Calculus Diagrams

Secant through P₁ and P₂: m = Δy/Δx = 1.93

Derivatives: Definition, Rules & Techniques › The Difference Quotient and Its Limit

Horizontal tangent: slope = 0.00 at a local maximum

Derivatives: Definition, Rules & Techniques › Derivative Graph Analysis

f″(c) = 0.00 at an inflection point

Derivatives: Definition, Rules & Techniques › Higher-Order Derivatives

f(x) = x² and f′(x) = 2x, marked at x = 1

Common Derivatives: Formulas & Examples › Power Functions

f(x) = sin x and f′(x) = cos x, marked at x = 1

Common Derivatives: Formulas & Examples › Trigonometric Functions — Sine and Cosine

f(x) = eˆx and f′(x) = eˆx coincide

Common Derivatives: Formulas & Examples › Exponential Functions

f(x) = ln x and f′(x) = 1/x, marked at x = 1

Common Derivatives: Formulas & Examples › Logarithmic Functions

Jump at x = 0: the two sides disagree

Differentiability: Corners, Cusps & Continuity › Discontinuities

The tangent line at P is the linear approximation

Differentials: Linear Approximation & Error › Differential Notation

f(x) = x³ above, f′(x) = 3x² below

Derivatives as Functions: Graphing & Properties › Derivative Notation: Function vs Point

Tangent at P with slope f′(c) = 1.89

Graph Analysis with Derivatives: Extrema & Curves › Tangent Lines

x³ − 3x on [−3, 3]: critical points at x = ±1

Graph Analysis with Derivatives: Extrema & Curves › Critical Points

f″(c) = 2.60: concave up, a cup

Graph Analysis with Derivatives: Extrema & Curves › The Second Derivative Test

f″(c) = −2.60: concave down

Graph Analysis with Derivatives: Extrema & Curves › Concavity

f″(c) = 0.00: the bending changes sides

Graph Analysis with Derivatives: Extrema & Curves › Inflection Points

x⁴ − 4x² on [−3, 3]: critical points and endpoints

Graph Analysis with Derivatives: Extrema & Curves › Optimization

f″(c) = 2.60: the slope is increasing

Higher Order Derivatives: Second, Third & nth › Notation for Higher-Order Derivatives

Secant over [a, b] and the parallel tangent at c

Derivative Rules: Formulas & Theorems › Mean Value Theorem

Identity, frozen at x0 = 1

Derivative Visualizer › Identity: a Constant Derivative

Quadratic, frozen at x0 = 1

Derivative Visualizer › Quadratic: a Derivative That Grows Linearly

Cubic, frozen at x0 = 1

Derivative Visualizer › Cubic: a Flat Spot That Is Not an Extremum

Sine, frozen at x0 = 1

Derivative Visualizer › Sine: the Derivative Is a Quarter Turn

Cosine, frozen at x0 = 1

Derivative Visualizer › Cosine: the Same Wave With a Sign

Exponential, frozen at x0 = 1

Derivative Visualizer › Exponential: Its Own Derivative

Logarithm, frozen at x0 = 1

Derivative Visualizer › Logarithm: a Derivative That Blows Up

Identity on [-2, 2], frozen

Mean Value Theorem Visualizer › Identity: Every Point Is a c

Quadratic on [-2, 2], frozen

Mean Value Theorem Visualizer › Quadratic: Rolle's Theorem in Disguise

Cubic on [-2, 2], frozen

Mean Value Theorem Visualizer › Cubic: Two Solutions at Once

Sine on [0, 2π], frozen

Mean Value Theorem Visualizer › Sine: a Full Period With a Flat Secant

Cosine on [0, π], frozen

Mean Value Theorem Visualizer › Cosine: a Secant That Is Not Flat

Exponential on [0, 1], frozen

Mean Value Theorem Visualizer › Exponential: Where the Average Rate Is Attained

Quadratic on [-3, 3], frozen

Optimization Visualizer › Quadratic: One Critical Point, No Ambiguity

x³ - 3x on [-3, 3], frozen

Optimization Visualizer › x³ − 3x: a Maximum and a Minimum

x³ on [-3, 3], frozen

Optimization Visualizer › x³: Where the Second-Derivative Test Fails

x⁴ - 4x² on [-3, 3], frozen

Optimization Visualizer › x⁴ − 4x²: Three Critical Points in a W

sin(x) on [-2π, 2π], frozen

Optimization Visualizer › sin(x): Four Critical Points Over Two Periods

e^(-x²) on [-3, 3], frozen

Optimization Visualizer › The Gaussian: One Maximum and Two Inflections That Are Not Critical

f(t) = t³ shaded from 0 to 2; F(2) = 4

Integrals: Definite, Indefinite & Techniques › Two Types of Integrals

Midpoint sum for x² on [0, 3] with n = 8

Integrals: Definite, Indefinite & Techniques › Definite Integrals

Left-endpoint sum for x² on [0, 3], n = 8

Definite Integrals: Riemann Sums & Properties › The Riemann Sum Construction

sin t from 0 to 2: area counted with sign

Definite Integrals: Riemann Sums & Properties › Definite Integral Notation

∫₀² t² dt = F(2) − F(0) = 8/3

Definite Integrals: Riemann Sums & Properties › Computing Definite Integrals

1/x² near x = 0: the integrand is unbounded

Improper Integrals: Convergence & Tests › What Makes an Integral Improper?

F(x) = ∫₀ˆx t² dt = x³/3 is an antiderivative of x²

Indefinite Integrals: Antiderivatives & Formulas › Antiderivatives

F(x) = ∫₀ˆx t³ dt with x = 2: F′(2) = f(2) = 8

Integration Rules & Fundamental Theorem › Fundamental Theorem of Calculus — Part 1

Left rule, frozen at n = 8

Riemann Sum Visualizer › Left Rule: Every Rectangle Falls Short

Right rule, frozen at n = 8

Riemann Sum Visualizer › Right Rule: Every Rectangle Overshoots

Midpoint rule, frozen at n = 8

Riemann Sum Visualizer › Midpoint Rule: Errors That Cancel Inside Each Strip

Trapezoid rule, frozen at n = 8

Riemann Sum Visualizer › Trapezoid Rule: Joining the Sample Points

Identity, frozen at a = 0, x = 2

Fundamental Theorem of Calculus Visualizer › Identity: Area That Grows Like a Triangle

Quadratic, frozen at a = 0, x = 2

Fundamental Theorem of Calculus Visualizer › Quadratic: the Tool's Opening State

Cubic, frozen at a = 0, x = 2

Fundamental Theorem of Calculus Visualizer › Cubic: a Faster Integrand, a Faster Accumulator

Sine, frozen at a = 0, x = 2

Fundamental Theorem of Calculus Visualizer › Sine: an Accumulator That Turns Around

Cosine, frozen at a = 0, x = 2

Fundamental Theorem of Calculus Visualizer › Cosine: Where the Two Curves Swap Roles

Exponential, frozen at a = 0, x = 2

Fundamental Theorem of Calculus Visualizer › Exponential: Accumulator and Integrand Almost Coincide

(x² − 1)/(x − 1) near x = 1: the hole

Limits: Definition, Rules & Evaluation Techniques › The Central Idea of a Limit

√x at x = 0: only the right-hand approach exists

Limits: Definition, Rules & Evaluation Techniques › One-Sided Limits

1/x² near x = 0: both sides run to +∞

Limits: Definition, Rules & Evaluation Techniques › Limits and Infinity

Wrong value at x = 1: the limit exists, f(1) misses it

Limits: Definition, Rules & Evaluation Techniques › Continuity

Condition 3 fails: lim f(x) ≠ f(1)

Continuity: Definition, Types & IVT › The Formal Definition

Hole at x = 1: removable

Continuity: Definition, Types & IVT › Removable Discontinuity

Jump at x = 0: left limit 0, right limit 1

Continuity: Definition, Types & IVT › Jump Discontinuity