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.
Differential and Integral Calculus by Feliciano and Uy remains a cornerstone textbook for engineering and mathematics students in the Philippines. Chapter 4 is particularly critical as it marks the transition from basic differentiation rules to the conceptual and practical applications of the derivative. This chapter bridges the gap between abstract formulas and real-world problem-solving.
The primary focus of Chapter 4 is the Application of Derivatives. While previous chapters teach you how to find the slope of a line, this chapter teaches you what that slope actually represents in physical and geometric contexts. Mastering this section is essential for passing subsequent courses like Integral Calculus and Differential Equations. Problem (paraphrased from Feliciano & Uy): A rectangular
One of the first major hurdles in Chapter 4 is Tangents and Normals. Students learn to find the equation of a line tangent to a curve at a specific point. The derivative gives the slope of the tangent line, while the normal line is simply the perpendicular counterpart. Understanding the geometric relationship between these two lines is foundational for visualizing how functions behave at local points.
Related Rates is often considered the most challenging section of the chapter. These problems involve variables that are changing with respect to time. For example, if water is being poured into a conical tank, the height of the water and the radius of the surface are both changing. Feliciano and Uy emphasize a systematic approach: identify the given rates, determine the required rate, and establish a geometric or algebraic relationship between the variables before differentiating implicitly.
Curvature and Radius of Curvature are also introduced here. These concepts describe how "sharply" a curve turns at any given point. This has significant implications in civil engineering, particularly in the design of highway curves and railway tracks where safety depends on the gradual change of direction.
The chapter also dives deep into Maxima and Minima. This is perhaps the most "useful" part of calculus for everyday optimization. Whether you are trying to minimize the material needed for a container or maximize the area of a fenced field, the principles remain the same. By setting the first derivative to zero, students locate critical points, and the second derivative test helps determine if those points are peaks or valleys. If you have a specific problem number from
Chapter 4 concludes with Concavity and Inflection Points. This section deals with the "shape" of the graph—whether it opens upward or downward. Finding the point where the concavity changes, known as the inflection point, provides a complete picture of the function’s behavior.
Studying Chapter 4 of Feliciano and Uy requires patience and a strong grasp of the chain rule from Chapter 3. The problems are designed to be rigorous, often requiring a blend of trigonometry and solid geometry. For students using this manual, the key to success is drawing clear diagrams for every word problem and maintaining consistent units throughout the calculation.
Chapter 4 assumes mastery. If you still struggle with the chain rule or product rule, stop. Go back. You cannot solve a related rates problem if you freeze up when differentiating ( \sin(x^2) ).
Strategy:
Example (classic):
A 5m ladder slides down a wall. Top slides down at 1 m/s. How fast is bottom moving when top is 3m from ground?
(x^2 + y^2 = 25), (\fracdydt = -1), find (\fracdxdt) when (y=3).
→ (x=4), (2x\fracdxdt + 2y\fracdydt = 0) → (8\fracdxdt + 6(-1)=0) → (\fracdxdt = 0.75) m/s.
Feliciano and Uy provide answers to odd-numbered problems in the back. Do not just check the answer; reverse-engineer your mistake. Specifically, focus on problems 4.1 (Tangents), 4.4 (Time Rates), and 4.7 (Optimization).
To use the first and second derivatives to analyze the behavior of functions, sketch their graphs accurately, and solve real-world optimization problems.