Unit 4 - Differential Calculus
Required Prior Knowledge
Questions
Find the stationary points for the function \(y=x\cos x\) in the interval \(-\pi\le x\le \pi\).
Determine the nature of each of these stationary points.
State the formulae for the volumes and surface areas for the following shapes:
Cylinder
Sphere
Cone
Solutions
Get Ready
Questions
You want to create the rectangle with the largest area that has a perimeter of 100 cm. What dimensions should the rectangle be?
Solutions
Notes
There are many scenarios where we want to find the optimum value of something.
Sometimes this is a maximum (such as profit, area) and sometimes this is a minimum (such as cost, materials required).
Since we are looking for maxima and minima we use differentiation.
There are 5 steps to solving an optimisation problem:
Identify the Variables (draw a diagram)
Identify the Constraints (the fixed parts of the problem)
Set up the model (the function for what we want to optimise)
Optimise (solve the model by finding the maximum or minimum value)
Interpret the Results (in the context of the problem)
Examples and Your Turns
Example
An open box is made by cutting congruent squares from the corners of a 4m by 4 m cardboard sheet. How large should the squares be so that the box has a maximum volume? What is the maximum capacity of the box?
Your Turn
A population of bacteria is modelled by \(P\left(t\right)=100te^{-0.5t}\) where \(t\) is the time in hours after midday.
Find the time at which the population is at its maximum.
Your Turn
You have been asked to design a cylindrical can to hold 1 litre of car oil, with the minimum surface area in order to minimize costs. Find the dimensions of the can.
Your Turn
Find the point on the curve \(y=\sqrt{x}\) that is closest to the point \(\left(2,0\right)\).
Your Turn (HL)
A point \(\left(x,y\right)\) moves along the ellipse \(x^{2}+4y^{2}=36\). Find the maximum possible value of the product \(P=xy\).
Your Turn (HL)
A rectangle has its base on the \(x\)-axis and its upper corners on the curve \(x^{2}+2y^{2}=6.
Find the maximum area of the rectangle.
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Why is it important to check the nature of any stationary points you find when optimising?
If a function has a single stationary point in the domain, and we know it is a local maximum, why do we still need to check if it is the global maximum?
Why is it important to check the end points of a function within the domain as well?
Common Mistakes / Misconceptions
The most common mistake is to ignore the constraints placed upon the situation and try to differentiate a function with two variables.
Another misconception is assuming that a LOCAL maximum is the GLOBAL maximum (without checking).
It is common to not answer the actual question, e.g. you are asked for the maximum AREA, but give the \(x\) value from the optimisation without using this to find the area.
Connecting This to Other Skills
The whole concept of optimisation is based on the idea of Stationary Points (4.14), but you need all the rules for differentiation such as Polynomials (4.5), Chain Rule (4.7) Product Rule (4.8), Quotient Rule (4.9), Implicit Differentiation (4.10), Exponentials and Logarithms (4.11) and Trigonometric Functions (4.12).
The next skill, Related Rates of Change (4.19) is the other main application of differentiation in the course.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?