Cheat Sheet
MATH 125
Navigation
Limits
Limit Laws
| Law | Equation |
|---|---|
| Sum Law | \(\lim_{x\to a} \left[f(x)+g(x)\right] = \lim_{x\to a} f(x) + \lim_{x\to a} g(x)\) |
| The Difference Law | \(\lim_{x\to a}\left[f(x)-g(x)\right] = \lim_{x\to a}f(x) - \lim_{x\to a}g(x)\) |
| Constant Multiple Law | \(\lim_{x\to a} \left[ cf(x) \right] = c \lim_{x\to a} f(x)\) |
| Product Law | \(\lim_{x\to a}\left[f(x) g(x)\right] = \lim_{x\to a}f(x) \cdot \lim_{x\to a}g(x)\) |
| Quotient Law | \(\lim_{x\to a}\frac{f(x)}{g(x)} = \frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)} \text{ if } \lim_{x\to a}g(x) \neq 0\) |
| Power Law | \(\lim_{x\to a}\left[f(x)\right]^n = \left[\lim_{x\to a}f(x)\right]^n\) |
| Constant Law | \(\lim_{x\to a}c = c\) |
| Linear Law | \(\lim_{x\to a} x = a\) |
| Exponential Law | \(\lim_{x\to a} x^n = a^n\) |
| Specific Root Law | \(\lim_{x\to a} \sqrt[n]{x} = \sqrt[n]{a}\) |
| General Root Law | \(\lim_{x\to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x\to a} f(x)}\) |
Limit Case Dynamics
| Limit Type | Result |
|---|---|
| \(c + \infty\) | \(\infty\) |
| \(c - \infty\) | \(-\infty\) |
| \(\infty + \infty\) | \(\infty\) |
| \(-\infty-\infty\) | \(-\infty\) |
| \(\infty-\infty\) | Needs further work |
| \(c\cdot \infty\), \(c>0\) | \(\infty\) |
| \(c\cdot \infty\), \(c<0\) | \(-\infty\) |
| \(0 \cdot \infty\) | Needs further work |
| \(\infty \cdot \infty\) | \(\infty\) |
| \(\infty \cdot -\infty\) | \(-\infty\) |
| \(-\infty\cdot-\infty\) | \(\infty\) |
| \(\frac{c}{\pm\infty}\) | \(0\) |
| \(\frac{\pm\infty}{\pm\infty}\) | Needs further work |
| \(\frac{0}{0}\) | Needs further work |
| \(\frac{c}{0}\) | Needs further work |
Methods to Solve Limits
- Distribution Substitution Property: if \(f(x)\) is an algebraic function and \(a\) is in the domain of \(f\), then \(\lim_{x\to a} f(x) = f(a)\). Much of solving for limits is changing the expression of a limit such that the DSP can be used.
- If \(\lim_{x\to a}\frac{P(x)}{Q(x)}\) is a rational function with \(P(a)=Q(a)=0\), then \((x-a)\) must be a factor of \(P(x)\) and \(Q(x)\). Factor it out and remove it from the rational function.
- In \(\frac{0}{0}\) cases, use rationalization. Multiply numerator/denominator with square root terms by the conjugate.
Asymptotes
To find horizontal asymptotes of \(f(x)\), compute \(\lim_{x\to\pm\infty} f(x)\).
Rules for computing \(x\to\pm\infty\):
- \(\lim_{x\to\infty} \frac{1}{x^r} = 0\) for \(r>0\).
- \(\lim_{x\to\infty} a^x = \infty\) for \(a > 1\); \(0\) for \(a < 1\).
- \(\lim_{x\to-\infty} a^x = 0\) for \(a > 1\); \(\infty\) for \(a < 1\).
- Polynomials: identify the term with the highest power; only that term plays the important role.
Limits to Infinity
- In a polynomial, the highest power domnates the long term behavior.
- In rational functions, the highest power dominates.
- If roots are involved in a rational function, like \(\lim_{x\to -\infty }\frac{\sqrt{x^2+4x}-2x}{x+4}\), make sure you take into account the negative sign of limits to negative infinity when pulling division by the highest polynomial term into the root.
- \(r^x\) dominates over every \(x^r\) (taking into account directionality of the exponential).
Technique: When taking a limit towards infinity or negative infinity for functions involving polynomial terms, find a way to divide all terms by the term by the highest polynomial term. For instance, when calculating \(\lim_{x\to \infty }\frac{5x^4+x^3}{4x^4-9x}\) one would divide the terms in the numerator and the denominator by \(x^4\):
\[\lim_{x\to \infty }\frac{\frac{5x^4}{x^4}+\frac{x^3}{x^4}}{\frac{4x^4}{x^4}-\frac{9x}{x^4}}=\li _{x\to \:\infty \:}\frac{5+x^{-1}}{4-9x^{-3}}=\frac{5}{4}\]Derivatives
Formal Definition
When \(x\to a\), the slope of the secant line will approach the slope of the tangent line.
\[f'(x) = \lim_{x\to a}\frac{f(x) - f(a)}{x-a}\]\(f'(a)\) is the derivative of \(f\) at \(x=a\). The units of the derivative are \(\frac{y \text{ unit}}{x \text{ unit}}\).
Derivative Patterns
- When \(f(x)\) is increasing, \(f'(x)\) is positive.
- When \(f(x)\) is decreasing, \(f'(x)\) is positive.
- When \(f(x)\) is at a local minima or maxima, \(f'(x) = 0\).
- When \(f(x)\) is convex (pointing up, smiling face), \(f'(x)\) is increasing.
- When \(f(x)\) is concave (pointing down, frowning face), \(f'(x)\) is decreasing.
Fundamental Differentiation Formulas
| Name | Rule |
|---|---|
| Constant Function Rule | \(\frac{d}{dx} c = 0\) |
| Power Rule | \(\frac{d}{dx} x^n = nx^{n-1}\) |
| Euler Number Rule | \(\frac{d}{dx} e^x = e^x\) |
| Constant Multiplier Rule | \((cf)' = cf'\) |
| Function Sum Rule | \((f + g)' = f' + g'\) |
| Function Difference Rule | \((f-g)' = f' - g'\) |
| Function Product Rule | \((fg)' = fg' + gf'\) |
| Function Quotient Rule | \(\left(\frac{f}{g}\right)' = \frac{gf' - fg'}{g^2}\) |
| Chain Rule | \(\frac{d}{dx} f(g(x)) = f'(g(x))\cdot g'(x)\) |
Note that the limit \(\lim_{h\to 0} \frac{\sin(h)}{h} = 1\).
Function Specific Differentiation Formulas
| Function | Derivative |
|---|---|
| \(\ln x\) | \(\frac{1}{x}\) |
| \(a^x\) | \(a^x \ln(a)\) |
| \(\log_a (x)\) | \(\frac{1}{x\ln (a)}\) |
| \(\sin x\) | \(\cos x\) |
| \(\cos x\) | \(-\sin x\) |
| \(\tan x\) | \(\frac{1}{\cos^2 x} = \sec^2 x\) |
| \(\csc x\) | \(-\frac{\cos x}{\sin^2 x} = -\cot x \cdot \csc x\) |
| \(\sec x\) | \(\frac{\sin x}{\cos^2 x} = \tan x \cdot\sec x\) |
| \(\cot x\) | \(-\frac{1}{\sin^2 x} = -\csc^2 x\) |
| \(\sin^{-1} x\) | \(\frac{1}{\sqrt{1-x^2}}\) |
| \(\cos^{-1} x\) | \(-\frac{1}{\sqrt{1-x^2}}\) |
| \(\tan^{-1} x\) | \(\frac{1}{1+x^2}\) |
Differentiation Techniques
- Implicit differentiation. If a curve cannot be written explicitly as \(y = ...\), differentiate both sides of the equation, treat \(y\) as a function of \(x\) and thus use chain rule to differentiate, solve for \(y'\).
- Parametric differentiation. If you are given \(x=x(t)\) and \(y=y(t)\), solve for \(\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\).
- Logarithmic differentiation. For complex derivatives as well as derivatives of functions in the form \(f(x)^{g(x)\), use logarithmic differentiation by taking the log of both sides and implicitly differentiating, then solving for \(y'\).
Derivative Applications
Real Life Rates
Often, in real-world problems we want to understand the dynamics of related rates - as one quantity changes, another desired quantity correspondingly changes.
- Use geometric tools to establish a mathematical relationship.
- Identify known variables and derivatives.
- Implicitly differentiate the mathematical relationship with respect to time or a different appropriate variable.
- Plug in knowns and solve for the desired derivative.
Linearization
We can approximate the value of a function by using the line tangent to a nearby convenient point.
Consider a convenient point \((c_x, c_y)\), a desired point \((d_x, d_y)\), and a function to be evaluated \(f\).
\[d_y = f'(d_x - c_x) + c_y\]Critical Points
- Critical numbers exist where the derivative is zero or does not exist.
- All extrema are critical numbers, but not all critical numbers are extrema.
- When solving for when the derivative is equal to 0, factor but do not get rid of solutions.
- When solving for absolute extrema within a defined domain \([a, b]\), make sure to consider \(x=a\) and \(x=b\) as candidates.
L’Hopital’s Rule
\[\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}\]- Works only for 0/0 or infty/infty.
- When working with roots, it may be easier not to use L’Hopital’s rule.
- Indeterminate limit types require more work.
- Rewrite \(fg\) as \(\frac{f}{\frac{1}{g}}\) or \(\frac{g}{\frac{1}{f}}\) to use L’Hopital’s.
- For \([\infty - \infty]\), factor out like \(f - g = f\left(1 - \frac{g}{f}\right)\)
- For exponential limits, use a logarithmic approach.