Year 6 Local Linearity and L'hopitals.Notebook December 04, 2018
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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? 3 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Example 1: Use local linear approximation to approximate . 4 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Example 2: Use local linear approximation to approximate . 5 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 6 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 7 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 8 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 9 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 10 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 11 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 12 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 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. 13 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 14 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 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? 15 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 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 16 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 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 ? 17 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 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: 18 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 L'Hopital's Rule for Form 0/0 19 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Example: Find the limit: 20 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Example: Evaluate each: 21 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Second Indeterminate Form for L'Hopital's Rule Evaluate: 22 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Evaluate: 23 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Evaluate: 24 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Evaluate: 25 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Evaluate: 26 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Evaluate: 27 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Evaluate: 28 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Evaluate: 29 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Evaluate: 30 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Last Three Indeterminate Forms What should we do??? EX: 31 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Evaluate: 32 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Evaluate: 33 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 Famous Limits Evaluate each: 1) 2) 34 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 35 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 36 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 37 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 38 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 39 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 40 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 41 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 42 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 43 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 44 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 45 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 46 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 47 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 48 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 49 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 50 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 51 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 52 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 53 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 54 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 55 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 56 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 57 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 58 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 59 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 60 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 61 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 62 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 63 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 64 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 65 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 66 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 67 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 68 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 69 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 70 Year 6 Local Linearity and L'Hopitals.notebook December 04, 2018 71.