MATH 100 & 180: What to Memorize Before the Exam

Limits at Infinity & Dominant Terms

• $\text{Who wins as } x \to \infty$
  $$\log x \ll x^p \ll e^x$$
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  Any positive power beats $\log x$; $e^x$ beats any power. This is for terms added or subtracted: multiplied or divided pieces all stay.
• $$\lim_{x \to -\infty} e^x = 0$$
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  So as $x \to -\infty$ the polynomial terms win.
• $$\lim_{x \to -\infty} e^{-x} = +\infty$$
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  $-x \to +\infty$, so $e^{-x}$ grows.
• $$\lim_{x \to -\infty} x^n = \begin{cases} +\infty & n \text{ even} \\ -\infty & n \text{ odd} \end{cases}$$
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  Odd powers keep the minus sign: $x^3 \to -\infty$, $x^2 \to +\infty$.
• $$\lim_{x \to \pm\infty} \dfrac{c}{x^n} = 0$$
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  For any $n > 0$, at both ends.
• $$\lim_{x \to -\infty} a^x = \begin{cases} 0 & a > 1 \\ +\infty & 0 < a < 1 \end{cases}$$
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  So $2^x \to 0$ but $(0.5)^x \to \infty$: a base below $1$ flips.
• $$\lvert x \rvert = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$
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  Negative inside: flip the sign. So $\lvert -3 \rvert = 3$, and $\lvert x - 2 \rvert = 2 - x$ when $x < 2$.
• $$\sqrt{x^2} = |x|$$
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  Equals $-x$ when $x \to -\infty$.

The Derivative from the Definition

• $$f'(x) = \lim_{h \to 0}\dfrac{f(x + h) - f(x)}{h}$$
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  When the question says “use the definition”.

Differentiation Rules

• $$\dfrac{d}{dx}\,x^n = n\,x^{n-1}$$
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  Any constant $n$: roots and fractions too. Rewrite $\sqrt{x} = x^{1/2}$, $\dfrac{1}{x^3} = x^{-3}$ first.
• $$\dfrac{d}{dx}\,e^x = e^x$$
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• $$\dfrac{d}{dx}\,a^x = a^x \log a$$
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  For a constant base like $2^x$. Not $x\,a^{x-1}$.
• $$\dfrac{d}{dx}\,\log x = \dfrac{1}{x}$$
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  $\log$ is the natural log. With an inside: $\dfrac{u'}{u}$.
• $$\dfrac{d}{dx}\,\sin x = \cos x$$
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• $$\dfrac{d}{dx}\,\cos x = -\sin x$$
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  Watch the minus sign.
• $$\dfrac{d}{dx}\,\tan x = \sec^2 x$$
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• $$(fg)' = f'g + fg'$$
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• $$\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$$
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  "Low d-high minus high d-low, over low squared."
• $$\dfrac{d}{dx}\,f(g(x)) = f'(g(x))\cdot g'(x)$$
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  Outside derivative, inside kept, times inside derivative.

Tangent Lines & Linear Approximation

• $\text{Tangent line at } x = a$
  $$y - f(a) = f'(a)(x - a)$$
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  Point-slope form: a point and a slope. The derivative $f'(a)$ is only the slope, not the line.
• $\text{Linear approximation at } x = a$
  $$L(x) = f(a) + f'(a)(x - a)$$
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  Exactly the tangent line at $x = a$, solved for $y$.

Inverse Trig Functions

• $$\dfrac{d}{dx}\,\arcsin x = \dfrac{1}{\sqrt{1 - x^2}}$$
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• $$\dfrac{d}{dx}\,\arctan x = \dfrac{1}{1 + x^2}$$
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Related Rates

• $\text{Right triangle}$
  $$a^2 + b^2 = c^2$$
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• $\text{Circle: area, circumference}$
  $$\pi r^2,\ \ 2\pi r$$
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• $\text{Sphere volume}$
  $$\tfrac{4}{3}\pi r^3$$
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• $\text{Cylinder volume}$
  $$\pi r^2 h$$
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  Closed surface area $2\pi r^2 + 2\pi r h$.
• $\text{Cone volume}$
  $$\tfrac{1}{3}\pi r^2 h$$
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  Use similar triangles to remove $r$ or $h$ first.

Curve Sketching

• $\text{When } f'(x) > 0$
  $$f \text{ is increasing}$$
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  The graph goes up, left to right.
• $\text{When } f'(x) < 0$
  $$f \text{ is decreasing}$$
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  The graph goes down, left to right.
• $\text{When } f''(x) > 0$
  $$f \text{ is concave up } (\cup)$$
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  The slope $f'$ is increasing.
• $\text{When } f''(x) < 0$
  $$f \text{ is concave down } (\cap)$$
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  The slope $f'$ is decreasing.

Optimization

• $\text{Right triangle}$
  $$a^2 + b^2 = c^2$$
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• $\text{Circle: area, circumference}$
  $$\pi r^2,\ \ 2\pi r$$
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• $\text{Sphere volume}$
  $$\tfrac{4}{3}\pi r^3$$
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• $\text{Cylinder volume}$
  $$\pi r^2 h$$
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  Closed surface area $2\pi r^2 + 2\pi r h$.
• $\text{Cone volume}$
  $$\tfrac{1}{3}\pi r^2 h$$
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  Use similar triangles to remove $r$ or $h$ first.

Taylor Polynomials & Newton's Method

• $\text{Taylor polynomial about } x = a$
  $$T_n(x) = \sum_{k=0}^{n} \dfrac{f^{(k)}(a)}{k!}(x - a)^k$$
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• $$e^x = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6} + \cdots$$
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• $$\sin x = x - \dfrac{x^3}{6} + \cdots$$
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• $$\cos x = 1 - \dfrac{x^2}{2} + \dfrac{x^4}{24} - \cdots$$
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• $$\log(1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \cdots$$
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Used in every topic

Log and exponent rules

• $$\log(ab) = \log a + \log b$$
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• $$\log\dfrac{a}{b} = \log a - \log b$$
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• $$\log(a^k) = k\log a$$
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• $$e^{\log x}\ \text{ and }\ \log(e^x) = x$$
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  $\log$ and $e^x$ undo each other. The first needs $x > 0$.
• $$\log 1 = 0$$
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  So $\log x = 0$ exactly when $x = 1$, and $\log e = 1$
• $\text{Get rid of a log: use } e$
  $$\log x = c \;\Longrightarrow\; x = e^{c}$$
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  Raise $e$ to the power of both sides, because $e^{\log x} = x$
• $\text{Get rid of } e\text{: use log}$
  $$e^{x} = c \;\Longrightarrow\; x = \log c$$
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  Take the log of both sides, because $\log(e^{x}) = x$ (needs $c > 0$)

Trig

• $$\tan x = \dfrac{\sin x}{\cos x}$$
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• $$\sec x = \dfrac{1}{\cos x}$$
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• $\text{sin at } 0,\ \tfrac{\pi}{4},\ \tfrac{\pi}{2},\ \tfrac{3\pi}{4},\ \pi$
  $$0,\ \tfrac{\sqrt{2}}{2},\ 1,\ \tfrac{\sqrt{2}}{2},\ 0$$
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• $\text{cos at } 0,\ \tfrac{\pi}{4},\ \tfrac{\pi}{2},\ \tfrac{3\pi}{4},\ \pi$
  $$1,\ \tfrac{\sqrt{2}}{2},\ 0,\ -\tfrac{\sqrt{2}}{2},\ -1$$
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  And $\tan\tfrac{\pi}{4} = 1$.
• $$\sin^2 x + \cos^2 x = 1$$
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• $$1 + \tan^2 x = \sec^2 x$$
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