Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018
New Divider! Application of Derivatives: Local Linearity and L'Hopitals Local Linear Approximation Do Now: For each sketch the function, write the equation of the tangent line at x = 0 and include the tangent line in your sketch.
1) 2)
In general, if a function f is differentiable at an x, then a sufficiently magnified portion of the graph of f centered at the point P(x, f(x)) takes on the appearance of a ______line segment.
For this reason, a function that is differentiable at x is sometimes said to be locally linear at x.
1 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018
How is this useful?
We are pretty good at finding the equations of tangent lines for various functions.
Question: Would you rather evaluate linear functions or crazy ridiculous functions such as higher order polynomials, trigonometric, logarithmic, etc functions?
Evaluate sec(0.3)
The idea is to use the equation of the tangent line to a point on the curve to help us approximate the function values at a specific x.
Get it??? Probably not....here is an example of a problem I would like us to be able to approximate by the end of the class.
Without the use of a calculator approximate .
2 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018
Local Linear Approximation General Proof Directions would say, evaluate f(a). If f(x) you find this impossible for some y reason, then that's how you would recognize we need to use local linear approximation! You would: 1) Draw in a tangent line at x = a. 2) Write the equation of the tangent line. You would need: point of tangency: ( , )
a x slope of tangent:
Tangent Line Equation: So now we recognize that our tangent line is easier to work with then our f(x). Pick a point on the tangent line that is close to your x = a. We call this point (a + Δx, f(a + Δx)).
Let's get a visual of this on our curve above!o Substitute this point into our tangent line equation, we can right? Why?
Simplify!
Based upon this general f(x), how would the approximation compare to the actual function value? What about the curve determines if the approximation is greater than or less than the actual function value?
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Example 1: Use local linear approximation to approximate .
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Example 2: Use local linear approximation to approximate .
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7.) Consider the curve defined by x2 + xy + y2 = 27. (This question is from the 1994 exam: #3.)
a. Write an expression for the slope of the curve at any point (x, y).
b. Determine whether the lines tangent to the curve at the xintercepts of the curve are parallel. c. Find the points on the curve where the lines tangent to the curve are vertical.
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Another use for local linear approximations! We are going to take a trip down memory lane....let's revisit limits!
What do all of these limits have in common?
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Evaluating Limits of an Indeterminate Form General Method:
Suppose is an indeterminate form of type in which f ' and
g' are continuous at x = a and g'(a) ≠ 0.
Since f and g can be closely approximated by their local linear approxmiations near a, then
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L'Hopital's Rule for Form 0/0 Graphic Understanding
f(x) = mf (x a) + 0 and
g(x) = mg (x a) + 0
What is ?
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L'Hopital's Rule for Form 0/0 Graphic Understanding Even if f(x) and g(x) are curved, by local linearity if we zoom in on the area that we are trying to approach with the limit then we would have the graph on the left:
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L'Hopital's Rule for Form 0/0
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Example: Find the limit:
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Example: Evaluate each:
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Second Indeterminate Form for L'Hopital's Rule
Evaluate:
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Evaluate:
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Evaluate:
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Evaluate:
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Evaluate:
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Evaluate:
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Evaluate:
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Evaluate:
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Evaluate:
30 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Last Three Indeterminate Forms What should we do???
EX:
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Evaluate:
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Evaluate:
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Famous Limits Evaluate each:
1) 2)
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