Differential And Integral Calculus By Feliciano And Uy Chapter 4

Find two numbers whose sum is 20 and product is maximum.
Solution: (x + y = 20), (P = xy = x(20-x) = 20x - x^2)
(P' = 20 - 2x = 0) → (x=10, y=10), max (P=100)

Example: ( y = \sin(5x^2) )

Feliciano & Uy famous problem types:

Test:

Procedure:

Example:
(f(x) = x^3 - 3x)
(f'(x) = 3x^2 - 3 = 3(x-1)(x+1))
Critical points: (x = -1, 1)
Sign:

This section introduces the concept of "turning points" where a function reaches a peak or a valley. Find two numbers whose sum is 20 and product is maximum

Based on forums and student feedback regarding Differential and Integral Calculus by Feliciano and Uy, Chapter 4 presents three specific challenges:

1. The Language Barrier of Word Problems The textbook uses formal, technical English. A problem that says "A man starts walking north at 4 ft/s from point P..." can confuse non-native English speakers. You must translate English into derivatives (( dx/dt )).

2. Algebraic Complexity You might understand the calculus (taking the derivative) but fail because of algebra. For example, optimizing tin cans (cylindrical surface area) requires solving ( dA/dr = 0 ) which involves fractions and radicals. One algebra mistake collapses the entire problem. Procedure:

3. Implicit Differentiation in Time Rates Students often forget when to plug in numbers. Rule of thumb from Feliciano and Uy: Differentiate first, then substitute. If you plug in values before differentiating, you will treat variables as constants and miss important terms.

The chapter opens with a review of geometric interpretation. You will learn how to find the slope of a curve at any given point, but more importantly, you will solve for:

Typical Problem: Find the equations of the tangent and normal to the curve ( y = x^3 - 2x^2 + 1 ) at ( x = 1 ). Example: (f(x) = x^3 - 3x) (f'(x) =