MATH 124 Calculus I
Contents
Limits and Continuity
Limits
| Description | Equations |
|---|---|
| Limit | $\lim\limits_{x\to a} f(x) = L$ |
| Left-hand limit | $\lim\limits_{x\to a^-} f(x) = L$ |
| Right-hand limit | $\lim\limits_{x\to a^+} f(x) = L$ |
| Infinite limit | $\lim\limits_{x\to a} f(x) = \pm\infin$ |
| Limit at infinity | $\lim\limits_{x\to \pm\infin} f(x) = L$ |
| Limit existence | $\lim\limits_{x\to a^-} f(x) = L = \lim\limits_{x\to a^+} f(x)$ $\iff \lim\limits_{x\to a} f(x) = L$ |
| Limit non-existence | 1. $\lim\limits_{x\to a^-} f(x) \not= \lim\limits_{x\to a^+} f(x)$ 2. $\lim\limits_{x\to a} f(x) = \pm\infin$ 3. oscillation |
Limit laws
| Description | Equations |
|---|---|
| Limit addition and subtraction | $\lim\limits_{x\to a} [f(x) \pm g(x)] = \lim\limits_{x\to a} f(x) \pm \lim\limits_{x\to a} g(x)$ |
| Limit constant multiplication | $\lim\limits_{x\to a} [cf(x)] = c \lim\limits_{x\to a} f(x)$ |
| Limit multiplication | $\lim\limits_{x\to a} [f(x)g(x)] = \lim\limits_{x\to a} f(x) \cdot \lim\limits_{x\to a} g(x)$ |
| Limit division | $\lim\limits_{x\to a} \dfrac{f(x)}{g(x)} = \dfrac{\lim\limits_{x\to a} f(x)}{\lim\limits_{x\to a} g(x)} \text{ if } \lim\limits_{x\to a} g(x) \not= 0$ |
| Limit power law ($n$ is positive integer) |
$\lim\limits_{x\to a} [f(x)]^n = [\lim\limits_{x\to a} f(x)]^n$ |
| Limit at infinity of inverse power $(r>0)$ |
$\lim\limits_{x\to \pm\infin} \dfrac{1}{x^r} = 0$ |
| Limit root law ($n$ is positive integer; limit > 0 for even $n$) |
$\lim\limits_{x\to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim\limits_{x\to a} f(x)}$ |
| Direct substitution property of limit | If $f$ is a polynomial or rational function, then $\lim\limits_{x\to a} f(x) = f(a)$ |
| Limit of functions with holes | If $f(x) = g(x)$ when $x\not= a$, then $\lim\limits_{x\to a} f(x) = \lim\limits_{x\to a} g(x)$ if it exists |
| The squeeze theorem | If $f(x) \le g(x) \le h(x)$ when $x$ is near $a$ and if $\lim\limits_{x\to a} f(x) = \lim\limits_{x\to a} h(x) = L$, then $\lim\limits_{x\to a} g(x) = L$ |
Continuity
| Description | Equations |
|---|---|
| Continuous at a number $a$ | $\lim\limits_{x\to a} f(x) = f(a)$ |
| Continuous from the left at a number $a$ | $\lim\limits_{x\to a^-} f(x) = f(a)$ |
| Continuous from the right at a number $a$ | $\lim\limits_{x\to a^+} f(x) = f(a)$ |
| Continuous operations | addition, subtraction, multiplication, division |
| Continuous functions in their domain | polynomials, rational, root, trigonometric, inverse trigonometric, exponential, logarithmic functions |
| Types of discontinuity | 1. removable discontinuity 2. jump discontinuity 3. infinite discontinuity |
| Continuity of function inputs and outputs | If $f$ is continuous at $b$ and $\lim\limits_{x\to a} g(x) = b$, then $\lim\limits_{x\to a} f(g(x)) = f(\lim\limits_{x\to a} g(x))$ |
| Continuity of composite functions | If $g$ is continuous at $a$ and if $f$ is continuous at $g(a)$, then $(f\circ g)(x) = f(g(x))$ is continuous |
| Intermediate value theorem | If $f$ is continuous on the closed interval $[a, b]$, and let N be any number between $f(a)$ and $f(b)$, where $f(a) \not= f(b)$, then there exist a number $c$ in $(a, b)$ such that $f(c) = N$ |
Differentiation
Derivatives
| Description | Equations |
|---|---|
| Derivative of a function $f$ at number $a$ | $f'(a) = \lim\limits_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$: $f'(a) = \lim\limits_{x \to a} \dfrac{f(x) - f(a)}{x-a}$ |
| Derivative as a function | $f'(x) = \lim\limits_{h \to 0}\dfrac{f(x+h) - f(x)}{h}$ |
| Geometric interpretation of derivatives | The tangent line of $y = f(x)$ at $(a, f(a))$ has a slope of $f'(a)$ $m = \lim\limits_{x \to a} \dfrac{f(x) - f(a)}{x-a} = f'(a)$ |
| Derivatives and instantaneous rate of change | The derivative $f'(a)$ is the instantaneous rate of change of $y=f(x)$ with respect to $x$ when $x=a$: $v(a) = \lim\limits_{h \to 0} \dfrac{f(a+h) - f(a)}{h} = f'(a)$ |
| Differentiation and continuity | If $f$ is differentiable at $a$, then $f$ is continuous at $a$. |
| Non-differentiable conditions | 1. a corner 2. a discontinuity 3. a vertical tangent |
Differentiation rules
| Description | Equations |
|---|---|
| Constant multiple rule | $\frac{d}{dx}[cf(x)] = c\frac{d}{dx}f(x)$ |
| Addition and subtraction rule | $\frac{d}{dx}[f(x)\pm g(x)] = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x)$ |
| Product rule | $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ |
| Quotient rule (best practice: use product rule) |
$\frac{d}{dx}\dfrac{f(x)}{g(x)} = \dfrac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$ |
| Constant rule | $\frac{d}{dx}c = 0$ |
| Power rule | $\frac{d}{dx} x^n = nx^{n-1}$ |
| Chain rule | $\dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \newline \frac{d}{dx} f(g(x)) = f'(g(x))\cdot g'(x)$ |
| Linear approximations | $f(x) \approx f(a) + f'(a)(x-a)$ |
| Differentials | $dy = f'(x) \ dx$ |
Special limits
| Description | Equations |
|---|---|
| Limit associated with sine | $\lim\limits_{\theta \to 0}\dfrac{\sin\theta}{\theta} = 1$ |
| Limit associated with cosine | $\lim\limits_{\theta\to 0}\dfrac{\cos\theta - 1}{\theta} = 0$ |
| Definition of $e$ | $\lim\limits_{h\to 0}\dfrac{e^h - 1}{h} = 1$ |
| $e$ as a limit | $e = \lim\limits_{x\to 0} (1+x)^{1/x}$ |
| $e$ as a limit | $e = \lim\limits_{n\to \infin} (1+\frac{1}{n})^{n}$ |
Table of derivatives
| Function $f(x)$ | Derivative $f'(x)$ | Function $f(x)$ | Derivative $f'(x)$ |
|---|---|---|---|
| $c$ | $0$ | $x^n$ | $nx^{n-1}$ |
| $x$ | $1$ | $\lvert x \rvert$ | $\dfrac{x}{\lvert x \rvert}$ |
| $e^x$ | $e^x$ | $\ln x$ | $\dfrac{1}{x}$ |
| $a^x$ | $a^x\ln(a)$ | $\log_{a}x$ | $\dfrac{1}{x\ln(a)}$ |
| $\sin x$ | $\cos x$ | $\sec x$ | $\sec x \tan x$ |
| $\cos x$ | $-\sin x$ | $\csc x$ | $-\csc x \cot x$ |
| $\tan x$ | $\sec^2 x$ | $\cot x$ | $-\csc^2 x$ |
| $\arcsin x$ | $\dfrac{1}{\sqrt{1-x^2}}$ | $\arctan x$ | $\dfrac{1}{1+x^2}$ |
| $\arccos x$ | $\dfrac{-1}{\sqrt{1-x^2}}$ |
Applications of Differentiation
Absolute extreme values
| Description | Equations |
|---|---|
| Extreme values | Maximum and minimum values of $f$ |
| Absolute maximum | $f(a) \ge f(x)$ for all $x \in D$ |
| Absolute maximum | $f(a) \le f(x)$ for all $x \in D$ |
| Local maximum | $f(a) \ge f(x)$ for $x$ near $a$ |
| Local minimum | $f(a) \le f(x)$ for $x$ near $a$ |
| Critical number $c$ | $f'(c) = 0$ or $f'(c)$ does not exist |
| Extreme value theorem | If $f$ is continuous on a closed interval $[a, b]$, then $f$ attains an absolute maximum $f(c)$ and absolute minimum $f(d)$ at some number $c, d \in [a, b]$ |
| Fermat’s theorem | If $f$ has a local maximum or minimum at $c$, then $c$ is a critical number of f |
| Closed interval method (Finding the absolute max and min in $[a,b]$) |
1. Find the values of $f$ at critical numbers of $f$ in $(a, b)$ 2. Find the values of $f$ at the endpoints of the interval 3. Compare the values. The largest is the abs max; the smallest is the abs min |
The mean value theorem
| Description | Equations |
|---|---|
| Rolle’s theorem | If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and endpoints $f(a) = f(b)$, then there is a number $c \in (a,b)$ such that $f'(c) = 0$ |
| The mean value theorem | If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, then there is a number $c \in (a,b)$ such that $f'(c) = \dfrac{f(b) - f(a)}{b-a}$ |
| Functions with derivative of zero are constants | If $f'(x) = 0$ for all $x \in (a,b)$, then $f$ is constant on $(a,b)$ |
| Functions with same derivatives are vertical translations of each other | If $f'(x) = g'(x)$ for all $x \in (a,b)$, then $f(x) = g(x) + c$ on $(a,b)$ |
Local extreme values
| Description | Equations |
|---|---|
| Increasing | $f(x_1) < f(x_2)$ for $x_1 < x_2$ in $I$ |
| Deceasing | $f(x_1) > f(x_2)$ for $x_1 < x_2$ in $I$ |
| Increasing/decreasing test | (a) If $f'(x)>0$ on an interval, then $f$ is increasing on that interval. (b) If $f'(x)<0$ on an interval, then $f$ is decreasing on that interval. |
| First derivative test | If $c$ is a critical value of continuous function $f$, then (a) If $f'$ changes $+ \to -$ at c, then $f$ has a local max at $c$ (b) If $f'$ changes $- \to +$ at c, then $f$ has a local min at $c$ (c) If $f'$ has no sign change at c, then $f$ has no local max/min at $c$ |
| Concave upward | If the graph of $f$ lies above all its tangents on an interval $I$ (slope increases) |
| Concave downward | If the graph of $f$ lies below all its tangents on an interval $I$ (slope decreases) |
| Inflection point | The point at which the curve changes concavity |
| Concavity test | (a) If $f''(x)>0$ on an interval, then $f$ is concave upward on that interval. (b) If $f''(x)<0$ on an interval, then $f$ is concave downward on that interval. |
| Second derivative test | If $f''$ is continuous near $c$, (a) If $f'(c) = 0$ and $f''(c)>0$, then $f$ has a local min at $c$ (b) If $f'(c) = 0$ and $f''(c)<0$, then $f$ has a local max at $c$ |
L’Hospital’s rule
| Description | Equations |
|---|---|
| Indeterminate forms | $\lim\limits_{x\to a}\dfrac{f}{g} = \dfrac{0}{0}, \dfrac{\infin}{\infin}$ |
| L’Hospital’s rule | If $f$ and $g$ are differentiable and $g'(x) \not= 0$ on open interval $I$ that contains $a$, and if the division has indeterminate form of $\frac{0}{0}$ or $\frac{\infin}{\infin}$, then $\lim\limits_{x \to a}\dfrac{f(x)}{g(x)} = \lim\limits_{x \to a}\dfrac{f'(x)}{g'(x)}$ |
| Indeterminate products | $\lim\limits_{x\to a}fg = 0 \cdot \infin \newline \lim\limits_{x\to a}fg = \dfrac{f}{1/g} = \dfrac{g}{1/f}$ |
| Indeterminate differences | $\lim\limits_{x\to a}(f-g) = \infin-\infin$ |
| Indeterminate powers | $\lim\limits_{x\to a}[f(x)]^{g(x)} = 0^0, \infin^0, 1^{\infin} \newline y \equiv [f(x)]^{g(x)} \newline \ln y = g(x) \ln f(x)$ |
Newton’s method
| Description | Equations |
|---|---|
| Newton’s method | $x_{n+1} = x_{n} - \dfrac{f(x_n)}{f'(x_n)}$ |