Summary calculus quick notes

Basic Calculus Quick Notes – Exam-Focused Guide




1. Limits and Continuity
Key Concepts:

 A limit finds the value that a function approaches as the input gets closer to a certain
point.
 Continuity means the function has no breaks, jumps, or holes.

Shortcut Trick:

 Direct Substitution: If lim⁡x→af(x)\lim_{x \to a} f(x) gives a number, it’s the limit.
 Factoring: If direct substitution gives 00\frac{0}{0}, factor and cancel.

Example:

Find lim⁡x→2x2−4x−2\lim_{x \to 2} \frac{x^2 - 4}{x - 2} Solution: Factor as (x−2)(x+2)x−2\
frac{(x-2)(x+2)}{x-2}, cancel (x−2)(x-2), limit = 4.




2. Derivatives
Key Concepts:

 Measures the rate of change of a function.
 f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.

Shortcut Trick:

 Power Rule: ddxxn=nxn−1\frac{d}{dx} x^n = nx^{n-1}
 Product Rule: (uv)′=u′v+uv′(uv)' = u'v + uv'
 Quotient Rule: (uv)′=u′v−uv′v2\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}
 Chain Rule: (f(g(x)))′=f′(g(x))g′(x)(f(g(x)))' = f'(g(x)) g'(x)

Example:

Find ddx(x3+5x)\frac{d}{dx} (x^3 + 5x) Solution: Using power rule, 3x² + 5.

,3. Integration
Key Concepts:

 Reverse of differentiation.
 ∫f(x)dx\int f(x)dx gives the area under the curve.

Shortcut Trick:

 Power Rule: ∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C
 Common Integrals:
o ∫exdx=ex+C\int e^x dx = e^x + C
o ∫sin⁡xdx=−cos⁡x+C\int \sin x dx = -\cos x + C
o ∫cos⁡xdx=sin⁡x+C\int \cos x dx = \sin x + C

Example:

Find ∫(3x2+4)dx\int (3x^2 + 4)dx Solution: x3+4x+Cx^3 + 4x + C.




4. Fundamental Theorem of Calculus
 Connects differentiation and integration: ddx∫axf(t)dt=f(x)\frac{d}{dx} \int_a^x f(t)dt =
f(x)
 Used to evaluate definite integrals.

Shortcut Trick:

 If F(x)F(x) is the antiderivative of f(x)f(x), then: ∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx =
F(b) - F(a)

Example:

Find ∫13(2x)dx\int_1^3 (2x)dx Solution: x2∣13=9−1=8x^2 |_{1}^{3} = 9 - 1 = 8.



📌 Exam Tips for Basic Calculus:

✅ Memorize basic derivative & integral rules. ✅ For limits, always try direct substitution first.
✅ Use differentiation to find max/min in word problems. ✅ Practice solving past paper
questions for speed.

, End of Basic Calculus Quick Notes ✅

Advanced Calculus Quick Notes – Exam-Focused Guide




1. Partial Derivatives
Key Concepts:

 If f(x,y)f(x, y) is a function of two variables, the partial derivatives are ∂f∂x\frac{\partial
f}{\partial x} and ∂f∂y\frac{\partial f}{\partial y}.
 Used in multivariable optimization and tangent plane calculations.

Shortcut Trick:

 Differentiate normally while treating other variables as constants.

Example:

Find ∂∂x(x2y+3y2)\frac{\partial}{\partial x} (x^2y + 3y^2) Solution: 2xy2xy.




2. Double & Triple Integrals
Key Concepts:

 Used to compute volume and area over a region.
 ∫ab∫cdf(x,y) dydx\int_{a}^{b} \int_{c}^{d} f(x,y)\, dy dx represents an area in xyxy-
plane.

Shortcut Trick:

 Reverse order of integration when limits are complex.

Example:

Find ∫01∫0xxy dydx\int_0^1 \int_0^x xy\, dy dx Solution: Solve inner integral first, then outer
integral.