Calculus II Equation Sheet

Riemann Sums and Trapezoidal Rule

Partition of an Interval

On the interval \([a,b]\) with \(n\) equal subintervals, \[ \Delta x = \frac{b-a}{n}, \qquad x_i = a + i\Delta x, \quad i = 0,1,\dots,n. \]

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General Riemann Sum

A Riemann sum for \(f(x)\) on \([a,b]\) is \[ \int_a^b f(x)\,dx \;\approx\; \sum_{i=1}^{n} f(x_i^*)\,\Delta x, \] where \(x_i^*\) is a sample point in the \(i\)th subinterval.

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Common Choices of Sample Points

• Left Riemann Sum \[ x_i^* = a + (i-1)\Delta x \]

• Right Riemann Sum \[ x_i^* = a + i\Delta x \]

• Midpoint Riemann Sum \[ x_i^* = a + \left(i-\tfrac{1}{2}\right)\Delta x \]

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Trapezoidal Rule

The trapezoidal rule approximates the integral by connecting function values at adjacent endpoints with straight line segments.

\[ \int_a^b f(x)\,dx \;\approx\; \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right]. \]

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Fundamental Theorem of Calculus

Part I (Accumulation)

If \[ F(x) = \int_a^x f(t)\,dt, \] then \[ F'(x) = f(x). \]

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Part II (Evaluation)

If \(F'(x) = f(x)\), then \[ \int_a^b f(x)\,dx = F(b) - F(a). \]

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Properties of Definite Integrals

Linearity \[ \int_a^b [f(x) + g(x)]\,dx = \int_a^b f(x)\,dx + \int_a^b g(x)\,dx \]

\[ \int_a^b c f(x)\,dx = c \int_a^b f(x)\,dx \]

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Additivity \[ \int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx \]

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Reversing Limits \[ \int_a^b f(x)\,dx = -\int_b^a f(x)\,dx \]

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Constant Functions \[ \int_a^b c\,dx = c(b-a) \]

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Indefinite Integrals

\[ \int f(x)\,dx = F(x) + C \quad \text{where } F'(x) = f(x). \]

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Basic Antiderivative Rules

Power Rule (\(n \neq -1\)) \[ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C \]

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Exponential \[ \int e^x\,dx = e^x + C \] \[ \int a^x\,dx = \frac{a^x}{\ln a} + C \]

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Logarithmic \[ \int \frac{1}{x}\,dx = \ln |x| + C \]

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Trigonometric \[ \int \sin x\,dx = -\cos x + C \] \[ \int \cos x\,dx = \sin x + C \]

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Substitution

If \(u = g(x)\) and \(du = g'(x)\,dx\), then \[ \int f(g(x))g'(x)\,dx = \int f(u)\,du \]

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Integration by Parts

\[ \int u\,dv = uv - \int v\,du \]

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Partial Fractions (General Forms)

Distinct Linear Factors \[ \frac{A}{x-a} + \frac{B}{x-b} \]

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Repeated Linear Factors \[ \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n} \]

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Irreducible Quadratic Factors \[ \frac{Ax + B}{x^2 + px + q} \]

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Repeated Irreducible Quadratics \[ \frac{A_1x + B_1}{x^2 + px + q} + \frac{A_2x + B_2}{(x^2 + px + q)^2} + \cdots \]

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Basic Derivative Rules

Rule	Function Form	Derivative Form
Constant	\(c\)	\(0\)
Power	\(x^n\)	\(n x^{n-1}\)
Constant Multiple	\(c \cdot f(x)\)	\(c \cdot f'(x)\)
Sum	\(f(x) + g(x)\)	\(f'(x) + g'(x)\)
Difference	\(f(x) - g(x)\)	\(f'(x) - g'(x)\)
Product	\(f(x)g(x)\)	\(f'(x)g(x) + f(x)g'(x)\)
Quotient	\(\dfrac{f(x)}{g(x)}\)	\(\dfrac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\)
Chain	\(f(g(x))\)	\(f'(g(x)) \cdot g'(x)\)
Inverse	\(f^{-1}(x)\)	\(\dfrac{1}{f'(f^{-1}(x))}\)

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Trigonometric Derivatives

Function	Derivative
\(\sin x\)	\(\cos x\)
\(\cos x\)	\(-\sin x\)
\(\tan x\)	\(\sec^2 x\)
\(\sec x\)	\(\sec x \tan x\)
\(\csc x\)	\(-\csc x \cot x\)
\(\cot x\)	\(-\csc^2 x\)

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Exponential & Logarithmic Derivatives

Function	Derivative
\(a^x\)	\(a^x \ln(a)\)
\(e^x\)	\(e^x\)
\(a^{g(x)}\)	\(g'(x)\, a^{g(x)} \ln(a)\)
\(e^{g(x)}\)	\(g'(x)\, e^{g(x)}\)
\(\log_a(x)\)	\(\dfrac{1}{x \ln(a)}\)
\(\ln(x)\)	\(\dfrac{1}{x}\)
\(\log_a(g(x))\)	\(\dfrac{g'(x)}{g(x)\ln(a)}\)
\(\ln(g(x))\)	\(\dfrac{g'(x)}{g(x)}\)

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Average Rate of Change

\[ AV_[a,b] = \frac{f(b) - f(a)}{b-a} \]

Limit Definition of a Derivative

\[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]

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General Functions & Forms

• Linear Forms:
  General: \(f(x) = ax+b\),
  Point-slope: \(y-y_1 = m(x-x_1)\),
  Slope-intercept: \(y=mx+b\)

• Quadratic Forms:
  General: \(ax^2+bx+c\),
  Vertex: \(a(x-h)^2+k\),
  Roots: \(\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)

• Polynomial (nth-order):
  \(f(x) = a_nx^n + \dots + a_1x + a_0\)

• Transformations:
  \(g(x) = a f(x-b) + c\)

• Sinusoid:
  \(f(x) = a \sin(kx+b)+c,\ \ T=\dfrac{2\pi}{k}\)

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Applications

• Slope of tangent line: \(m = f'(a)\)
  
• Tangent line equation: \(y = f'(a)(x-a)+f(a)\)
  
• Linear approximation: \(L(x)=f(a)+f'(a)(x-a)\)
  
• Related rates: \(\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}\)
  
• Concavity: \(f''>0 \Rightarrow\) concave up, \(f''<0 \Rightarrow\) concave down
  
• Extrema: solve \(f'(x)=0\), check sign of \(f'\) or use \(f''\)

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