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# MAT 1221 Survey of Calculus - PowerPoint PPT Presentation

MAT 1221 Survey of Calculus. Section B.1, B.2 Implicit Differentiation, Related Rates. http://myhome.spu.edu/lauw. Expectations. Use equal signs Simplify answers Double check the algebra. HW. WebAssign HW B.1, B.2

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### MAT 1221Survey of Calculus

Section B.1, B.2

Implicit Differentiation, Related Rates

http://myhome.spu.edu/lauw

• Use equal signs

• Double check the algebra

• 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.

• 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

• Extended Power Rule Revisit

• The needs for new differentiation technique –Implicit Differentiation

• The needs to know the relation between two rates – Related Rates

We now free the variable, which we need for the next formula.

• If is a function in , then

• If y is a function in x, then

• If is a function in , then

• If is a function in , then

• Find the slopes of the tangent line on the graph

• i.e. find

y

x

• Make as the subject of the equation:

y

x

• Make as the subject of the equation:

y

x

• Make y as the subject of the equation:

y

x

Suppose the point is on the upper half circle

y

x

Suppose the point is on the lower half circle

y

x

• Two disadvantages of Method I:

• ???

• ???

Implicit Differentiation:

Differentiate both sides of the equation.

Find the slope of the tangent line at

• If and are related by an equation, their derivatives (rate of changes)

and

are also related.

• If and are related by an equation, their derivatives (rate of changes)

and

are also related.

• Note that the functions are time dependent

• Extended Power Rule will be used frequently, e.g.

• Consider a “growing” circle.

• Both the radius and the area are increasing.

• What is the relation between and ?

• 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?

• 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 5: Use differentiation to find the related rate variables defined

When the radius is 5 feet, the area is changing at a rate of …

Steps for Word Problems variables defined

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 variables defined

To save time, problem number 2 does not required all the steps.

Expectations variables defined

• Use equal signs correctly.

• Use and notations correctly.

• Pay attention to the independent variables: Is it or ?

• Pay attention to the units.