MAT 1221 Survey of Calculus Section B.1, B.2 Implicit Differentiation, Related Rates

Expectations Use equal signs Simplify answers Double check the algebra

HW WebAssign HW B.1, B.2 Additional HW listed at the end of the handout (need to finish, but no need to turn in) Need to do your homework soon. Do not wait until tomorrow afternoon.

Exam 1 Bring your Tutoring Bonus paper to class on Thursday.

Exam 1 You should have already started reviewing for Exam 1 Proficiency: You need to know how to do your HW problem on your own You need to understand how to solve problems Memorizing the solutions of all the problems is not a good idea

Preview Extended Power Rule Revisit The needs for new differentiation technique – Implicit Differentiation The needs to know the relation between two rates – Related Rates

Extended Power Rule

If y is a function in x, then

Extended Power Rule

Example 0

The Needs for Implicit Differentiation…

Example 1 Find the slopes of the tangent line on the graph i.e. find x y

Example 1: Method I x y

x y

Make y as the subject of the equation: x y

Example 1: Method I Suppose the point is on the upper half circle x y

Example 1: Method I Suppose the point is on the lower half circle x y

Example 1: Method I Two disadvantages of Method I: 1. ??? 2. ???

Example 1: Method II Implicit Differentiation: Differentiate both sides of the equation.

Expectation You are required to show the implicit differentiation step

Notations

Example 2 Find the slope of the tangent line at

Notations

Correct or Incorrect because

B.2. Related Rates

Related Rates

Example 3 Consider a “growing” circle.

Example 3 Both the radius and the area are increasing.

Example 3 What is the relation between and ?

Example 1 GO NUTS!

Example 1 GO NUTS!

Example 3 A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 1 Draw a diagram A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 2: Define the variables A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 3: Write down all the information in terms of the variables defined A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 4: Set up a relation between the variables

Step 5: Use differentiation to find the related rate Formal Answer When the radius is 5 feet, the area is changing at a rate of …

Steps for Word Problems 1. Draw a diagram 2. Define the variables 3. Write down all the information in terms of the variables defined 4. Set up a relation between the variables 5. Use differentiation to find the related rate. Formally answer the question.

Remark on Classwork #2 To save time, problem number 2 does not required all the steps.

Expectations