Calculus
Calculus Complete Cheatsheet
Limits, derivatives, integrals, and the Fundamental Theorem — complete calculus reference for A-Level and university.
01Limits▼
Limit
lim[x->a] f(x)=L
Value f approaches as x approaches a.
Direct sub
Try f(a) first
Works if f is continuous at a.
L'Hopital
lim f/g = lim f'/g'
Use for 0/0 or inf/inf forms.
CALCULUSLimit techniques
# Direct substitution lim[x->3](x2+1) = 10 # Factoring (0/0 form) lim[x->2](x2-4)/(x-2) = lim(x+2)=4 # Limits at infinity lim[x->inf](3x2+2)/(5x2-1) = 3/5 (leading coefficients) # L Hopital: lim[x->0] sin(x)/x d/dx(sinx)=cosx, d/dx(x)=1 -> cos(0)/1=1
02Derivatives▼
| Function | Derivative |
|---|---|
| x^n | nx^(n-1) |
| e^x | e^x |
| ln x | 1/x |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec^2 x |
| sin^-1 x | 1/sqrt(1-x^2) |
| tan^-1 x | 1/(1+x^2) |
CALCULUSDerivative rules
# Power: d/dx[x^5]=5x^4 # Chain: d/dx[sin(x^2)]=2x*cos(x^2) # Product: d/dx[x*e^x]=e^x+x*e^x=e^x(1+x) # Quotient: d/dx[sin x/x]=(x*cosx-sinx)/x^2
03Applications of Derivatives▼
Critical points
f'(x)=0 or undefined
Max/Min
f''(x)>0: min. f''(x)<0: max
Increasing
f'(x)>0
Concave up
f''(x)>0
Inflection
f'' changes sign
Related rates
Differentiate both sides w.r.t time t
CALCULUSOptimization example
f(x)=x^3-3x^2+4 f(x)=3x^2-6x=3x(x-2) Critical: x=0 and x=2 f(0): f''(0)=-6<0 -> LOCAL MAX at x=0 f(2): f''(2)=6>0 -> LOCAL MIN at x=2
❓ Quiz
d/dx[sin(x^2)]=?
Chain rule: outer=sin->cos, inner=x^2->2x. Answer: 2x*cos(x^2).
04Integration▼
| Function | Integral |
|---|---|
| x^n | x^(n+1)/(n+1)+C |
| e^x | e^x+C |
| 1/x | ln|x|+C |
| sin x | -cos x+C |
| cos x | sin x+C |
| sec^2 x | tan x+C |
| 1/sqrt(1-x^2) | sin^-1 x+C |
CALCULUSIntegration techniques
# Substitution: integral x*e^(x^2)dx u=x^2, du=2x dx = (1/2)*integral e^u du = e^(x^2)/2+C # By parts: integral x*e^x dx u=x, dv=e^x dx. du=dx, v=e^x = x*e^x - integral e^x dx = e^x(x-1)+C
05Fundamental Theorem▼
FTC Part 2
integral[a to b]f(x)dx=F(b)-F(a)
Where F'=f.
Area between
integral[a to b](f-g)dx
When f(x)>=g(x) on [a,b].
CALCULUSDefinite integral
integral[1 to 3](x^2+2x)dx F(x)=x^3/3+x^2 F(3)=9+9=18. F(1)=1/3+1=4/3 Result = 18-4/3 = 50/3 approx 16.67 Area under sin from 0 to pi: [-cos x] from 0 to pi = 1+1=2