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Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 78:35 AM
Some Limit Rules Before Beginning Trig Limits:
1. The limit of a constant is the constant
2. The limit of sum/difference is the sum/difference of the limits.
Oct 78:53 AM
4. The limit of product is the product of the limits.
5. The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not 0.
3. The limit of a constant times a function is the constant times the limit of the function.
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 51:53 PM
Limits of Trigonometric Functions
Ex 1: Evaluate
a. b.
Ex 2: Are there any values of c for which ordoes not exist? Explain.
Before we begin, please make sure your calculator is in radian mode.
The Squeeze TheoremIf two functions squeeze together at a particular point, then any function trapped between them will get squeezed to that same point.
The Squeeze Theorem deals with limit values, rather than function values.
In the graph below, the upper and lower functions have the same limit value at x = a. The middle function has the same limit value because it is trapped between the two outer functions.
Suppose f(x) < g(x) < h(x) for all x in an open interval about a (except possibly at a itself). Further suppose
Then,
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
https://www.khanacademy.org/math/apcalculusab/ablimitsnew/ab18/v/squeezesandwichtheorem
Ex 3:
Oct 52:04 PM
Ex 4: Evaluate
a. b.
These two limits will be used in conjunction with algebraic manipulation to evaluate all other trigonometric limits of the form 0/0.
Hint: Graph it :)
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 711:31 AM
Ex 5:
Ex 6:
Ex 7:
Compute all the following limits without a calculator. (handout)
Oct 711:44 AM
Ex 8:
Ex 9:
Ex 10:
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 711:46 AM
Ex 11:
Ex 12:
Ex 13:
Oct 711:48 AM
Ex 14:
Ex 15:
Ex 16:Homework: Limits of Trig Functions Handout
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 52:08 PM
What is a derivative?https://www.khanacademy.org/math/apcalculusab/abdifferentiation1new/ab21/v/derivativeasaconcept
Oct 52:01 PM
A derivative is the instantaneous rate of change of a function at a specific value of x.Graphically, it is the slope of the tangent line to a curve at a specific value of x.We can represent this algebraically using limits:
Note two different types of notation to represent the derivative. Why do you think this is so?
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 98:29 AM
Ex 1: Find the derivative of y = 2x + 3 using the limit definition.
Oct 1010:55 AM
Ex 2: Find f '(x) for f(x) = x2 + 2x using the limit definition.
Ex 3: Write the equation of the line tangent to f(x) = x2 + 2x at x = 4.
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 112:32 PM
Ex 4: Find dy/dx for y = 1/x.
Ex 5: Write the equation of the tangent line to f(x) = 1/x at x = 2.
Oct 101:37 PM
The definition of the derivative that we have used thus far produces a derivative function that has to then be evaluated at a specific value of x. If we know this specific value in advance, then we can evaluate a less complicated limit that will produce a numeric value for the derivative immediately.
The Derivative at a Point
To find the derivative of f(x) at x = c, evaluate:
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 118:30 AM
Ex 1: Find the derivative of f(x) = x2 3x 4 when x = 5.
Oct 118:33 AM
Ex 2: Write the equation for the tangent line to f(x) = x2 + 2x at x = 3.
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 112:36 PM
Ex 3: Find the derivative of f(x) = √x + 1 when x = 3.
Oct 51:33 PM
Limit Definition of Continuity
A function is said to be CONTINUOUS at x = c if 1. the twosided limit at x = c exists2. the function value at x = c is defined3. and these two values are equivalent.
Graphically, when is a function NOT continuous?
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 51:45 PM
We can see if a function is continuous from the graph, but how do we determine if a function is continuous from the equation?
Ex 1: Is g(x) continuous when x = 1?
Ex 2: Is g(x) continuous when x = 1?
Oct 51:57 PM
Ex 3: For what value of k will g(x) be continuous at x = 2?
Homework: p64 Q1 Q10p74 T2 T5
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 267:18 AM
Ex 4: Is f(x) continuous when x = 3 and x = 3?
Oct 267:22 AM
Ex 5: For what value (s) of A will f(x) be continuous at x = 3?
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 118:44 AM
DifferentiabilityIn order for a function to be differentiable at a specific value of x, the function must be continuous there and the limit which defines the derivative must exist.
Geometrically, a function f is differentiable at x0 if the graph of f has a tangent line at x0. Thus f is not differentiable at any point x0 where the secant lines do not approach a unique nonvertical limiting position as x approaches x0. These cases can be described informally as corner points (cusps) and points of vertical tangency.
What is a cusp?
Oct 118:54 AM
Trig Limits, Continuity, and Differentiability.notebook
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October 03, 2019
Oct 167:06 AM
Determine if the following function is continuous at x = 2:
Is f(x) differentiable at x = 2? Explain.