State Standard – 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity

State Standard 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity.Objective To be able to find the limit of a function.

Definition:

And say the limit of f(x), as x approaches a, equals L

This says that the values of f(x) get closer and closer to the number L as x gets closer to the number a (from either side)

Example 1 Find the Limit of f(x) for these three cases:222

Example 2 Find the Limit of f(x) for these four cases:222DNE

Example 3 Find the Limit of f(x) for these four cases:13DNE1.5

Example 4 Find the Limit of f(x) for these three cases:- + DNE

Example 5Find the Limit of f(x) for these three cases:+ + +

Example 6 Find the Limit of f(x) for these two cases:-14

Example 7 Find the Limit of f(x) for these two cases:+ -2

WS on Limits

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State Standard 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity.Objective To be able to find the limit of a function.

Example 1 + + 0+ DNE-1

Example 2 2 12

Example 3 00DNE00DNE

DNE0+ DNE0

State Standard 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity.Objective To be able to find the limit of a function.

Example 1Use a table of values to estimate the value of the limit. 0.750.900.99?0.9991.0011.011.11.25x approaches 1 from the LEFTx approaches 1 From the RIGHT= 1

Example 2Use a table of values to estimate the value of the limit. 100 4001x106?1x106 400100x approaches + from the LEFTx approaches + From the RIGHT= +

Example 3Use a table of values to estimate the value of the limit. 63.2063.9263.998?64.00164.2465x approaches 64 from the LEFTx approaches 64 From the RIGHT= 64

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Use a table of values to estimate the value of the limit. -9-19-999?1001 216x approaches from the LEFTx approaches + From the RIGHT= DNE

State Standard 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity.Objective To be able to find the limit of a function.

Example for case 1:Ex for case 2:= 16= 8= 2= -4Ex for case 3:= 32= 9

Limit Laws Suppose that c is a constant:and

Example 1aFind the Limit:3(4) 10 + 4= 6

Example 1bFind the Limit:

Example 2Find the Limit:

Pg. 111-1121 9

Evaluate the limit. = 3/2= 9

State Standard 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity.Objective To be able to find the limit of a function.

Example 1Find:Functions with the Direct Substitution Property are called continuous at a. However, not all limits can be evaluated by direct substitution, as the following example shows:= 1+1= 2

Example 2Find the Limit:

Example 3Find the Limit:

Example 4Find the Limit:

Example 5Find the Limit:

Pg. 11211 29 odd

Evaluate the limit. = DNE = 16

State Standard 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity.Objective To be able to find the limit of a function.

If a function f is not continuous at a point c, we say that f is discontinuous at c or c is a point of discontinuity of f.

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.A function is continuous at a point if the limit is the same as the value of the function.This function has discontinuities at x=1 and x=2.It is continuous at x=0, x=3, and x=4, because the one-sided limits match the value of the function

jumpinfiniteoscillatingEssential Discontinuities:Removable Discontinuities:(You can fill the hole.)

Removing a discontinuity:

Removing a discontinuity:

ExampleFind the value of x which f is not continuous, which of the discontinuities are removable?Removable discontinuity is at:Where as x 1 is NOT a removable discontinuity.

Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.

WS 1 10, 13 17 oddPg. 1331-6, 10-12, and 15 20

Describe the continuity of the graph.

State Standard 1.0 Students demonstrate knowledge of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity.Objective To be able to find the limit of a function.

Definition: The line y = L is called a horizontal asymptote of the curve y = f(x) if either or

Example 1Case 1: Numerator and Denominator of Same Degree

Divide numerator and denominator by x2

Example 2Case 2: Degree of Numerator Less than Degree of Denominator

Divide numerator and denominator by x3

Example 3Case 3: Degree of Numerator Greater Than Degree of Denominator

Divide numerator and denominator by x

Example 4a)b)

Example 5Find the Limit:

Example 6Find the Limit:

WS 1 8andPg. 14711 18, and 20 22

Solve and show work!

State Standard 4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function. Objective To be able to find the tangent line.

Definition of a Tangent Line:

PQTangent Line

Slope:

P(a,f(a))Q (x,f(x))x a f(x) f(a)

Definition The tangent line to the curve y = f(x) at the point P(a,f(a)) is the line through P with the slope:

Provided that this limit exists.

Example 1Find an equation of the tangent line to the parabola y=x2 at the point (2,4).Use Point Slope y y1 = m (x x1)y 4 = 4(x 2) y 4 = 4x 8 +4+4y = 4x 4

Provided that this limit exists.For many purposes it is desirable to rewrite this expression in an alternative form by letting:h = x aThen x = a + h

y y1 = m (x x1)y 1 = -1/3(x 3) y 1 = -1/3x + 1 +1+1y = -1/3x + 2

Example 3Find an equation of the tangent line to the parabola y = x2 at the point (3,9).y y1 = m (x x1)y 9 = 6(x 3) y 9 = 6x 18 +9+9y = 6x 9

Example 4Find an equation of the tangent line to the parabola y = x24 at the point (1,-3).y y1 = m (x x1)y -3 = 2(x 1) y + 3 = 2x 2 -3-3y = 2x 5

Pg. 1565a, 5b, 6a, 6b, 7 10, 11a, 12a, 13b, and 14b

State Standard 4.0 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function. Objective To be able to find the derivative of a function.

is called the derivative of at .The derivative of f with respect to a is

f prime xorthe derivative of f with respect to xy primedee why dee ecksorthe derivative of y with respect to xdee eff dee ecksorthe derivative of f with respect to xdee dee ecks of eff of ecksorthe derivative of f of x

Example 1Find the derivative of the function f(x) = x2 8x + 9 at the number a.

Example 2Find the derivative of the function (x 9) (x 9) (x +h 9) (x +h 9)

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