7.4.1 Applications of the Second Derivative

The second derivative of a function provides important information about the curvature and concavity of the function's graph. It is a powerful tool in calculus, used in various applications ranging from finding local maxima and minima to analysing the shape of curves and solving practical optimization problems. Here's an overview of the main applications of the second derivative.

1. Concavity and Convexity:

The second derivative tells us whether a function is concave up or concave down:

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Concave Up (Convex): If $\ f''(x) > 0\ for\ all \ x$ in an interval, the function $\ f(x)$ is concave up on that interval. The graph of the function bends upwards, resembling the shape of a U.
This implies that the slope of the tangent line (the first derivative) is increasing.

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Concave Down (Concave): If $\ f''(x) < 0 \ for\ all \ x$ in an interval, the function $\ f(x)$ is concave down on that interval. The graph of the function bends downwards, resembling the shape of an upside-down U.
This implies that the slope of the tangent line (the first derivative) is decreasing.

2. Inflection Points:

An inflection point is a point on the graph of a function where the concavity changes from concave up to concave down, or vice versa. It occurs where the second derivative changes sign:

To find an inflection point, solve $\ f''(x) = 0$ or where $\ f''(x)$ is undefined.
Verify that the concavity changes by checking the sign of $\ f''(x)$ on either side of the point.

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Example:

If $\ f(x) = x^3 - 3x^2 + 4x$ , then:

First derivative: $\ f'(x) = 3x^2 - 6x + 4$
Second derivative: $\ f''(x) = 6x - 6$ Setting the second derivative equal to zero gives: $6x - 6 = 0 \quad \Rightarrow \quad x = 1$ Check the sign of $\ f''(x) \ around \ x = 1$ to determine if it is an inflection point.

3. Second Derivative Test:

The second derivative test helps determine whether a critical point (where $\ f'(x) = 0$ ) is a local maximum, local minimum, or neither:

Local Minimum: If $\ f'(x) = 0 \ and \ f''(x) > 0 \ at \ x = c \ , then \ f(x)$ has a local minimum at $\ x = c .$
Local Maximum: If $\ f'(x) = 0 \ and \ f''(x) < 0 \ at \ x = c \ , then \ f(x)$ has a local maximum at $\ x = c .$
Inconclusive: If $\ f'(x) = 0 \ and \ f''(x) = 0 ,$ the test is inconclusive, and further analysis is needed.

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Example:

Consider $\ f(x) = x^3 - 3x^2 + 2$ :

First derivative: $\ f'(x) = 3x^2 - 6x$
Critical points: Solve $\ 3x^2 - 6x = 0 \ to\ find \ x = 0 \ and \ x = 2 \ .$
Second derivative: $\ f''(x) = 6x - 6$ At $\ x = 0 :$ $f''(0) = 6(0) - 6 = -6 \quad \text{(Local Maximum)}$ At $\ x = 2$ : $f''(2) = 6(2) - 6 = 6 \quad \text{(Local Minimum)}$

4. Curve Sketching:

The second derivative is essential in curve sketching, as it provides insight into the shape of the graph:

Identify intervals where the function is concave up or down by analysing the sign of the second derivative.
Find inflection points where the concavity changes.
Combine this information with the first derivative analysis to sketch the graph.

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Example:

For $\ f(x) = \frac{1}{3}x^3 - x^2 + 2x$ :

First derivative: $\ f'(x) = x^2 - 2x + 2$
Second derivative: $\ f''(x) = 2x - 2$ By analysing the signs of $\ f'(x)$ and $\ f''(x)$ , you can determine the increasing/decreasing behaviour and concavity to sketch the graph accurately.

5. Optimization in Economics and Engineering:

In economics, the second derivative is used to determine the concavity of cost, revenue, or profit functions, helping to identify points of maximum profit or minimum cost.

Maximizing Profit: If the profit function $\ P(x)$ is concave down (i.e., $\ P''(x) < 0$ ), then any critical point will be a maximum profit point.
Minimizing Cost: If the cost function $\ C(x)$ is concave up (i.e., $\ C''(x) > 0$ ), then any critical point will be a minimum cost point. In engineering, the second derivative helps in designing systems that optimize efficiency, stability, and other key parameters.

6. Motion and Acceleration in Physics:

In physics, the second derivative of the position function $\ s(t)$ with respect to time $\ t$ gives the acceleration $\ a(t) :$

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$\ v(t) = \frac{ds(t)}{dt}$ is the velocity.
$\ a(t) = \frac{dv(t)}{dt} = \frac{d^2s(t)}{dt^2}$ is the acceleration.

Understanding how acceleration varies with time helps in analysing motion, forces, and energy in mechanical systems.

7. Testing for Concavity and Convexity:

The second derivative is also used to determine whether a function is convex or concave:

A function is convex if $\ f''(x) \geq 0$ for all $\ x$ in its domain.
A function is concave if $\ f''(x) \leq 0$ for all $\ x$ in its domain. This property is crucial in optimization problems, where convex functions have global minima at their lowest points.

Summary:

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The second derivative provides information about the curvature and concavity of a function, helping to identify inflection points, local maxima, and minima.
It is widely used in optimization, curve sketching, motion analysis, and economic models to understand the behaviour of functions.
Mastering the applications of the second derivative is essential for solving advanced problems in calculus and its applications across various fields.

Applications of the Second Derivative Simplified Revision Notes for A-Level OCR Maths Pure

7.4.1 Applications of the Second Derivative

1. Concavity and Convexity:

2. Inflection Points:

Example:

3. Second Derivative Test:

Example:

4. Curve Sketching:

Example:

5. Optimization in Economics and Engineering:

6. Motion and Acceleration in Physics:

7. Testing for Concavity and Convexity:

Summary:

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