AS Level Biology 9700
Chapter 8 - Further Differentiation
Written by: Tharun Athreya
Formatted by: Tharun Athreya
Index
8.1 Increasing and Decreasing Function
- Increasing function: A function \( f(x) \) is increasing where the derivative \( f'(x) > 0 \) (the slope of the tangent is positive).
- Decreasing function: A function \( f(x) \) is decreasing where the derivative \( f'(x) < 0 \) (the slope of the tangent is negative).
Tip: To determine if a function is increasing or decreasing:
Method-1:
Method-1:
- Differentiate the function to find \( f'(x) \).
- Analyse the sign of \( f'(x) \) over the interval of interest.
- If \( f'(x) > 0 \), then the function is increasing.
- If \( f'(x) = 0 \), then the function is constant.
- If \( f'(x) < 0 \), then the function is decreasing.
- Look at the graph to see if the function is increasing or decreasing:
- Increasing: If the graph goes up from left to right, then the function is increasing.
- Constant: If the graph is a horizontal line, then the function is constant.
- Decreasing: If the graph goes down from left to right, then the function is decreasing.
8.2 Stationary Points
- A point where the gradient is zero is called a stationary point or turning point.
- Maximum point: A type of stationary point where the value of \( y \) at this point is greater than the value of \( y \) at nearby points.
- \( \frac{dy}{dx} = 0 \)
- The gradient is positive to the left of the maximum and negative to the right.
- Minimum point: The point where the curve stops going down and starts to go up again.
- \( \frac{dy}{dx} = 0 \)
- The gradient is negative to the left of the minimum and positive to the right.
- Point of inflection: A stationary point where the graph of a function changes from concave to convex, or convex to concave.
- At a point of inflection, the slope of the function is neither increasing nor decreasing.
- \( \frac{dy}{dx} = 0 \)
- The gradient changes from positive to zero and then to positive again, or from negative to zero and then to negative again.
- How to determine the nature of the stationary point:
- First, find the second derivative.
- If \( \frac{dy}{dx} = 0 \) and \( \frac{d^2y}{dx^2} < 0 \), then the point is a maximum.
- If \( \frac{dy}{dx} = 0 \) and \( \frac{d^2y}{dx^2} > 0 \), then the point is a minimum.
- First Derivative Test: Compare the signs of the first derivative (positive or negative) a little bit to either side of the stationary point.
- Find the first derivative.
- For each stationary point, find the values of the first derivative slightly to the left and right.
- Compare the signs of the derivatives:
- If negative on the left and positive on the right, the point is a minimum.
- If positive on the left and negative on the right, the point is a maximum.
- If the signs are the same on both sides, the point is a point of inflection.
- If \( \frac{dy}{dx} = 0 \) and \( \frac{d^2y}{dx^2} = 0 \), then use the first derivative test to determine the nature of the stationary point.
- Sketching graphs using the gradient function:
- The behaviour of a function gives information about the behaviour of its gradient function.
- While sketching graphs, keep in mind:
- Show the \( y \)-intercept.
- Show the stationary points.
8.3 Practical maximum and minimum problems
- Derivatives can be calculated for any variables, not only for x and y.
- In every case the derivative is a formula giving the rate of change of one variable with respect to the other variable.
- There are many problems for which we need to find the maximum or minimum value of an expression. For which, you can use differentiation as a method.
8.4 Rates of change
- In situations involving more than two variables, you can use the chain rule to connect multiple rates of change into a single equation.
- General rule:
\( \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} \) - You can also use:
\( \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} \)