1# Differential calculus
2
3## Limits
4
5$$\lim_{x \rightarrow a}f(x)$$
6
7$L^-$ - limit from below
8
9$L^+$ - limit from above
10
11$\lim_{x \to a} f(x)$ - limit of a point
12
13- Limit exists if $L^-=L^+$
14- If limit exists, point does not.
15
16Limits can be solved using normal techniques (if div 0, factorise)
17
18## Limit theorems
19
201. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
212. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
223. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
234. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
24
25Corollary: $\lim_{x \rightarrow a} c \times f(x)=cF$ where $c=$ constant
26
27## Solving limits for $x\rightarrow\infty$
28
29Factorise so that all values of $x$ are in denominators.
30
31e.g.
32
33$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$
34
35
36## Continuous functions
37
38A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
39
40## Gradients of secants and tangents
41
42Secant (chord) - line joining two points on curve
43
44Tangent - line that intersects curve at one point
45
46given $P(x,y) \quad Q(x+\delta x, y + \delta y)$:
47gradient of chord joining $P$ and $Q$ is ${m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x}$
48
49As $Q \rightarrow P, \delta x \rightarrow 0$. Chord becomes tangent (two infinitesimal points are equal).
50
51Can also be used with functions, where $h=\delta x$.
52
53## First principles derivative
54
55$$f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx}$$
56
57$$m_{\tan}=\lim_{h \rightarrow 0}f^\prime(x)$$
58
59
60
61$$m_{\vec{PQ}}=f^\prime(x)$$
62
63first principles derivative:
64$${m_{\text{tangent at }P} =\lim_{h \rightarrow 0}}{{f(x+h)-f(x)}\over h}$$
65
66## Gradient at a point
67
68Given point $P(a, b)$ and function $f(x)$, the gradient is $f^\prime(a)$
69
70
71## Derivatives of $x^n$
72
73$${d(ax^n) \over dx}=anx^{n-1}$$
74
75If $x=$ constant, derivative is $0$
76
77If $y=ax^n$, derivative is $a\times nx^{n-1}$
78
79If $f(x)={1 \over x}=x^{-1}, \quad f^\prime(x)=-1x^{-2}={-1 \over x^2}$
80
81If $f(x)=^5\sqrt{x}=x^{1 \over 5}, \quad f^\prime(x)={1 \over 5}x^{-4/5}={1 \over 5 \times ^5\sqrt{x^4}}$
82
83If $f(x)=(x-b)^2, \quad f^\prime(x)=2(x-b)$
84
85$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
86
87## Derivatives of $u \pm v$
88
89$${dy \over dx}={du \over dx} \pm {dv \over dx}$$
90where $u$ and $v$ are functions of $x$
91
92## Euler's number as a limit
93
94$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
95
96## Chain rule for $(f\circ g)$
97
98If $f(x) = h(g(x)) = (h \circ g)(x)$:
99
100$$f^\prime(x) = h^\prime(g(x)) \cdot g^\prime(x)$$
101
102If $y=h(u)$ and $u=g(x)$:
103
104$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
105$${d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}$$
106
107Used with only one expression.
108
109e.g. $y=(x^2+5)^7$ - Cannot reasonably expand
110Let $u-x^2+5$ (inner expression)
111${du \over dx} = 2x$
112$y=u^7$
113${dy \over du} = 7u^6$
114
115## Product rule for $y=uv$
116
117$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
118
119## Quotient rule for $y={u \over v}$
120
121$${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$$
122
123$$f^\prime(x)={{v(x)u^\prime(x)-u(x)v^\prime(x)} \over [v(x)]^2}$$
124
125## Logarithms
126
127$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
128
129Wikipedia:
130
131> the logarithm of a given number $x$ is the exponent to which another fixed number, the base $b$, must be raised, to produce that number $x$
132
133### Logarithmic identities
134
135$\log_b (xy)=\log_b x + \log_b y$
136$\log_b x^n = n \log_b x$
137$\log_b y^{x^n} = x^n \log_b y$
138
139### Index identities
140
141$b^{m+n}=b^m \cdot b^n$
142$(b^m)^n=b^{m \cdot n}$
143$(b \cdot c)^n = b^n \cdot c^n$
144${a^m \div a^n} = {a^{m-n}}$
145
146### $e$ as a logarithm
147
148$$\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y$$
149$$\ln x = \log_e x$$
150
151### Differentiating logarithms
152$${d(\log_e x)\over dx} = x^{-1} = {1 \over x}$$
153
154## Derivative rules
155
156| $f(x)$ | $f^\prime(x)$ |
157| ------ | ------------- |
158| $\sin x$ | $\cos x$ |
159| $\sin ax$ | $a\cos ax$ |
160| $\cos x$ | $-\sin x$ |
161| $\cos ax$ | $-a \sin ax$ |
162| $\tan f(x)$ | $f^2(x) \sec^2f(x)$ |
163| $e^x$ | $e^x$ |
164| $e^{ax}$ | $ae^{ax}$ |
165| $ax^{nx}$ | $an \cdot e^{nx}$ |
166| $\log_e x$ | $1 \over x$ |
167| $\log_e {ax}$ | $1 \over x$ |
168| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
169| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
170| $\sin^{-1} x$ | $1 \over {\sqrt{1-x^2}}$ |
171| $\cos^{-1} x$ | $-1 \over {sqrt{1-x^2}}$ |
172| $\tan^{-1} x$ | $1 \over {1 + x^2}$ |
173
174## Reciprocal derivatives
175
176$${1 \over {dy \over dx}} = {dx \over dy}$$
177
178## Differentiating $x=f(y)$
179
180Find $dx \over dy$. Then ${dx \over dy} = {1 \over {dy \over dx}} \implies {dy \over dx} = {1 \over {dx \over dy}}$.
181
182$${dy \over dx} = {1 \over {dx \over dy}}$$
183
184## Second derivative
185
186$$f(x) \longrightarrow f^\prime (x) \longrightarrow f^{\prime\prime}(x)$$
187
188$$\therefore y \longrightarrow {dy \over dx} \longrightarrow {d({dy \over dx}) \over dx} \longrightarrow {d^2 y \over dx^2}$$
189
190Order of polynomial $n$th derivative decrements each time the derivative is taken
191
192### Points of Inflection
193
194*Stationary point* - point of zero gradient (i.e. $f^\prime(x)=0$)
195*Point of inflection* - point of maximum $|$gradient$|$ (i.e. $f^{\prime\prime} = 0$)
196
197* if $f^\prime (a) = 0$ and $f^{\prime\prime}(a) > 0$, then point $(a, f(a))$ is a local min (curve is concave up)
198* if $f^\prime (a) = 0$ and $f^{\prime\prime} (a) < 0$, then point $(a, f(a))$ is local max (curve is concave down)
199* if $f^{\prime\prime}(a) = 0$, then point $(a, f(a))$ is a point of inflection
200 + if also $f^\prime(a)=0$, then it is a stationary point of inflection
201
202![](graphics/second-derivatives.png)
203
204## Implicit Differentiation
205
206**On CAS:** Action $\rightarrow$ Calculation $\rightarrow$ `impDiff(y^2+ax=5, x, y)`. Returns $y^\prime= \dots$.
207
208Used for differentiating circles etc.
209
210If $p$ and $q$ are expressions in $x$ and $y$ such that $p=q$, for all $x$ nd $y$, then:
211
212$${dp \over dx} = {dq \over dx} \quad \text{and} \quad {dp \over dy} = {dq \over dy}$$
213
214## Integration
215
216$$\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)$$
217
218$$\int x^n \cdot dx = {x^{n+1} \over n+1} + c$$
219
220- area enclosed by curves
221- $+c$ should be shown on each step without $\int$
222
223### Integral laws
224
225$\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx$
226$\int k f(x) dx = k \int f(x) dx$
227
228| $f(x)$ | $\int f(x) \cdot dx$ |
229| ------------------------------- | ---------------------------- |
230| $k$ (constant) | $kx + c$ |
231| $x^n$ | ${x^{n+1} \over {n+1}} + c$ |
232| $a x^{-n}$ | $a \cdot \log_e x + c$ |
233| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
234| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
235| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
236| $e^k$ | $e^kx + c$ |
237| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
238| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
239| $\sec^2 kx$ | ${1 \over k} \tan(kx) + c$ |
240| $1 \over \sqrt{a^2-x^2}$ | $\sin^{-1} {x \over a} + c \>\vert\> a>0$ |
241| $-1 \over \sqrt{a^2-x^2}$ | $\cos^{-1} {x \over a} + c \>\vert\> a>0$ |
242| $a \over {a^2-x^2}$ | $\tan^{-1} {x \over a} + c$ |
243| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
244| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
245| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
246
247Note $\sin^{-1} {x \over a} + \cos^{-1} {x \over a}$ is constant for all $x \in (-a, a)$.
248
249### Definite integrals
250
251$$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)$$
252
253- Signed area enclosed by: $\> y=f(x), \quad y=0, \quad x=a, \quad x=b$.
254- *Integrand* is $f$.
255- $F(x)$ may be any integral, i.e. $c$ is inconsequential
256
257#### Properties
258
259$$\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx$$
260
261$$\int^a_a f(x) \> dx = 0$$
262
263$$\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx$$
264
265$$\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx$$
266
267$$\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx$$
268
269### Integration by substitution
270
271$$\int f(u) {du \over dx} \cdot dx = \int f(u) \cdot du$$
272
273Note $f(u)$ must be one-to-one $\implies$ one $x$ value for each $y$ value
274
275e.g. for $y=\int(2x+1)\sqrt{x+4} \cdot dx$:
276let $u=x+4$
277$\implies {du \over dx} = 1$
278$\implies x = u - 4$
279then $y=\int (2(u-4)+1)u^{1 \over 2} \cdot du$
280Solve as a normal integral
281
282#### Definite integrals by substitution
283
284For $\int^b_a f(x) {du \over dx} \cdot dx$, evaluate new $a$ and $b$ for $f(u) \cdot du$.
285
286### Trigonometric integration
287
288$$\sin^m x \cos^n x \cdot dx$$
289
290**$m$ is odd:**
291$m=2k+1$ where $k \in \mathbb{Z}$
292$\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x$
293Substitute $u=\cos x$
294
295**$n$ is odd:**
296$n=2k+1$ where $k \in \mathbb{Z}$
297$\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x$
298Subbstitute $u=\sin x$
299
300**$m$ and $n$ are even:**
301Use identities:
302
303- $\sin^2x={1 \over 2}(1-\cos 2x)$
304- $\cos^2x={1 \over 2}(1+\cos 2x)$
305- $\sin 2x = 2 \sin x \cos x$
306
307## Partial fractions
308
309On CAS: Action $\rightarrow$ Transformation $\rightarrow$ `expand/combine`
310or Interactive $\rightarrow$ Transformation $\rightarrow$ `expand` $\rightarrow$ Partial
311
312## Graphing integrals on CAS
313
314In main: Interactive $\rightarrow$ Calculation $\rightarrow$ $\int$ ($\rightarrow$ Definite)
315Restrictions: `Define f(x)=...` $\rightarrow$ `f(x)|x>1` (e.g.)
316
317## Applications of antidifferentiation
318
319- $x$-intercepts of $y=f(x)$ identify $x$-coordinates of stationary points on $y=F(x)$
320- nature of stationary points is determined by sign of $y=f(x)$ on either side of its $x$-intercepts
321- if $f(x)$ is a polynomial of degree $n$, then $F(x)$ has degree $n+1$
322
323To find stationary points of a function, substitute $x$ value of given point into derivative. Solve for ${dy \over dx}=0$. Integrate to find original function.
324
325## Solids of revolution
326
327Approximate as sum of infinitesimally-thick cylinders
328
329### Rotation about $x$-axis
330
331\begin{align*}
332 V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
333 &= \pi \int^b_a (f(x))^2 \> dx
334\end{align*}
335
336### Rotation about $y$-axis
337
338\begin{align*}
339 V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
340 &= \pi \int^b_a (f(y))^2 \> dy
341\end{align*}
342
343### Regions not bound by $y=0$
344
345$$V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx$$
346where $f(x) > g(x)$
347
348## Length of a curve
349
350$$L = \int^b_a \sqrt{1 + ({dy \over dx})^2} \> dx \quad \text{(Cartesian)}$$
351
352$$L = \int^b_a \sqrt{{dx \over dt} + ({dy \over dt})^2} \> dt \quad \text{(parametric)}$$
353
354Evaluate on CAS. Or use Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ `arcLen`.
355
356## Rates
357
358### Related rates
359
360$${da \over db} \quad \text{(change in } a \text{ with respect to } b)$$
361
362### Gradient at a point on parametric curve
363
364$${dy \over dx} = {{dy \over dt} \div {dx \over dt}} \> \vert \> {dx \over dt} \ne 0$$
365
366$${d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \div {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}$$
367
368## Rational functions
369
370$$f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}$$
371
372### Addition of ordinates
373
374- when two graphs have the same ordinate, $y$-coordinate is double the ordinate
375- when two graphs have opposite ordinates, $y$-coordinate is 0 i.e. ($x$-intercept)
376- when one of the ordinates is 0, the resulting ordinate is equal to the other ordinate
377
378## Fundamental theorem of calculus
379
380If $f$ is continuous on $[a, b]$, then
381
382$$\int^b_a f(x) \> dx = F(b) - F(a)$$
383
384where $F$ is any antiderivative of $f$
385
386## Differential equations
387
388One or more derivatives
389
390**Order** - highest power inside derivative
391**Degree** - highest power of highest derivative
392e.g. ${\left(dy^2 \over d^2 x\right)}^3$: order 2, degree 3
393
394### Verifying solutions
395
396Start with $y=\dots$, and differentiate. Substitute into original equation.
397
398### Function of the dependent variable
399
400If ${dy \over dx}=g(y)$, then ${dx \over dy} = 1 \div {dy \over dx} = {1 \over g(y)}$. Integrate both sides to solve equation. Only add $c$ on one side. Express $e^c$ as $A$.
401
402### Mixing problems
403
404$$\left({dm \over dt}\right)_\Sigma = \left({dm \over dt}\right)_{\text{in}} - \left({dm \over dt}\)_{\text{out}}$$
405
406### Separation of variables
407
408If ${dy \over dx}=f(x)g(y)$, then:
409
410$$\int f(x) \> dx = \int {1 \over g(y)} \> dy$$