Quick Reference

This page provides quick syntax lookups for R, Mathematica, and common math formulas used throughout Math 119.

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See the Common Errors FAQ for copy-paste fixes to frequent issues.

Solving Equations with uniroot()

Basic pattern:

# Define function (what you want to equal zero)
f <- function(x){ ... - ... }

# Solve: find where f(x) = 0
uniroot(f, c(a, b))$root

Example: Solve \(3x - 5 = e^{-x}\)

# Rewrite as: (3x - 5) - e^(-x) = 0
f <- function(x){ 3*x - 5 - exp(-x) }

# Plot to find interval containing zero
x <- seq(-5, 10, 0.1)
plot(x, f(x), type="l")
abline(h=0, col="red")

# Solve
uniroot(f, c(0, 5))$root

Common error: “f() values at end points not of opposite sign” - Your interval [a,b] doesn’t contain a zero - Plot the function first to find the right interval

Mathematical Functions

Math Notation	R Function	Example
\(e^x\)	`exp(x)`	`exp(2)` → 7.389
\(\ln(x)\)	`log(x)`	`log(10)` → 2.303
\(\log_2(x)\)	`log2(x)`	`log2(8)` → 3
\(\log_{10}(x)\)	`log10(x)`	`log10(100)` → 2
\(\sqrt{x}\)	`sqrt(x)`	`sqrt(16)` → 4
\(\\|x\\|\)	`abs(x)`	`abs(-5)` → 5

Defining Functions

Single variable:

f <- function(x){ x^2 + 3*x - 5 }
f(2)  # Evaluate at x=2

Multiple variables:

f <- function(x, a, b){ a*x^2 + b*x }
f(2, a=3, b=-1)  # Evaluate at x=2, a=3, b=-1

Piecewise functions (using ifelse):

f <- function(x){
  ifelse(x >= 0, x^2, -x)
}

Plotting

Basic plot:

# Plot function f(x) from x=a to x=b
x <- seq(a, b, 0.1)
plot(x, f(x), type="l")

Add elements:

# Add horizontal line at y=0
abline(h=0, col="red")

# Add vertical line at x=5
abline(v=5, col="blue")

# Add another function to same plot
lines(x, g(x), col="green")

# Add points
points(2, f(2), pch=16, col="red")

Multiple plots side-by-side:

par(mfrow=c(1,2))  # 1 row, 2 columns
plot(x, f(x), type="l")
plot(x, g(x), type="l")

Common R Errors

“Error: object ‘f’ not found”

You forgot to define the function before using it. Add:

f <- function(x){ ... }

“Error in f(x): could not find function ‘f’”

Same as above - define the function first.

“Error: unexpected ‘}’”

Missing opening brace { or mismatched braces. Check your function definition.

Integration

Definite integral (exact symbolic):

Integrate[f, {x, a, b}]

Example: \(\int_0^{15} k(15-x) \, dx\)

Integrate[k*(15-x), {x, 0, 15}]

Numerical integration (when symbolic fails):

NIntegrate[f, {x, a, b}]

Example: Normal distribution probability

NIntegrate[1/Sqrt[50*Pi]*Exp[-(1/50)*(x-21)^2], {x, -Infinity, 22}]

Solving Equations

Solve symbolically:

Solve[equation, variable]

Example: Solve \(\frac{30x - x^2}{225} = 0.6\)

Solve[30*x - x^2 == 135, x, Reals]

Solve numerically:

NSolve[equation, variable]

Example: Find 80th percentile

NSolve[NIntegrate[1/Sqrt[50*Pi]*Exp[-(1/50)*(x-21)^2],
                  {x, -Infinity, xp}] == 0.80, xp]

Expected Value and Variance

Expected value: \(E[X] = \int x \cdot f(x) \, dx\)

Integrate[x * f[x], {x, a, b}]

Second moment: \(E[X^2] = \int x^2 \cdot f(x) \, dx\)

Integrate[x^2 * f[x], {x, a, b}]

Variance: \(\text{Var}(X) = E[X^2] - (E[X])^2\)

ex2 = Integrate[x^2 * f[x], {x, a, b}]
ex = Integrate[x * f[x], {x, a, b}]
variance = ex2 - ex^2

Common Mathematica Errors

Wrong brackets

Use [] for functions: Integrate[...], not Integrate(...)
Use {} for lists: {x, 0, 15}

Multiplication

Use * explicitly: 3*x not 3x

Comparison

Use == for equations: x^2 == 4, not x^2 = 4

Derivative Rules

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

Constant Multiple: \[\frac{d}{dx}[cf(x)] = c \cdot f'(x)\]

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

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

Quotient Rule: \[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}\]

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

Example: \(\frac{d}{dx}[(3x^2 + 1)^5]\)

Outer function: \(f(u) = u^5\) → \(f'(u) = 5u^4\)
Inner function: \(g(x) = 3x^2 + 1\) → \(g'(x) = 6x\)
Chain rule: \(f'(g(x)) \cdot g'(x) = 5(3x^2 + 1)^4 \cdot 6x\)

Common Derivatives

Function	Derivative
\(x^n\)	\(nx^{n-1}\)
\(e^x\)	\(e^x\)
\(e^{kx}\)	\(ke^{kx}\)
\(\ln(x)\)	\(\frac{1}{x}\)
\(\log_a(x)\)	\(\frac{1}{x \ln(a)}\)

Logarithm Properties

Product: \[\ln(ab) = \ln(a) + \ln(b)\]

Quotient: \[\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\]

Power: \[\ln(a^b) = b \ln(a)\]

Sum of logs (used in maximum likelihood): \[\ln\left(\prod_{i=1}^{n} x_i\right) = \sum_{i=1}^{n} \ln(x_i)\]

Example: \(\ln\left(\prod_{i=1}^{44} (y_i + ax_i)^2\right) = 2\sum_{i=1}^{44} \ln(y_i + ax_i)\)

Second Derivative Test

At a critical point where \(f'(c) = 0\):

If \(f''(c) > 0\) → local minimum (concave up)
If \(f''(c) < 0\) → local maximum (concave down)
If \(f''(c) = 0\) → test inconclusive

Probability Rules

Complement: \[P(A^c) = 1 - P(A)\]

Addition Rule (mutually exclusive): \[P(A \cup B) = P(A) + P(B)\]

Multiplication Rule (independent): \[P(A \cap B) = P(A) \cdot P(B)\]

Conditional Probability: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\]

Bayes’ Theorem: \[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\]

Expected Value and Variance

Discrete random variable: \[E[X] = \sum_{i} x_i \cdot p(x_i)\] \[\text{Var}(X) = E[X^2] - (E[X])^2\]

Continuous random variable: \[E[X] = \int x \cdot f(x) \, dx\] \[\text{Var}(X) = \int x^2 \cdot f(x) \, dx - \left(\int x \cdot f(x) \, dx\right)^2\]

Solving Equations with uniroot()

Mathematical Functions

Defining Functions

Plotting

Common R Errors

Integration

Solving Equations

Expected Value and Variance

Common Mathematica Errors

Derivative Rules

Common Derivatives

Logarithm Properties

Second Derivative Test

Probability Rules

Expected Value and Variance

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