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 1221 survey of calculus

MAT 1221Survey of Calculus

Section B.1, B.2

Implicit Differentiation, Related Rates

http://myhome.spu.edu/lauw


Expectations

Expectations

  • Use equal signs

  • Simplify answers

  • Double check the algebra


Mat 1221 survey of calculus

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

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

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 rule1

Extended Power Rule


Extended power rule2

Extended Power Rule


Extended power rule3

Extended Power Rule

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


Extended power rule4

Extended Power Rule

  • If is a function in , then

  • If y is a function in x, then


Extended power rule5

Extended Power Rule

  • If is a function in , then

  • If is a function in , then


Example 0

Example 0


The needs for implicit differentiation

The Needs for Implicit Differentiation…


Example 1

Example 1

  • Find the slopes of the tangent line on the graph

  • i.e. find

y

x


Example 1 method i

Example 1: Method I

  • Make as the subject of the equation:

y

x


Example 1 method i1

Example 1: Method I

  • Make as the subject of the equation:

y

x


Example 1 method i2

Example 1: Method I

  • Make y as the subject of the equation:

y

x


Example 1 method i3

Example 1: Method I

Suppose the point is on the upper half circle

y

x


Example 1 method i4

Example 1: Method I

Suppose the point is on the lower half circle

y

x


Example 1 method i5

Example 1: Method I

  • Two disadvantages of Method I:

  • ???

  • ???


Example 1 method ii

Example 1: Method II

Implicit Differentiation:

Differentiate both sides of the equation.


Example 2

Example 2

Find the slope of the tangent line at


Notations

Notations


B 2 related rates

B.2. Related Rates


Related rates

Related Rates

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

    and

    are also related.


Related rates1

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

Example 3

  • Consider a “growing” circle.


Example 31

Example 3

  • Both the radius and the area are increasing.


Example 32

Example 3

  • What is the relation between and ?


Example 33

Example 3


Example 34

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

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

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

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 4: Set up a relation between the variables


Step 5 use differentiation to find the related rate

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

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

Remark on Classwork #2

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


Expectations1

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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