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MAT 1221 Survey of Calculus

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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
- Simplify answers
- 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?

- 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?

Formal Answer

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

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.

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

- Use equal signs correctly.
- Use and notations correctly.
- Pay attention to the independent variables: Is it or ?
- Pay attention to the units.