Differential And Integral Calculus By Feliciano And Uy Chapter 4 | VERIFIED |

This is a custom study and solution guide for Chapter 4: Applications of Differential Calculus (commonly titled Applications of the First Derivative) in the textbook Differential and Integral Calculus by Feliciano and Uy (a standard reference in Philippine engineering and math curricula).

Since I do not have the exact 1983/1998 edition text, this guide is reconstructed based on the standard content of Chapter 4 in that specific book, covering: Tangents and Normals, Increasing/Decreasing Functions, Maxima/Minima, Concavity, Points of Inflection, and Applied Optimization.

4. Implicit Differentiation

Though sometimes treated as a separate advanced topic, many standard texts, including Feliciano and Uy, introduce implicit differentiation in the context of the Chain Rule. This technique is used when a function is not isolated as $y = f(x)$. This is a custom study and solution guide

• Technique: One differentiates both sides of the equation with respect to $x$, treating $y$ as a function of $x$ (and thus applying the Chain Rule to terms containing $y$) and subsequently solving for $dy/dx$.

Cheat Sheet: Common Derivatives Needed in Chapter 4

Keep this list handy while working through Feliciano and Uy Chapter 4:

• Power Rule: ( \fracddx x^n = n x^n-1 )
• Trigonometric: ( \fracddx \sin u = \cos u \cdot \fracdudx )
• Logarithmic: ( \fracddx \ln u = \frac1u \cdot \fracdudx )
• Parametric: ( \fracdydx = \fracdy/dtdx/dt )

4. Concavity and Second Derivative Test

Definitions:

• (f''(x) > 0) → concave up (cup-shaped)
• (f''(x) < 0) → concave down (cap-shaped)
• Inflection point: where (f''(x) = 0) or undefined and concavity changes.

Second Derivative Test for Extrema:

• (f'(c) = 0) and (f''(c) > 0) → local minimum
• (f'(c) = 0) and (f''(c) < 0) → local maximum
• (f''(c) = 0) → test fails (use first derivative test)

Example:
(f(x) = x^4 - 4x^2)
(f'(x) = 4x^3 - 8x = 4x(x^2 - 2)) → CP: (x = 0, \pm\sqrt2)
(f''(x) = 12x^2 - 8) Technique: One differentiates both sides of the equation

• (f''(\sqrt2) = 16 > 0) → local min
• (f''(-\sqrt2) = 16 > 0) → local min
• (f''(0) = -8 < 0) → local max

C. Fencing/Cost problems

Rectangular field with 600m fencing, one side against river (no fence). Max area.
Let (x) = side parallel to river, (y) = other side. (x + 2y = 600) → (A = xy = y(600-2y)) → derivative → (y=150), (x=300).