Application of Derivatives Class 12 NCERT Solutions Explained

Understanding the Basics of Application of Derivatives

The application of derivatives in Class 12 NCERT focuses on using derivatives to solve practical problems. A derivative represents the rate of change of a function with respect to a variable. Key concepts include:

• Rate of change: How one quantity changes relative to another.
• Tangents and normals: Finding slopes of curves at a point.
• Increasing and decreasing functions: Using the first derivative test.
• Concavity and points of inflection: Using the second derivative.

Mastering these basics helps in tackling all exercise problems confidently.

How to Find Tangents and Normals Using Derivatives

One important application is finding the equation of tangents and normals to curves.

• The slope of the tangent at point $x = a$ is $f'(a)$.
• Equation of tangent: $$y - f(a) = f'(a)(x - a)$$
• Equation of normal: $$y - f(a) = -\frac{1}{f'(a)}(x - a)$$

Example: Find the tangent to $y = x^2$ at $x=2$.

• $f'(x) = 2x$, so $f'(2) = 4$.
• Tangent: $y - 4 = 4(x - 2)$ or $y = 4x - 4$.

This method is frequently tested in Class 12 exams.

Using Derivatives to Identify Increasing and Decreasing Functions

Derivatives help determine where a function is increasing or decreasing:

• If $f'(x) > 0$, the function is increasing on that interval.
• If $f'(x) < 0$, the function is decreasing.

Steps to analyze:

1. Find $f'(x)$. 2. Solve $f'(x) = 0$ to find critical points. 3. Test intervals around critical points.

Example: For $f(x) = x^3 - 3x^2 + 4$:

• $f'(x) = 3x^2 - 6x$.
• Set $f'(x) = 0$: $3x(x - 2) = 0$ gives $x=0$ or $2$.
• Test intervals:
• $x < 0$: $f'(x) > 0$ (increasing)
• $0 < x < 2$: $f'(x) < 0$ (decreasing)
• $x > 2$: $f'(x) > 0$ (increasing)

This analysis helps in sketching graphs and solving optimization problems.

Finding Maxima and Minima Using the First and Second Derivative Tests

Maxima and minima points are crucial in optimization problems:

• First derivative test: If $f'(x)$ changes from positive to negative at $x=c$, then $f(c)$ is a local maximum.
• If $f'(x)$ changes from negative to positive, then $f(c)$ is a local minimum.

• Second derivative test: If $f'(c) = 0$ and $f''(c) > 0$, $f(c)$ is a local minimum.
• If $f''(c) < 0$, $f(c)$ is a local maximum.

Example: Find maxima/minima of $f(x) = x^3 - 3x + 1$.

• $f'(x) = 3x^2 - 3$, set to zero: $3x^2 = 3 ightarrow x = \\pm1$.
• $f''(x) = 6x$.
• At $x=1$: $f''(1) = 6 > 0$ (local minimum).
• At $x=-1$: $f''(-1) = -6 < 0$ (local maximum).

This technique is vital for exam questions.

Concavity and Points of Inflection Explained

Concavity tells us how the curve bends:

• If $f''(x) > 0$, the graph is concave upward (like a cup).
• If $f''(x) < 0$, it is concave downward.

A point of inflection is where concavity changes sign, i.e., where $f''(x) = 0$ and changes sign around that point.

Example: For $f(x) = x^3$:

• $f''(x) = 6x$.
• $f''(0) = 0$ and concavity changes from negative to positive.
• So, $x=0$ is a point of inflection.

Understanding concavity helps in graph sketching and interpreting function behavior.

Summary Table: First vs Second Derivative Tests

| Usefulness | More general, works even if $f''(c) = 0$ | Faster but requires $f''(c) eq 0$ |

This table helps clarify when to use each test effectively.