Mat 1221 survey of calculus
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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 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


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

  • 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


Extended Power Rule


Extended Power Rule


Extended Power Rule

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


Extended Power Rule

  • If is a function in , then

  • If y is a function in x, then


Extended Power Rule

  • If is a function in , then

  • If is a function in , then


Example 0


The Needs for Implicit Differentiation…


Example 1

  • Find the slopes of the tangent line on the graph

  • i.e. find

y

x


Example 1: Method I

  • Make as the subject of the equation:

y

x


Example 1: Method I

  • Make as the subject of the equation:

y

x


Example 1: Method I

  • Make y as the subject of the equation:

y

x


Example 1: Method I

Suppose the point is on the upper half circle

y

x


Example 1: Method I

Suppose the point is on the lower half circle

y

x


Example 1: Method I

  • Two disadvantages of Method I:

  • ???

  • ???


Example 1: Method II

Implicit Differentiation:

Differentiate both sides of the equation.


Example 2

Find the slope of the tangent line at


Notations


B.2. Related Rates


Related Rates

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

    and

    are also related.


Related Rates

  • 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.


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 3


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

  • Use equal signs correctly.

  • Use and notations correctly.

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

  • Pay attention to the units.


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