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Summary Calculus Notes and Exercises
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  • Course
  • ECO 3101
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  • University Of The Philippines

Description Comprehensive Calculus Course Material with Notes, Exercises, and Solutions Dive deep into the world of Calculus with this extensive and meticulously crafted document designed for college and university students. This six-page document provides a thorough overview of key calculus conc...

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  • July 21, 2024
  • 5
  • 2023/2024
  • Summary

Subjects

  • differential calculus
  • integral calculus
  • limits
  • derivatives
  • integrals
  • mathematics
  • college course
  • university course
  • math exercises

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  • University of the Philippines
  • ECO 3101
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Calculus Notes and Exercises

Introduction to Derivatives
Notes:

Definition: The derivative of a function f at a point x is given by:

f'(x) = lim_{h \to 0} (f(x+h) - f(x))/h

Interpretation: The derivative represents the rate of change or the slope of the tangent line
to the function at a point.

Basic Rules:

Power Rule: d/dx x^n = nx^{n-1}

Sum Rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)

Product Rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Quotient Rule: d/dx [f(x)/g(x)] = (f'(x)g(x) - f(x)g'(x))/g(x)^2

Chain Rule: d/dx f(g(x)) = f'(g(x))g'(x)

Exercises:

1. Exercise 1: Find the derivative of f(x) = 3x^4 + 2x^2 - x + 7.

Solution: f'(x) = 12x^3 + 4x - 1

2. Exercise 2: Differentiate g(x) = (5x^3 - 2x + 1)/x^2.

Solution: g'(x) = (15x^2 - 2x^2 - 10x + 4)/(x^4) = 15 - 10/x - 2/x^2 + 4/x^3

3. Exercise 3: If h(x) = sin(x^2), find h'(x).

Solution: h'(x) = 2x cos(x^2)

Applications of Derivatives
Notes:

Critical Points: Points where f'(x) = 0 or f'(x) does not exist.

Increasing/Decreasing Functions: If f'(x) > 0 on an interval, f is increasing. If f'(x) < 0, f is
decreasing.

Concavity and Inflection Points:

, f is concave up if f''(x) > 0.

f is concave down if f''(x) < 0.

An inflection point is where f changes concavity.

Exercises:

4. Exercise 1: Find the critical points of f(x) = x^3 - 3x^2 + 4x - 2.

Solution: f'(x) = 3x^2 - 6x + 4
Set f'(x) = 0:
3x^2 - 6x + 4 = 0
x=1
Critical point: x = 1

5. Exercise 2: Determine the intervals where g(x) = x^4 - 4x^2 is increasing or decreasing.

Solution: g'(x) = 4x^3 - 8x = 4x(x^2 - 2)
Set g'(x) = 0:
4x(x^2 - 2) = 0
x = 0, ±sqrt(2)
Intervals:
(-∞, -sqrt(2)): g'(x) < 0
(-sqrt(2), 0): g'(x) > 0
(0, sqrt(2)): g'(x) < 0
(sqrt(2), ∞): g'(x) > 0

6. Exercise 3: Identify the inflection points of h(x) = x^4 - 4x^3 + 6x^2.

Solution: h''(x) = 12x^2 - 24x + 12
Set h''(x) = 0:
12x^2 - 24x + 12 = 0
x=1
Inflection point: x = 1

Introduction to Integrals
Notes:

Definition: The integral of a function f over an interval [a, b] is given by:

∫_a^b f(x) dx

Interpretation: The integral represents the area under the curve of f from a to b.

Basic Rules:

Power Rule: ∫ x^n dx = x^{n+1}/(n+1) + C (for n ≠ -1)

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