Differentiation Gradients from parametric equations Implicit differentiation Rates of change Applications of differential equations

Differentiation Esme is about to set off for University to study Biology after she completes her A Levels. She is faced with several situations which she must find the solution to in order to settle into university life.

Problem 1 Esme wishes to purchase an MX5 and wants to work out the rate of depreciation of her car using an appropriate differential equation. Here for a picture

Problem 2 Esmee wants to study at The University of Sydney in Australia still however the disease Revoltus is still growing in population at a rate of 9% each year. What is the equation that describes this population at time t if P0 is the initial population? Find an expression for t in terms of P and P0. Find the number of years T when the population doubles from t = 0. Lastly find as a multiple pf P0 the rate of change of the population dP/dt at time t =T.

Problem 3 Esme wants to learn how to mix cocktails for uni parties and she wants to calculate the rate of change of the volume for an ice cube to melt. The rate of decrease of the surface area is -2cm2/s. Find an expression for dV/dt.

Problem 4 Ernest her brother is not going with Esme and will stay in HK. He loves to eat ice cream however he is an extremely slow eater and it the ice cream melts through the bottom before he gets a chance to eat it. Esme wants to calculate the rate of change of the height of the ice cream with respect to time. Here is some more information to help you solve this. The ice cream fills the cone with no overflow and melts through a small hole at the bottom at a rate of 6 cm3/s The angle between the cone and slanting edge is 30 0 hence show that the volume of ice cream is 1/9 h3 where h is the height of the ice cream at time t. Show that the rate of change of the height of the ice cream is inversely proportional to h2. write down the differential equation relation h and t.

Tools We need some tools to help us solve these types of equations. Finding the gradient function when given parametric equations

Example 1

Implicit Differentiation Equations which contain x and y in a form such as this x2+y2 = 8x are implicitly expressed. Here we use the chain rule and product rules when necessary

Examples

Using Implicit differentiation for exponential functions!

Y = ex

Y=xax

Y = 2x+2-x

Another example A particular radioactive isotope has an activity R millicuries at time t given by the equation R=200(0.9)t. Find the value of dR/dt when t=8.

Rates of Change Given the area of a circle A sqcm is related to its radius r cm by the formula A= r2, and the rate of change of its radius in cm/s is given by dr/dt = 5 find dA/dt when r=3.

Applications of Differential Equations