^t f(x) dx.

• if f(x) is continuous on (-infinity, b], then ∫_-infinity^b f(x) dx = lim_t→−∞ ∫_t^b f(x) dx.

• if f(x) is continuous on [a, b) and has an infinite discontinuity at b, then ∫_a^b f(x) dx = lim_t→b- ∫_a^t f(x) dx.

• if f(x) is continuous on (a, b] and has an infinite discontinuity at a, then ∫_a^b f(x) dx = lim_t→a+ ∫_t^b f(x) dx.

Directly Proportional & Inversely Proportional

• Directly: if a is proportional to b, then a = kb, where k is a constant

• Inversely: if a is inversely proportional to b, then a = k/b, where k is a constant

Verifying Solutions

• The process of checking whether a given solution satisfies the original equation or conditions of a problem, often involving substitution back into the equation to confirm correctness.

Sketching Slope Fields

• A method for visualizing solutions to differential equations by plotting tangent lines at various points in the plane, representing the slope of the solution curve.

• a slope field represents a differential equation on an xy-plane. It shows the “slope” of all the particular solutions to the differential equation

Reasoning using Slope Fields

• Involves interpreting the behavior of solutions to differential equations by analyzing the direction and steepness of the tangent lines within a slope field. This approach helps predict the general shape and trends of solution curves.

Euler’s Method

• A numerical technique for approximating solutions to first-order differential equations by using tangent line segments to step forward in small increments.

• delta x = (x₂-x₁)/total steps

• y-y₁= m(x-x₁)

• y= y₁+m(x-x₁) ← new y

Separation of Variables (General & Particular Solutions)

• General: integrate x and y separately and come up with a solution (usually explicitly) by using algebra

• Particular: solve for the specific solution by applying initial conditions to the general solution.

Exponential Models with Differential Equations

• y = a(b)^t

• rate of change of a quantity is proportional to the size of the quantity

• The solution to dy/dt = ky is y = Ce^kt, where C represents the initial value of the model

• A growth model will have a positive exponent

• A decay model will have a negative exponent

• Doubling Time/Half-Life: doubling time is the time required for a quantity to double in size, while half-life is the time required for it to reduce to half its initial value.

Logistic Models

• dy/dt = ky(1-y/L), where L is the limiting value and k is a constant

• dy/dt = (k/L)y(L-y) → ky(a-y), where L is the limiting value and k is a constant

Average Value of a Function

The average value of a function over an interval [a, b] is defined as $\frac{1}{b-a} \int_a^b f(x) \, dx$. This gives the mean value of the function's outputs on that interval.

Velocity with integrals; different meanings

• ∫velocity = |∫velocity| = displacement

• ∫|velocity| = total distance

• |velocity| = speed

Accumulation and integrals

• when you integrate a rate, you get net change

• ∫ rate of change = net change

Area between curves (with respect to x)

• A = ∫_a^b [f(x) - g(x)] dx

• f(x) ≥ g(x) for all x in [a, b]

Area between curves (with respect to y)

• A = ∫_c^d [f(y) - g(y)] dy

• f(y) ≥ g(y) for all y in [c, d]

• must perform implicit differentiation if function is given with respect to x

Area - More than Two Intersections

• To find the area between curves with more than two intersections, first identify the points of intersection.

• Then, break the integral into separate intervals where you can evaluate the area using the correct upper and lower functions for each segment.

• The final area is the sum of these integrals across all intervals.

Volume of a solid - Squares and Rectangles

• V = ∫_a^b A(x) dx

• A(x) is the area of a cross section perpendicular to the x-axis

• For squares, A(x) = s² where s is f(x) - g(x)

• For rectangles, A(x) = width x height, where width = f(x) - g(x) and height is given in the problem

Volumes of a Solid - Triangles and Semicircles

• V = ∫_a^b A(x) dx

• A(x) is the area of a cross section perpendicular to the x-axis

• For equilateral triangles, A(x) = (√3 /4 s²), where s = f(x) - g(x)

• For isosceles right triangles, A(x) = (1/2 s²), where s = f(x) - g(x)

• For semicircles, A(x) = (1/2πr²), where r = [f(x) - g(x)]/2

Disc Method - revolve around x or y axis

• V = ∫_a^b π[R(x)]² dx, where R(x) is the “distance” between the axis of revolution and the outside of the solid

• Area of the cross section looks like a circle, after revolution, it should have a cone shape

• The radius is whatever f(x) is, and it is also R(x)

Disc Method - Revolve around other axes

• V = ∫_a^b π[R(x)]² dx, where R(x) is the “distance” between the axis of revolution and the outside of the solid

• Requires finding the boundaries by setting the function equal to the axes you are revolving around

Washer Method - Revolve around x or y axis

• V = ∫_a^b π[R(x)]² dx - ∫_a^b π[r(x)]² dx or V = ∫_a^b[R(x)]²-[r(x)]² dx, where R(x) is the radius to the outside of the object and r(x) is the radius to the inside of the object

• Gives a tube-like shape after revolved around the axes

• Usually involves two different functions instead of one

Washer Method - Revolve around other axes

• V = ∫_a^b π[R(x)]² dx - ∫_a^b π[r(x)]² dx or V = ∫_a^b[R(x)]²-[r(x)]² dx, where R(x) is the radius to the outside of the object and r(x) is the radius to the inside of the object

• Often requires two steps to determine the volume of the tube created

Arc Length

• if a function y = f(x) represents a smooth continuous curve on the closed interval [a, b], then the arc length of f between a and b is given by the formula below

• ∫_a^b√(1+[f’(x)]²) dx

• it is basically how long the curve is on the graph

Parametric Equations

• if f and g are continuous functions of t on an interval, then the equations x = f(t) and y = g(t) are parametric equations and t is the parameter

• derivative of a parametric equation: dy/dx = (dy/dt)/(dx/dt), dx/dt ≠ 0

• 2nd derivative of a parametric equation: d²y/dx² = [d/dx(dy/dx)]/(dx/dt)

Arc Length (Parametric Form)

• L = ∫_a^b√[(dx/dt)² + (dy/dt)²] dt

Vector-valued Functions

• Magnitude is designated by ||v||

• f(t) and g(t) are component functions with the parameter t

• Differentiation: r’(t) = <f’(t), g’(t)>

• Integration: ∫r(t) dt = <∫f(t) dt, ∫g(t) dt>

Motion using Parametric and Vectors

• speed: √{[(x’(t)]² + [y’(t)]²}

• total distance: ∫_a^b √{[(x’(t)]² + [y’(t)]²} dt

Differentiating in Polar Form

• (r, theta) is used for polar coordinate system

• r is a directed distance from the origin to a point

• theta is the directed angle

• x = r * cos theta

• y = r * sin theta

• tan theta = y/x

• r² = x² + y²

• Slope of a Curve in Polar Form: dy/dx = [y’(theta)]/[x’(theta)]

Area Bounded by a Polar Curve

• if f is continuous and nonnegative on the interval [alpha, beta], then the area of the region bounded by the graph of r = f(theta) between the radial lines theta = alpha and theta = beta

• A = (1/2) ∫_alpha^beta r² d(theta)

Area Bounded by Two Polar Curves

• A = (1/2) ∫_a^b (r₂²) d(theta) - (1/2) ∫_a^b (r₁)² d(theta) or A = (1/2) ∫_a^b(r₂² - r₁²) d(theta)