Calculus 1 – Complete Reference Guide

Limits • Derivatives • Integrals • Applications • Problem-Solving Strategies

Limits: Definition & Notation

Informal Definition

$\lim_{x \to a} f(x) = L$ means as $x$ approaches $a$, $f(x)$ approaches $L$.

One-Sided Limits

Left: $\lim_{x \to a^-} f(x)$ (approach from left)

Right: $\lim_{x \to a^+} f(x)$ (approach from right)

Limit exists ⟺ both one-sided limits exist and are equal

Infinite Limits

$\lim_{x \to a} f(x) = \infty$ — vertical asymptote at $x=a$

$\lim_{x \to \infty} f(x) = L$ — horizontal asymptote at $y=L$

Memory Tip: Think of a limit as "What value is $f(x)$ getting close to?" NOT "What is $f(a)$?"

Limit Laws

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then:

Law	Formula
Constant	$\lim_{x \to a} c = c$
Sum	$\lim [f \pm g] = L \pm M$
Product	$\lim [f \cdot g] = L \cdot M$
Quotient	$\lim [f/g] = L/M$ (if $M \neq 0$)
Power	$\lim [f]^n = L^n$
Root	$\lim \sqrt[n]{f} = \sqrt[n]{L}$

Special Limits (Memorize!)

Sine limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Cosine limit $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$

e definition $\lim_{n \to \infty} (1 + \frac{1}{n})^n = e$

Exponential $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

Logarithm $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$

⚠ Caution: These only work with $x$ in radians!

Continuity

Definition

$f$ is continuous at $x = c$ if ALL three conditions hold:

1. $f(c)$ is defined

2. $\lim_{x \to c} f(x)$ exists

3. $\lim_{x \to c} f(x) = f(c)$

Types of Discontinuity

Removable: Limit exists but ≠ $f(a)$

Jump: Left & right limits differ

Infinite: Limit is ±∞

Intermediate Value Theorem

If $f$ is continuous on $[a,b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c \in (a,b)$ where $f(c) = N$.

Use for: Proving roots exist! If $f(a) < 0$ and $f(b)> 0$, there's a root in $(a,b)$.

Squeeze Theorem

If $g(x) \leq f(x) \leq h(x)$ near $a$ and

$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$

then $\lim_{x \to a} f(x) = L$

Example: $\lim_{x \to 0} x^2 \sin(1/x) = 0$ since $-x^2 \leq x^2\sin(1/x) \leq x^2$

Derivative Definition

Limit Definition

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Notation

$f'(x), \frac{dy}{dx}, \frac{d}{dx}[f(x)], Df(x), y'$

Interpretation

Geometric: Slope of tangent line at $(a, f(a))$

Physical: Instantaneous rate of change

Velocity: $v(t) = s'(t)$

Acceleration: $a(t) = v'(t) = s''(t)$

⚠ Key Fact: Differentiable ⟹ Continuous, but Continuous ⟹̸ Differentiable (e.g., $|x|$ at $x=0$)

Basic Derivative Rules

Rule	Formula
Constant	$\frac{d}{dx}[c] = 0$
Power	$\frac{d}{dx}[x^n] = nx^{n-1}$
Const. Mult.	$\frac{d}{dx}[cf] = c \cdot f'$
Sum/Diff.	$\frac{d}{dx}[f \pm g] = f' \pm g'$
Product	$(fg)' = f'g + fg'$
Quotient	$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$
Chain	$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

Product Rule: "First times derivative of second, plus second times derivative of first"

Quotient Rule: "Low d-high minus high d-low, over low squared"

Trigonometric Derivatives

$\frac{d}{dx}[\sin x]$	$= \cos x$
$\frac{d}{dx}[\cos x]$	$= -\sin x$
$\frac{d}{dx}[\tan x]$	$= \sec^2 x$
$\frac{d}{dx}[\cot x]$	$= -\csc^2 x$
$\frac{d}{dx}[\sec x]$	$= \sec x \tan x$
$\frac{d}{dx}[\csc x]$	$= -\csc x \cot x$

Pattern: "Co" functions have negative derivatives. Derivatives of sec/csc include themselves.

Inverse Trig Derivatives

$\frac{d}{dx}[\sin^{-1} x]$	$= \frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}[\cos^{-1} x]$	$= \frac{-1}{\sqrt{1-x^2}}$
$\frac{d}{dx}[\tan^{-1} x]$	$= \frac{1}{1+x^2}$
$\frac{d}{dx}[\cot^{-1} x]$	$= \frac{-1}{1+x^2}$
$\frac{d}{dx}[\sec^{-1} x]$	$= \frac{1}{\|x\|\sqrt{x^2-1}}$
$\frac{d}{dx}[\csc^{-1} x]$	$= \frac{-1}{\|x\|\sqrt{x^2-1}}$

Exponential & Logarithmic

$\frac{d}{dx}[e^x]$	$= e^x$
$\frac{d}{dx}[a^x]$	$= a^x \ln a$
$\frac{d}{dx}[\ln x]$	$= \frac{1}{x}$
$\frac{d}{dx}[\log_a x]$	$= \frac{1}{x \ln a}$
$\frac{d}{dx}[e^{g(x)}]$	$= e^{g(x)} \cdot g'(x)$
$\frac{d}{dx}[\ln g(x)]$	$= \frac{g'(x)}{g(x)}$

Logarithmic Diff: For $y = x^x$, take $\ln$ of both sides: $\ln y = x \ln x$, then differentiate implicitly.

Implicit Differentiation

1. Differentiate both sides w.r.t. $x$

2. Apply chain rule: $\frac{d}{dx}[y] = \frac{dy}{dx}$

3. Collect all $\frac{dy}{dx}$ terms on one side

4. Factor out $\frac{dy}{dx}$ and solve

Example: $x^2 + y^2 = 25$
$2x + 2y\frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}$

Related Rates Strategy

1. Draw diagram, label variables

2. Identify given rates (known $\frac{d}{dt}$)

3. Identify unknown rate (find what?)

4. Write equation relating variables

5. Differentiate w.r.t. time $t$

6. Substitute known values and solve

⚠ Critical: Always differentiate BEFORE substituting values!

Common Formulas

Circle: $A = \pi r^2$, $C = 2\pi r$

Sphere: $V = \frac{4}{3}\pi r^3$, $S = 4\pi r^2$

Cone: $V = \frac{1}{3}\pi r^2 h$

Cylinder: $V = \pi r^2 h$

Pythagorean: $a^2 + b^2 = c^2$

L'Hôpital's Rule

If $\lim_{x \to a} \frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then:

$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

Other Indeterminate Forms

$0 \cdot \infty$: Rewrite as $\frac{f}{1/g}$ or $\frac{g}{1/f}$

$\infty - \infty$: Find common denominator

$0^0, 1^\infty, \infty^0$: Take ln, use $e^{\ln(...)}$

⚠ Verify: Always check it's indeterminate first! Can apply repeatedly if still indeterminate.

Curve Sketching Checklist

□ Domain

□ Intercepts ($x$ and $y$)

□ Symmetry (even/odd/periodic)

□ Asymptotes (V, H, oblique)

□ $f'(x)$: increasing/decreasing

□ Critical points → local extrema

□ $f''(x)$: concavity

□ Inflection points

First & Second Derivative Tests

First Derivative Test

$f'$ Changes	Conclusion
$+ \to -$	Local maximum
$- \to +$	Local minimum
Same sign	Not an extremum

Second Derivative Test

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

$f''(c)$	Conclusion
$> 0$	Local minimum
$< 0$	Local maximum
$= 0$	Inconclusive (use 1st test)

Concavity

$f''(x) > 0$: Concave up (cup ∪)

$f''(x) < 0$: Concave down (cap ∩)

$f''(x) = 0$: Possible inflection point

Optimization Strategy

1. Identify quantity to maximize/minimize

2. Draw diagram, assign variables

3. Write primary equation

4. Use constraint to get 1 variable

5. Find critical points ($f' = 0$)

6. Test critical pts & endpoints

7. Verify answer makes sense

Closed Interval Method: Compare $f$ at all critical points AND both endpoints. Largest = abs max, smallest = abs min.

Basic Antiderivatives

$f(x)$	$\int f(x)\,dx$
$k$	$kx + C$
$x^n$ $(n \neq -1)$	$\frac{x^{n+1}}{n+1} + C$
$\frac{1}{x}$	$\ln\|x\| + C$
$e^x$	$e^x + C$
$a^x$	$\frac{a^x}{\ln a} + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\csc^2 x$	$-\cot x + C$
$\sec x \tan x$	$\sec x + C$
$\csc x \cot x$	$-\csc x + C$
$\frac{1}{\sqrt{1-x^2}}$	$\sin^{-1} x + C$
$\frac{1}{1+x^2}$	$\tan^{-1} x + C$

⚠ Don't forget the $+C$! All antiderivatives include an arbitrary constant.

U-Substitution

1. Choose $u = g(x)$ (inner function)

2. Compute $du = g'(x)\,dx$

3. Rewrite integral in terms of $u$

4. Integrate with respect to $u$

5. Substitute back (or change limits)

Example: $\int 2x \cos(x^2)\,dx$
Let $u = x^2$, $du = 2x\,dx$
$= \int \cos u\,du = \sin u + C = \sin(x^2) + C$

Definite integrals: Either change limits when you substitute, OR substitute back before evaluating.

Fundamental Theorem of Calculus

Part 1 (Derivative of Integral)

If $F(x) = \int_a^x f(t)\,dt$, then $F'(x) = f(x)$

Chain rule: $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)$

Part 2 (Evaluation)

$\int_a^b f(x)\,dx = F(b) - F(a)$

where $F'(x) = f(x)$ (F is any antiderivative of f)

Net Change: $\int_a^b f'(x)\,dx = f(b) - f(a)$
The integral of a rate gives total change.

Definite Integral Properties

$\int_a^a f(x)\,dx = 0$

$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$

$\int_a^b cf(x)\,dx = c\int_a^b f(x)\,dx$

$\int_a^b [f \pm g]\,dx = \int_a^b f\,dx \pm \int_a^b g\,dx$

$\int_a^b f\,dx + \int_b^c f\,dx = \int_a^c f\,dx$

Average Value

$f_{avg} = \frac{1}{b-a}\int_a^b f(x)\,dx$

Area & Volume

Area Between Curves

$A = \int_a^b |f(x) - g(x)|\,dx$ (vertical)

$A = \int_c^d |f(y) - g(y)|\,dy$ (horizontal)

Volume by Revolution

Method	Formula
Disk	$V = \pi\int_a^b [R(x)]^2\,dx$
Washer	$V = \pi\int_a^b [R^2 - r^2]\,dx$
Shell	$V = 2\pi\int_a^b x \cdot f(x)\,dx$

Disk/Washer: Perpendicular to axis. Shell: Parallel to axis.

⚠ Common Pitfalls

Product Rule: $(fg)' \neq f' \cdot g'$

Chain Rule: $\frac{d}{dx}[f(g(x))] \neq f'(g(x))$
Must multiply by $g'(x)$!

Powers: $\frac{d}{dx}[x^n] \neq x^{n-1}$
Don't forget the coefficient $n$!

Integration: $\int fg\,dx \neq \int f\,dx \cdot \int g\,dx$

Related Rates: Differentiate THEN substitute!

CALCULUS 1 Complete Reference