Unit 4 - Differential Calculus
Required Prior Knowledge
Questions
Find the gradient of \(f\left(x\right)=e^{\sin x}\) at \(x=\frac{5\pi}{6}\).
Solutions
Get Ready
Questions
Imagine a leaky pipe, dripping water into a circular puddle on the floor.
If we know that the water is leaking at a rate of 1 litre per hour:
a) How quickly is the volume of water in the puddle increasing?
b) Can we tell how quickly the radius of the puddle is increasing? What about the area?
c) Why is it more difficult to find the rates of change for the radius than for the volume of water in the puddle?
Solutions
Notes
Differentiating with respect to time gives us a rate of change of a variable.
For example, \(\frac{dA}{dt}\) tells us how fast the area, \(A\) m², is changing with respect to time.
Sometimes we know the rate at which one thing is changing, and want to find how fast a related variable is changing.
The steps to achieve this are:
Identify the Variables (draw a diagram if needed)
Identify the Constraints (the given rate of change AND the equation connecting the variables)
Set up the Model (differentiate the equation and use the Chain Rule to connect the various derivatives)
Solve the Model (substitute all the derivatives you know and rearrange)
Interpret the Answer (substitute the end time you are asked about)
Examples and Your Turns
Example
Some oil is spilt onto a level surface and spreads out in the shape of a circle.
The radius \(r\) cm of the circle is increasing at a rate of \(0.5\text{cm s}^{-1}\).
At what rate is the area of the circle increasing when the radius is \(5\) cm?
Your Turn
A stone is thrown into a still lake, creating a circular ripple. The radius \(r\) increases at a rate of \(2\) cm/s. Find the rate at which the area \(A\) is increasing when the radius is \(10\) cm.
Your Turn
A spherical balloon is being inflated at a rate of \(100\) cm³/s. Find the rate at which the radius is increasing when the diameter is \(20\) cm.
Your Turn
A \(10\) m long ladder is leaning against a wall on a building site. It starts to slip down the wall at a rate of \(0.5\) m/s.
How fast is the foot of the ladder moving along the ground when it is \(6\) m from the wall?
Your Turn
A man \(2\) m tall walks away from a lamppost \(6\) m high at a speed of \(1.5\) m/s. How fast is the length of his shadow increasing? (Hint: use similar triangles)
Your Turn
Water is poured into a conical tank at a rate of \(3\) m³/ min. The tank stands with the point downward. How fast is the water level rising when the depth is \(2\) m and the radius of the water surface is \(1.5\) m?
Your Turn
A \(10\) m long ladder is leaning against a wall on a building site. It starts to slip down the wall at a rate of \(0.5\) m/s.
How fast is the angle between the ladder and the ground changing when the vertical height of the ladder is \(8\) m?
Your Turn
There are two ships at sea, Zadar and Rab. At a given moment Zadar is \(40\) km south and \(50\) km east of Rab. Zadar sails north at a rate of \(12\) km/h, whilst Rab sails east at a rate of \(15\) km/h.
a) How fast are the two ships approaching each other after \(2\) hours?
b) How fast is the bearing of Zadar from Rab changing after \(2\) hours?
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 the Chain Rule so important for Related Rates of Change problems?
If the volume of a cube is increasing at a constant rate, does the side length also increase at a constant rate? Why or why not?
In the "ladder" example, why does the speed of the top of the ladder increase as the base gets further from the wall?
Common Mistakes / Misconceptions
This is a HL application of differentiation, and most students find it very difficult.
A common misconception is substituting the ‘end’ value before setting up the model. This is only used in the very last step.
Another common difficulty is using Implicit Differentiation when needed.
Connecting This to Other Skills
The basic principle that is essential or this topic is the Chain Rule (4.7).
You also need to be able to use your formulae for Volumes and Surface Areas (PK7).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?