100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Previously searched by you
Previously searched by you
logo
Summary Calculus Notes and Exercises $3.44   Add to cart
Quickly navigate to
Preview
  2.  
  3. University Of The Philippines
  4.  
  5. ECO 3101

Summary

Summary Calculus Notes and Exercises
 9 views  0 purchase
  • Course
  • ECO 3101
  • Institution
  • 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...

[Show more]

Preview 2 out of 5  pages

Document information

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

Subjects

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

Written for

  • University of the Philippines
  • ECO 3101
All documents for this subject (1)

Seller

avatar-seller
notyou1

Content preview

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)

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller notyou1. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $3.44. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

75860 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling
$3.44
  • (0)
  Add to cart