4 — Differential And Integral Calculus By Feliciano And Uy Chapter
This is often a faster way to classify maxima and minima than the First Derivative Test:
Problem (paraphrased from Feliciano & Uy):
A rectangular sheet of paper 24 cm × 9 cm is to be made into a box with an open top by cutting equal squares from the corners and folding up the sides. Find the side of the square to be cut so that the volume is maximum.
Solution:
Let (x) = side of square cut.
Length after cut = (24 - 2x)
Width after cut = (9 - 2x)
Height = (x)
Volume (V = x(24-2x)(9-2x))
(V = 4x^3 - 66x^2 + 216x)
(V' = 12x^2 - 132x + 216 = 12(x^2 - 11x + 18) = 12(x-2)(x-9))
Critical points: (x=2, 9) (discard (x=9) → no width left)
Check (V''(2) < 0) → maximum.
Answer: Cut (2) cm squares.
If you have a specific problem number from Feliciano & Uy Chapter 4, paste it here and I can solve it step-by-step.
In the textbook Differential and Integral Calculus by Feliciano and Uy
, Chapter 4 is titled "Differentiation of Transcendental Functions". This chapter expands beyond algebraic functions to cover the rules and techniques for finding derivatives of trigonometric, logarithmic, exponential, and hyperbolic functions. Core Topics in Chapter 4
The chapter is structured to introduce specific transcendental functions and their corresponding differentiation formulas:
Trigonometric Functions: Differentiation of the six basic functions (sine, cosine, tangent, cotangent, secant, and cosecant).
Inverse Trigonometric Functions: Finding derivatives for functions like , and others.
Logarithmic Functions: Differentiation rules for natural logarithms ( ) and common logarithms ( logaulog base a of u Exponential Functions: Formulas for eue to the u-th power aua to the u-th power
, including the use of Logarithmic Differentiation to simplify complex products or powers. This is often a faster way to classify
Hyperbolic Functions: Introduction and differentiation of hyperbolic sine ( sinhhyperbolic sine ), cosine ( coshhyperbolic cosine ), and related functions. Key Concepts & Formulas
While the text provides many variations, the fundamental formulas discussed typically include: Trigonometric: Exponential: Logarithmic: Typical Problems Exercises in this chapter often involve:
Finding the derivative of composite transcendental functions (e.g.,
Using logarithmic differentiation for functions where the variable appears in both the base and the exponent.
Applications of these derivatives in optimization problems, such as finding dimensions for inscribed figures.
For step-by-step walkthroughs of specific problems, you can find a complete solution manual for Chapter 4 online.
Mastering Derivatives: A Deep Dive into Chapter 4 of Feliciano and Uy
For many students in the Philippines and abroad, "Differential and Integral Calculus" by Feliciano and Uy is the definitive "blue book" of mathematics. While the early chapters set the stage with limits and continuity, Chapter 4 is where the real work begins.
This chapter focuses on the Derivatives of Algebraic Functions, serving as the bridge between theoretical limits and practical calculus application. 1. The Core Objective: Moving Beyond the Limit Definition
In Chapter 3, you likely spent hours calculating derivatives using the "Increment Method" (the Problem (paraphrased from Feliciano & Uy): A rectangular
or delta method). Chapter 4 is a relief; it introduces Differentiation Rules. These rules allow you to find the slope of a tangent line or the rate of change without the tedious algebraic expansion of limits. 2. Essential Rules to Memorize
Chapter 4 breaks down the mechanics of calculus into several "shortcuts" that you will use for the rest of your academic career: The Power Rule: The bread and butter of calculus. If , then .
The Product Rule: Crucial for functions multiplied together (
). Feliciano and Uy emphasize the pattern: the first times the derivative of the second, plus the second times the derivative of the first.
The Quotient Rule: Used for fractions. A common mnemonic for this is "Low d-High minus High d-Low, over Low-Low."
The Chain Rule: This is often the "make or break" section of Chapter 4. It teaches you how to differentiate composite functions—functions within functions. 3. Why This Chapter Matters
Feliciano and Uy’s approach is uniquely structured with a heavy emphasis on drill problems. Chapter 4 isn't just about understanding the theory; it’s about building muscle memory.
The problems in this chapter start simple but quickly escalate into complex algebraic simplifications. Succeeding here requires not just calculus skills, but strong algebraic foundation. Many students find that they understand the "calculus" part (the derivative), but struggle with the "simplification" part (the algebra) required to match the answers in the back of the book. 4. Study Tips for Chapter 4
Show Every Step: Don't skip steps when applying the Quotient Rule. One missed sign in the numerator will ruin the entire result.
Master the Chain Rule Early: Most errors in later chapters (like Transcendental Functions) stem from a weak grasp of the Chain Rule in Chapter 4. If you have a specific problem number from
Check the Odd Numbers: Use the provided answers for odd-numbered problems to verify your simplification techniques. Conclusion
Chapter 4 of Feliciano and Uy is the cornerstone of differential calculus. By mastering these algebraic rules, you transition from a student who calculates math to a student who understands how variables change in relation to one another.
If you have specific problems from Chapter 4, you can type them here (or describe them), and I’ll explain the solutions step-by-step.
If you need a summary of the chapter’s concepts, let me know, and I’ll provide a concise, original write-up based on standard calculus content that matches that textbook’s level (typical for engineering/STEM in the Philippines).
If you need a reviewer – I can generate practice problems similar to those in Chapter 4.
Just let me know which of these would help you most.
Since you requested a "paper" on this specific textbook chapter, I have structured this as a comprehensive chapter summary and study guide. This is designed to mimic the style of an academic review or a supplemental lecture note often used in calculus courses.
Title: A Comprehensive Review of Differential Calculus: The Rules of Differentiation Source Material: Differential and Integral Calculus by Feliciano and Uy Subject Area: Mathematics (Calculus I)
The derivative operator is linear. It can be distributed across addition and subtraction.