Single Variable Calculus

Limits and Continuity

Definition of a Limit: \[ \lim_{x \to c} f(x) = L \]

One-Sided Limits: \[ \lim_{x \to c^+} f(x) = L \] \[ \lim_{x \to c^-} f(x) = L \]

Limit Laws: \[ \lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) \] \[ \lim_{x \to c} [f(x) - g(x)] = \lim_{x \to c} f(x) - \lim_{x \to c} g(x) \] \[ \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) \] \[ \lim_{x \to c} \left[ \frac{f(x)}{g(x)} \right] = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \]

Continuity: A function ( f(x) ) is continuous at ( x = c ) if:

1. ( f(c) ) is defined.
2. ( \lim_{x \to c} f(x) ) exists.
3. ( \lim_{x \to c} f(x) = f(c) ).

Derivatives

Definition of the Derivative: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

Power Rule: \[ \frac{d}{dx} [x^n] = nx^{n-1} \]

Product Rule: \[ \frac{d}{dx} [uv] = u'v + uv' \]

Quotient Rule: \[ \frac{d}{dx} \left[ \frac{u}{v} \right] = \frac{u'v - uv'}{v^2} \]

Chain Rule: \[ \frac{d}{dx} f(g(x)) = f'(g(x)) g'(x) \]

Higher-Order Derivatives: \[ f''(x) = \frac{d^2 f}{dx^2} \]

Applications of Derivatives

Critical Points: Points where ( f'(x) = 0 ) or ( f'(x) ) does not exist.

Increasing/Decreasing Test:

• If ( f'(x) > 0 ) on an interval, ( f(x) ) is increasing on that interval.
• If ( f'(x) < 0 ) on an interval, ( f(x) ) is decreasing on that interval.

First Derivative Test:

• If ( f' ) changes from positive to negative at ( c ), ( f(c) ) is a local maximum.
• If ( f' ) changes from negative to positive at ( c ), ( f(c) ) is a local minimum.

Concavity and Inflection Points:

• ( f''(x) > 0 ) on an interval, ( f ) is concave up on that interval.
• ( f''(x) < 0 ) on an interval, ( f ) is concave down on that interval.
• Inflection points occur where ( f''(x) = 0 ) and changes sign.

Second Derivative Test:

• If ( f''(c) > 0 ), ( f(c) ) is a local minimum.
• If ( f''(c) < 0 ), ( f(c) ) is a local maximum.

Optimization: Use derivatives to find the maximum and minimum values of a function in a given interval.

Related Rates: Use derivatives to relate the rates of change of two or more related variables.

Integrals

Indefinite Integral (Antiderivative): \[ \int f(x) , dx = F(x) + C \] where ( F'(x) = f(x) ) and ( C ) is the constant of integration.

Basic Integration Rules: \[ \int x^n , dx = \frac{x^{n+1}}{n+1} + C ] \[ \int e^x , dx = e^x + C \] \[ \int \sin x , dx = -\cos x + C \] \[ \int \cos x , dx = \sin x + C \]

Definite Integral: \[ \int_a^b f(x) , dx \] represents the area under the curve ( f(x) ) from ( x = a ) to ( x = b ).

Fundamental Theorem of Calculus:

1. \[ F(x) = \int_a^x f(t) , dt \implies F'(x) = f(x) \]
2. \[ \int_a^b f(x) , dx = F(b) - F(a) \] where ( F ) is an antiderivative of ( f ).

Integration by Substitution: \[ \int f(g(x))g'(x) , dx = \int f(u) , du \] where ( u = g(x) ).

Integration by Parts: \[ \int u , dv = uv - \int v , du \]

Applications of Integrals

Area Between Curves: \[ \int_a^b [f(x) - g(x)] , dx \] where ( f(x) ) is the upper function and ( g(x) ) is the lower function.

Volume of Solids of Revolution (Disk Method): \[ V = \pi \int_a^b [f(x)]^2 , dx \]

Volume of Solids of Revolution (Washer Method): \[ V = \pi \int_a^b \left[ [R(x)]^2 - [r(x)]^2 \right] , dx \]

Arc Length: \[ L = \int_a^b \sqrt{1 + [f'(x)]^2} , dx \]

Surface Area: \[ SA = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} , dx \]

Calculus 2 Formulas

Techniques of Integration

Integration by Partial Fractions: For a rational function ( \frac{P(x)}{Q(x)} ), decompose into partial fractions before integrating.

Trigonometric Integrals: \[ \int \sin^m x \cos^n x , dx \] Use reduction formulas or trigonometric identities.

Trigonometric Substitution: Use substitutions like ( x = a \sin \theta ), ( x = a \tan \theta ), or ( x = a \sec \theta ) to simplify integrals.

Improper Integrals: \[ \int_a^\infty f(x) , dx = \lim_{b \to \infty} \int_a^b f(x) , dx \] \[ \int_{-\infty}^\infty f(x) , dx = \lim_{a \to -\infty} \lim_{b \to \infty} \int_a^b f(x) , dx \]

Series

Sequences: A sequence is an ordered list of numbers defined by a formula ( a_n ).

Series: A series is the sum of the terms of a sequence. \[ \sum_{n=1}^\infty a_n \]

Geometric Series: \[ \sum_{n=0}^\infty ar^n = \frac{a}{1-r} \quad \text{for} \quad |r| < 1 \]

Convergence Tests:

• Integral Test: \[ \sum_{n=1}^\infty a_n \quad \text{and} \quad \int_1^\infty f(x) , dx \quad \text{either both converge or both diverge} \]
• Comparison Test: If ( 0 \le a_n \le b_n ) and ( \sum b_n ) converges, then ( \sum a_n ) converges.
• Ratio Test: \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L ]
  • If ( L < 1 ), the series converges.
  • If ( L > 1 ), the series diverges.
  • If ( L = 1 ), the test is inconclusive.
• Root Test: \[ \lim_{n \to \infty} \sqrt[n]{|a_n|} = L ]
  • If ( L < 1 ), the series converges.
  • If ( L > 1 ), the series diverges.
  • If ( L = 1 ), the test is inconclusive.

Power Series: \[ \sum_{n=0}^\infty c_n (x - a)^n \] with radius of convergence ( R ).

Taylor Series: \[ f(x) = \sum_{n=0}^\infty \frac{f^n(a)}{n!} (x - a)^n \]

Maclaurin Series: \[ f(x) = \sum_{n=0}^\infty \frac{f^n(0)}{n!} x^n \]

Parametric Equations and Polar Coordinates

Parametric Equations:

• A curve is defined by ( x = f(t) ) and ( y = g(t) ).

Slope of a Parametric Curve: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \]

Arc Length of a Parametric Curve: \[ L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} , dt \]

Polar Coordinates:

• Points are defined by ( (r, \theta) ).

Converting Polar to Cartesian: \[ x = r \cos \theta \] \[ y = r \sin \theta \]

Converting Cartesian to Polar: \[ r = \sqrt{x^2 + y^2} \] \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]

Area in Polar Coordinates: \[ A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 , d\theta \]

Arc Length in Polar Coordinates: \[ L = \int_{\alpha}^{\beta} \sqrt{\left( \frac{dr}{d\theta} \right)^2 + r^2} , d\theta \]

Differential Equations

Separable Differential Equations: \[ \frac{dy}{dx} = g(x)h(y) \] Separate and integrate: \[ \int \frac{1}{h(y)} , dy = \int g(x) , dx \]

Linear First-Order Differential Equations: \[ \frac{dy}{dx} + P(x)y = Q(x) \] Solve using the integrating factor: \[ \mu(x) = e^{\int P(x) , dx} \]

General Solution: \[ y(x) = \frac{1}{\mu(x)} \left( \int \mu(x) Q(x) , dx + C \right) \]

Applications of Differential Equations

Exponential Growth and Decay: \[ \frac{dy}{dt} = ky \] Solution: \[ y(t) = y_0 e^{kt} \]

Newton's Law of Cooling: \[ \frac{dT}{dt} = k(T - T_{\text{env}}) \] Solution: \[ T(t) = T_{\text{env}} + (T_0 - T_{\text{env}}) e^{kt} \]

Logistic Growth: \[ \frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right) \] Solution: \[ P(t) = \frac{KP_0 e^{rt}}{K + P_0 (e^{rt} - 1)} \]