Differentiation for Graph Sketching

Key Features of Graphs

Differentiation is a powerful tool for analyzing and sketching graphs of functions. By finding the first and second derivatives, we can identify key features such as:

• Stationary Points: Points where the gradient of the function is zero ($f’(x) = 0$). These can be local maxima, local minima, or stationary points of inflection.
• Points of Inflection: Points where the concavity of the function changes. These occur where the second derivative is zero or undefined ($f’‘(x) = 0$).
• Intervals of Increasing and Decreasing: Intervals where the function is strictly increasing ($f’(x) > 0$) or strictly decreasing ($f’(x) < 0$).

KEY TAKEAWAY: Differentiation allows us to move beyond plotting points and truly understand the behavior of a function.

Stationary Points

Definition

A stationary point of a function $f(x)$ is a point where the derivative $f’(x) = 0$. At these points, the tangent to the curve is horizontal.

Types of Stationary Points

1. Local Maximum: A point where the function reaches a maximum value within a small interval. To the left of the maximum, the function is increasing ($f’(x) > 0$), and to the right, it is decreasing ($f’(x) < 0$).
2. Local Minimum: A point where the function reaches a minimum value within a small interval. To the left of the minimum, the function is decreasing ($f’(x) < 0$), and to the right, it is increasing ($f’(x) > 0$).

3. Stationary Point of Inflection: A point where the gradient is zero, but the function does not change direction (i.e., it doesn’t go from increasing to decreasing or vice-versa). The concavity changes at this point.

   • Gradient positive to the left and right: gradient + 0 +
   • Gradient negative to the left and right: gradient - 0 -

Determining the Nature of Stationary Points

First Derivative Test

1. Find the $x$-values where $f’(x) = 0$. These are the $x$-coordinates of the stationary points.
2. Choose test values of $x$ slightly to the left and right of each stationary point.
3. Evaluate $f’(x)$ at these test values.
4. Determine the nature of the stationary point based on the sign change of $f’(x)$:
   • Positive to Negative: Local Maximum
   • Negative to Positive: Local Minimum
   • Positive to Positive or Negative to Negative: Stationary Point of Inflection

Second Derivative Test

1. Find the second derivative, $f’‘(x)$.
2. Evaluate $f’‘(x)$ at each stationary point (where $f’(x) = 0$).
3. Determine the nature of the stationary point based on the value of $f’‘(x)$:
   • $f’‘(x) > 0$: Local Minimum
   • $f’‘(x) < 0$: Local Maximum
   • $f’‘(x) = 0$: The test is inconclusive; use the first derivative test.

EXAM TIP: The second derivative test is often easier to apply, but remember it’s inconclusive when $f’‘(x) = 0$.

Points of Inflection

Definition

A point of inflection is a point on a curve where the concavity changes. This means the curve changes from being concave up to concave down, or vice versa.

Finding Points of Inflection

1. Find the second derivative, $f’‘(x)$.
2. Solve $f’‘(x) = 0$ for $x$. These are potential points of inflection.
3. Check that the concavity changes at these points. This can be done by:
   • Evaluating $f’‘(x)$ on either side of the potential point of inflection. If the sign of $f’‘(x)$ changes, then it is a point of inflection.
   • Checking that $f’‘’(x) \neq 0$ at the point. (If the third derivative exists and is non-zero, it’s a point of inflection.)

Concavity

• Concave Up: The curve is shaped like a cup opening upwards. $f’‘(x) > 0$.
• Concave Down: The curve is shaped like a cup opening downwards. $f’‘(x) < 0$.

COMMON MISTAKE: For a stationary point of inflection, both $f’(x) = 0$ and $f’‘(x) = 0$. However, not all points where $f’‘(x) = 0$ are stationary points of inflection.

Intervals of Increasing and Decreasing Functions

Increasing Function

A function $f(x)$ is strictly increasing on an interval if $f’(x) > 0$ for all $x$ in that interval. As $x$ increases, $y$ also increases.

Decreasing Function

A function $f(x)$ is strictly decreasing on an interval if $f’(x) < 0$ for all $x$ in that interval. As $x$ increases, $y$ decreases.

Finding Intervals of Increasing and Decreasing

1. Find the first derivative, $f’(x)$.
2. Find the critical points where $f’(x) = 0$ or $f’(x)$ is undefined. These points divide the domain into intervals.
3. Choose a test value within each interval and evaluate $f’(x)$ at that value.
4. Determine whether the function is increasing or decreasing in each interval based on the sign of $f’(x)$:
   • $f’(x) > 0$: Increasing
   • $f’(x) < 0$: Decreasing

STUDY HINT: Create a sign diagram for $f’(x)$ to visually represent the intervals of increasing and decreasing.

Graph Sketching

To sketch a graph of a function $f(x)$ using differentiation, follow these steps:

1. Find the domain and range (if possible).
2. Find the $x$ and $y$ intercepts.
3. Find the stationary points and determine their nature (local max, local min, or stationary point of inflection).
4. Find the points of inflection.
5. Determine the intervals of increasing and decreasing.
6. Determine the concavity of the function.
7. Consider the end behavior of the function (as $x$ approaches positive and negative infinity).
8. Sketch the graph, plotting all key points and ensuring the graph reflects the information gathered in the previous steps.

APPLICATION: Graph sketching is used in many real-world scenarios, such as modeling population growth, analyzing financial data, and designing engineering structures.

Example

Let’s sketch the graph of $f(x) = x^3 - 3x^2 + 2$.

1. Domain: $R$
2. y-intercept: $f(0) = 2$
3. First derivative: $f’(x) = 3x^2 - 6x$
4. Stationary points: $f’(x) = 0 \Rightarrow 3x(x - 2) = 0 \Rightarrow x = 0, 2$
   • $f(0) = 2$, $f(2) = -2$
5. Second derivative: $f’‘(x) = 6x - 6$
6. Nature of stationary points:
   • $f’‘(0) = -6 < 0 \Rightarrow$ Local maximum at $(0, 2)$
   • $f’‘(2) = 6 > 0 \Rightarrow$ Local minimum at $(2, -2)$
7. Point of inflection: $f’‘(x) = 0 \Rightarrow 6x - 6 = 0 \Rightarrow x = 1$
   • $f(1) = 0$
8. Intervals of increasing/decreasing:
   • $x < 0$: $f’(x) > 0$ (increasing)
   • \$0 < x < 2$: $f’(x) < 0$ (decreasing)
   • $x > 2$: $f’(x) > 0$ (increasing)
9. Concavity:
   • $x < 1$: $f’‘(x) < 0$ (concave down)
   • $x > 1$: $f’‘(x) > 0$ (concave up)

Using this information, you can accurately sketch the graph of $f(x)$.

VCAA FOCUS: Exam questions often require you to sketch graphs and identify key features. Practice sketching a variety of functions, including polynomials, trigonometric functions, and exponentials.