Scribd
Upload
10 views6 pages

Calculus I Lecture Notes

The document contains lecture notes for an undergraduate Calculus I course, focusing on limits and derivatives. It includes definitions, techniques for calculating limits, the interpretation of derivatives, differentiation rules, worked examples, and practice problems with solutions. The notes serve as a concise study pack for the first week of the course.

Uploaded by

jattsaab17022005
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
10 views6 pages

Calculus I Lecture Notes

The document contains lecture notes for an undergraduate Calculus I course, focusing on limits and derivatives. It includes definitions, techniques for calculating limits, the interpretation of derivatives, differentiation rules, worked examples, and practice problems with solutions. The notes serve as a concise study pack for the first week of the course.

Uploaded by

jattsaab17022005
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 6

Calculus I — Lecture Notes (Limits &

Derivatives)
Prepared for: Undergraduate Calculus — Week 1

Contents
1. Limits — definition and techniques
2. Continuity
3. Derivative — definition and interpretation
4. Rules of differentiation
5. Worked examples
6. Practice problems (with answers)
1. Limits — definition
Limit of f(x) as x approaches a is the value L that f(x) approaches as x gets arbitrarily close to a.
Notation: \(\lim_{x\to a} f(x) = L\).

Techniques: algebraic simplification, factoring, rationalization, squeeze theorem.


3. Derivative — definition and interpretation
The derivative f'(x) is the instantaneous rate of change: \(f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\).
Geometrically, slope of tangent.

Common rules: constant rule, power rule, sum rule, product rule, quotient rule, chain rule.
5. Worked examples

Example 1: Differentiate f(x) = 3x^4 - 5x + 7


Solution: f'(x)=12x^3 - 5.

Example 2: Differentiate g(x) = (2x^3)(sin x)


Solution: Product rule: g' = 6x^2 sin x + 2x^3 cos x.

Example 3: Find \(\lim_{x\to2}\frac{x^2-4}{x-2}\).


Solution: Factor numerator: (x-2)(x+2)/(x-2) = x+2; limit = 4.
6. Practice problems (answers on next page)
1. Compute derivative of f(x) = x^5 - 4x^2 + 3x - 1
2. Evaluate \(\lim_{x\to0}\frac{\sin x}{x}\)
3. Differentiate h(x) = e^{2x} * x^2
Answers
1. 5x^4 - 8x + 3
2. 1
3. h' = 2e^{2x}x^2 + 2x e^{2x} = 2xe^{2x}(x+1)

End of Calculus I notes — concise 2-page study pack.

You might also like