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MATA33H3 Lecture Notes - Lecture 9: Natural Number, Power Law

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School
UTSC
Department
Mathematics
Course
MATA33H3
Professor
Nick Cheng

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Chapter 11 - Differentiation
Motivation Bombordier 10 Day Average
Linear Approximation
You can extend the graph linearly after zooming in.
Question: What is the equation of this line after we zoom in?
The best line approximation is the tangent line.
How to obtain the tangent line?
- In general the equation of a line is y=mx+b
We need another equation to solve for m and b.
- To do this we attempt to find m independently of b.
f(x)
Definition: (Derivative of a) Let f be a function on (Real Line). The derivative of f at x=a is defined to be
The slope of f at x=a is the derivative of f at x=a.
Remark: We often write
2. Note that’s can also write
Similarly, can write
Remark: The process of taking derivatives is called differentiation. In other words, to differentiate means "to take the
derivative."
Remark: It is possible that f’(a) does not exist. In this case, f is not differentiate at a, otherwise f is differentiable.
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Related Questions

1) Use implicit differentiation to find the slope of the tangentline to the curve

at the point .


"Just need the answer"




2) Use implicit differentiation to find the equation of the tangentline to the curve at the point. The equation ofthis tangent line can be written in the form where is:

and where is:

"Just need the answers"



3) Use implicit differentiation to find the slope of the tangent lineto the curve

atthe point .


"Just need the answers"

Let .
Find the differential whenand

Find the differential whenand


Recall that linear approximation uses thevalues of a tangent line to approximate valuesof a function. Use this method to approximate asfollows:
Let . The equation of the tangent line toat can be written in the formwhere: = and b=,Using this, we find our approximation foris:


Recall that linear approximation uses thevalues of a tangent line to approximate valuesof a function. Use this method to approximate asfollows:
Let . The equation of the tangent line toat can be written in the formwhere =: and b= ,Usingthis, we find our approximation for is

NOTE: For this part, give your answer to at least 5 placesafter the decimal, or use fractions in your answer.


The function f is twice differentiable for x > 0 with f (1) =15 and f '' (1)=20. values of f'. the derivative of f, are givne for selected.

a. Write an equation for the line tangent to the graph of at x=1.

Use this line to approximate f (1.4).

b. Use a midpoint Riemann sum two subintervals of equal length and values from the table to approximate  Use the approximation for to estimate the value of f (1.4). Show the computations that lead to your answer.