Sec311LinearApproximationLectureMath151Done.notebook
Sec. 3.10 Linear Approximations and Differentials
Given a function and its derivative We can think of its differential as the "rise" that results from a small "run"
Sec. 3.10 Linear Approximations and Differentials
Given a point on a curve of y=f(x), the tangent line through that point
Is a good approximation for the curve (at least neat the point). It is often easier to calculate and so we can use it as an estimate for the function near that point.
For values of ‘x’ that are near ‘a’, we have Or find a similar formula for f(x) using "x+h" as the nearby point.
February 21, 2012
Sec311LinearApproximationLectureMath151Done.notebook
Practical example…Suppose the population of a city is now 75,000
And it is increasing at a rate of 3000 people per year. What will the population be in 2015?
Estimate using differentials.
What is the real value?
February 21, 2012
Sec311LinearApproximationLectureMath151Done.notebook
Estimate using differentials.
What is the real value?
Estimate and using differentials.
What are the real values?
February 21, 2012
Sec311LinearApproximationLectureMath151Done.notebook
Estimate and using differentials.
What are the real values?
February 21, 2012
Sec311LinearApproximationLectureMath151Done.notebook
February 21, 2012
Sec311LinearApproximationLectureMath151Done.notebook
February 21, 2012
A Linear approximation uses the "point­slope" form to approximate the function (near a certain value of x) by its tangent line.
Find the Linear Approximation for
Use it to estimate f(3.97) and f(4.07). Also f(10).
at a=4
Sec311LinearApproximationLectureMath151Done.notebook
February 21, 2012
There are ways to approximate any function by quadratic functions instead of Linear functions.
Generally they are called Taylor polynomials.
As they show in the Lab Project at the end of this section, it is not hard to make a polynomial that matches any graph at a point.
To have the same value, derivative, second derivative, third derivative... etc. As much as you want. Each give a better approximation for the function.
GraphsOfTaylorPolynomials.nbp
Find a quadratic function that matches the values of 2Sin(x)­Cos(3x) at x=0 for the value, the derivative and the second derivative.
Attachments
GraphsOfTaylorPolynomials.nbp