Calculus Key Concepts: Limits, Continuity, Derivatives, and Applications

Absolute Convergence

If ∑|𝑎𝑛| converges, then ∑𝑎𝑛 also converges.

Alternate Definition of the Derivative

f'(c) = lim x→c (f(x) - f(c)) / (x - c).

Alternating Series

The series converges if 𝑎𝑛+1 ≤𝑎𝑛 for all n and lim 𝑛→∞𝑎𝑛= 0.

Antiderivative

If 𝐹′(𝑥) = 𝑓(𝑥) for all x, 𝐹(𝑥) is an antiderivative of f.

Arc Length (Function x=f(y))

Length of the arc on [a, b] is ∫√1 + [𝑓′(𝑦)]2 𝑑𝑦.

Arc Length (Function y=f(x))

Length of the arc on [a, b] is ∫√1 + [𝑓′(𝑥)]2 𝑑𝑥.

Area Between Two Curves

Area = ∫[𝑓(𝑥) −𝑔(𝑥)] 𝑑𝑥 from 𝑎 to 𝑏.

Area in Polar Coordinates

Area = 1/2 ∫[𝑓(𝜃)]²𝑑𝜃 from 𝛼 to 𝛽.

Average Rate of Change

The average rate of change, m, of a function f on the interval [a, b] is given by m = (f(b) - f(a)) / (b - a).

Both Infinite Limits

∫𝑓(𝑥)𝑑𝑥 = lim 𝑎→−∞∫𝑓(𝑥)𝑑𝑥 + lim 𝑏→∞∫𝑓(𝑥)𝑑𝑥.

Concave Down

If 𝑓′′(𝑥) < 0, the graph of 𝑓(𝑥) is concave down.

Concavity

If 𝑓′′(𝑥) > 0, the graph of 𝑓(𝑥) is concave up.

Conditional Convergence

If ∑|𝑎𝑛| diverges, but ∑𝑎𝑛 converges.

Constant Functions

If 𝑓′(𝑥) = 0 in (a, b), then f is constant on (a, b).

Convergence condition for p-series

If 𝑝> 1, then the series converges.

Convergence of a series

If lim 𝑛→∞𝑆𝑛= 𝑆, then ∑𝑎𝑛 converges to 𝑆.

Convergence of geometric series

If |𝑟| < 1, then the series converges to 𝑎/(1−𝑟).

Critical Value

When f(c) is defined, if f ' (c) = 0 or f ' is undefined at x = c, the values of the x - coordinate at those points are called critical values.

Decreasing Functions

If 𝑓′(𝑥) < 0 in (a, b), then f is decreasing on (a, b).

Definite Integral

A definite integral is an integral with upper and lower limits, a and b, respectively.

Definition of Continuity

A function f(x) is continuous at c if: I. lim x→c f(x) exists; II. f(c) exists; III. lim x→c f(x) = f(c).

Definition of the Derivative

The derivative f'(x) = lim h→0 (f(x+h) - f(x))/h.

Derivative of a function

Gives the value of the slope of the function at each point (x, y).

Derivative of an Inverse

If f and its inverse g are differentiable, then 𝑑/𝑑𝑥[𝑔(𝑥)] = 1/𝑓′(𝑓−1(𝑥)).

Differentiability and Continuity Properties (A)

If f(x) is differentiable at x = c, then f(x) is continuous at x = c.

Differentiability and Continuity Properties (B)

If f(x) is not continuous at x = c, then f(x) is not differentiable at x = c.

Differentiability and Continuity Properties (C)

The graph of f is continuous but not differentiable at x = c if: I. The graph has a cusp or sharp point at x = c; II. The graph has a vertical tangent line at x = c; III. The graph has an endpoint at x = c.

Differential Equation

An equation involving an unknown function and one or more of its derivatives.

Differential equation example

𝑑𝑦/𝑑𝑥 = 𝑥/𝑦

Differential Equation for Exponential Growth

𝑑𝑦/𝑑𝑡= 𝑘𝑦.

Differential Equation for Logistic Growth

𝑑𝑃/𝑑𝑡= 𝑘𝑃(1 − 𝑃/𝐿).

Direct Comparison Test

If 𝑎𝑛≤𝑏𝑛 for all n, then if ∑𝑎𝑛 diverges, ∑𝑏𝑛 diverges.

Divergence of a series

If the terms of a sequence do not converge to 0, then the series must diverge.

Euler's Method

Uses a linear approximation with increments for approximating the solution to a differential equation.

Exponential Growth

When the rate of change of a variable y is directly proportional to the value of y.

Extrema of a Function (A)

Absolute Extrema: An absolute maximum is the highest y-value of a function on a given interval.

Extrema of a Function (B)

Relative Maximum: The y-value of a function where the graph changes from increasing to decreasing.

Extrema of a Function (C)

Relative Minimum: The y-value of a function where the graph changes from decreasing to increasing.

Extreme Value Theorem

If the function f is continuous on the closed interval [a, b], then the absolute extrema of the function f on the closed interval will occur at the endpoints or critical values of f.

First Derivative Test

After calculating any discontinuities of a function f and calculating the critical values of a function f, create a sign chart for f '.

First Fundamental Theorem of Calculus

If 𝐹(𝑥) is the antiderivative of a continuous function 𝑓(𝑥), then ∫𝑏𝑎𝑓(𝑥)𝑑𝑥= 𝐹(𝑏) −𝐹(𝑎).

General Solution for Exponential Growth

𝑦= 𝐶𝑒𝑘𝑡.

General Solution for Logistic Growth

𝑃(𝑡) = 𝐿/(1+𝑏𝑒−𝑘𝑡).

General Solution of a Differential Equation

Left with the constant of integration, C, undefined.

Geometric Series

A series of the form ∞∑𝑎𝑟𝑛 with specific convergence conditions.

Improper Integral

Characterized by having limits of integration that is infinite or the function having an infinite discontinuity.

Increasing Functions

If 𝑓′(𝑥) > 0 in (a, b), then f is increasing on (a, b).

Infinite Discontinuity

∫𝑓(𝑥)𝑑𝑥 = lim 𝑥→𝑘−∫𝑓(𝑥)𝑑𝑥 + lim 𝑥→𝑘+∫𝑓(𝑥)𝑑𝑥.

Infinite Lower Limit

∫𝑓(𝑥)𝑑𝑥 = lim 𝑎→−∞∫𝑓(𝑥)𝑑𝑥 from 𝑎 to 𝑏.

Infinite Upper Limit

∫𝑓(𝑥)𝑑𝑥 = lim 𝑏→∞∫𝑓(𝑥)𝑑𝑥 from 𝑎 to 𝑏.

Initial value

(𝑥0, 𝑦0)

Integral Test

If f is positive, continuous, and decreasing for 𝑥≥1, then ∑𝑎𝑛 and ∫𝑓(𝑥)𝑑𝑥 either both converge or both diverge.

Integration by Parts

∫ln 𝑥𝑑𝑥= 𝑥ln 𝑥 −𝑥+ 𝐶

Intermediate Value Theorem

If f is a continuous function on [a, b] and k is any number between f(a) and f(b), then there exists at least one value of c on [a, b] such that f(c) = k.

L'Hospital's Rule

If lim x→c f(x) results in an indeterminate form (0/0, ∞/∞, etc.), then lim x→c f(x) = lim x→c (p(x)/q(x)) = lim x→c (p'(x)/q'(x)).

Left Riemann Sum

𝐴𝑟𝑒𝑎≈∆𝑥 [𝑓(𝑥1) + 𝑓(𝑥2) + 𝑓(𝑥3) + ⋯+ 𝑓(𝑥𝑛)].

Limit Comparison Test

If lim 𝑛→∞ 𝑎𝑛/𝑏𝑛= 𝐿, where L is finite and positive, then ∑𝑎𝑛 and ∑𝑏𝑛 either both converge or both diverge.

Limit of a Continuous Function

If f(x) is a continuous function for all real numbers, then lim x→c f(x) = f(c)

Limits of a Function as x Approaches Infinity (A)

If f(x) is a rational function given by f(x) = p(x)/q(x), such that p(x) and q(x) are both polynomial functions, and the degree of p(x) > q(x), then lim x→∞ f(x) = ∞.

Limits of a Function as x Approaches Infinity (B)

If the degree of p(x) < q(x), then lim x→∞ f(x) = 0; y = 0 is a horizontal asymptote.

Limits of a Function as x Approaches Infinity (C)

If the degree of p(x) = q(x), then lim x→∞ f(x) = c, where c is the ratio of the leading coefficients; y = c is a horizontal asymptote.

Limits of Rational Functions (A)

If f(x) is a rational function given by f(x) = p(x)/q(x), such that p(x) and q(x) have no common factors, and c is a real number such that q(c) = 0, then lim x→c f(x) does not exist or lim x→c f(x) = ±∞; x = c is a vertical asymptote.

Limits of Rational Functions (B)

If f(x) is a rational function given by f(x) = p(x)/q(x), such that reducing a common factor between p(x) and q(x) results in the agreeable function k(x), then lim x→c f(x) = lim x→c k(x) = k(c); Hole at the point (c, k(c)).

Logistic Growth

A population that experiences a limit factor in growth based on available resources.

Maclaurin Series

A Taylor series centered at 0.

Mean Value Theorem for Derivatives

If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one number c between a and b such that (f(b) - f(a)) / (f'(c)) = (b - a).

Midpoint Riemann Sum

𝐴𝑟𝑒𝑎≈∆𝑥 [𝑓(𝑥1/2) + 𝑓(𝑥3/2) + 𝑓(𝑥5/2) + ⋯+ 𝑓(𝑥(2𝑛−1)/2)].

Normal Lines

The equation of the normal line at a point (a, f(a)) is y - f(a) = -1/f'(a)(x - a).

Nth Term Test for Divergence

If lim 𝑛→∞𝑎𝑛≠0, then ∑𝑎𝑛 diverges.

Optimization

Finding the largest or smallest value of a function subject to some kind of constraints.

P-Series

The form of a p-series is ∑1/𝑛𝑝.

Parametric Arc Length

Arc length over the interval 𝑎≤𝑡≤𝑏 is ∫√(𝑑𝑥/𝑑𝑡)2 + (𝑑𝑦/𝑑𝑡)2 𝑑𝑡.

Parametric Equations

The equations 𝑥= 𝑓(𝑡) and 𝑦= 𝑔(𝑡) are called parametric equations.

Partial Fractions

Used to rewrite R(x) as the sum or difference of simpler rational functions.

Particle Motion

A velocity function is found by taking the derivative of position. An acceleration function is found by taking the derivative of a velocity function.

Particular Solution of a Differential Equation

Uses the initial condition to calculate the value of C.

Point of Inflection

If f is continuous at x = c, f ''(c) = 0 or f ''(c) is undefined, and f ''(x) changes sign at x = c, then the point (𝑐, 𝑓(𝑐)) is a point of inflection.

Power Series

A series of the form ∞∑𝑎𝑛𝑥𝑛.

Ratio Test

If lim 𝑛→∞|𝑎𝑛+1/𝑎𝑛| < 1, then ∑𝑎𝑛 converges.

Recurrence relation for x

𝑥1 = 𝑥0 + ℎ

Recurrence relation for y

𝑦1 = 𝑦0 + ℎ∙𝐹(𝑥0, 𝑦0)

Related Rates (A)

Identify the known variables, including their rates of change and the rate of change that is to be found.

Related Rates (B)

Implicitly differentiate both sides of the equation with respect to time.

Relative Maximum (First Derivative Test)

If 𝑓′(𝑥) changes sign from positive to negative at 𝑥= 𝑐, then 𝑓(𝑐) is a relative maximum of f.

Relative Maximum (Second Derivative Test)

If 𝑓′(𝑐) = 0 and 𝑓′′(𝑐) < 0, then 𝑓(𝑐) is a relative maximum.

Relative Minimum

The y-value of a function where the graph of the function changes from decreasing to increasing.

Relative Minimum (First Derivative Test)

If 𝑓′(𝑥) changes sign from negative to positive at 𝑥= 𝑐, then 𝑓(𝑐) is a relative minimum of f.

Riemann Sum

A Riemann Sum is the use of geometric shapes to approximate the area under a curve.

Right Riemann Sum

𝐴𝑟𝑒𝑎≈∆𝑥 [𝑓(𝑥0) + 𝑓(𝑥1) + 𝑓(𝑥2) + ⋯+ 𝑓(𝑥𝑛−1)].

Rolle's Theorem

If f is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there exists at least one number c between a and b such that (f(b) - f(a)) / (f'(c)) = (b - a) = 0.

Root Test

If lim 𝑛→∞√|𝑎𝑛| < 1, then ∑𝑎𝑛 converges.

Second Derivative Test

If 𝑓′(𝑐) = 0 and 𝑓′′(𝑐) > 0, then 𝑓(𝑐) is a relative minimum.

Separation of Variables

Technique to solve differential equations by separating variables.

Shelf Point

If there is no sign change of 𝑓′(𝑥), there exists a shelf point.

Slope Field

A graphical representation of all possible solutions to a differential equation.

Slope of Parametric Curve

The slope of the curve at the point (x, y) is 𝑑𝑦/𝑑𝑥= 𝑑𝑦/𝑑𝑡 / 𝑑𝑥/𝑑𝑡.

Special Trig Limits (A)

lim x→0 (sin(ax)/ax) = 1.

Special Trig Limits (B)

lim x→0 (x/(1-cos(ax))) = 0.

Tangent Lines

The equation of the tangent line at a point (a, f(a)) is y - f(a) = f'(a)(x - a).

Taylor Series

A series that represents a function f with derivatives of all orders at x = c.