monday, september 7, 2015mat 145. monday, september 7, 2015mat 145
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Calculus I (MAT 145)Dr. Day Monday Sept 7, 2015
Limits Involving Infinity (2.6)
Continuity (2.5)
Assignments
Labor Day Holiday: No Classes!
Monday, September 7, 2015
MAT 145
MAT 145
Transitioning from . . .
Approachment
Monday, September 7, 2015
As the values of some set of inputs tend to get closer and closer to a number, the corresponding list of outputs get closer and closer to some number.… to …
an equivalent
mathematical representation
MAT 145Monday, September 7, 2015
This says that the values of f(x) tend to
get closer and closer to the number L as
x gets closer and closer to the number a
(from either side of a) but x ≠ a.
The concept of . . . LIMIT!
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
The line y = L is called a
horizontal asymptoteof the curve y = f(x) if either:
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
Sketch the graph of by finding its intercepts and its limits as and as .
MAT 145
What Makes a Function Continuous? (PIP pp 13-14)
Informal Perspective
Formal Definition
Types of Discontinuities
Continuity on an Interval
Connections: Limits and Continuity
Monday, September 7, 2015
MAT 145Monday, September 7, 2015
Here is a graph of a function f. At which values is f discontinuous? Why?
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
Removable DiscontinuityInfiniteDiscontinuity
JumpDiscontinuity
Removable Discontinuity
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
The following types of functions are continuous at every number in their domains.PolynomialsRational functionsRoot functionsTrigonometric functions Inverse trigonometric functionsExponential functions Logarithmic functions
MAT 145Monday, September 7, 2015
The Intermediate Value Theorem
Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) ≠ f(b).
Then, there exists a number c in (a, b) such that f(c) = N.
“A continuous function takes on every intermediate value between the function values f(a) and f(b).”
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
Match the expression in the left column (A thru F) with its correct description in the right column (1 thru 6). Explain your determination of each correct match!
MAT 145Monday, September 7, 2015
Match the expression in the left column (A thru F) with its correct description in the right column (1 thru 6). Explain your determination of each correct match!
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
A
B
C
D
E
MAT 145Monday, September 7, 2015
Calculate the slope at x = −2.
Calculate the slope at x = −1.
Calculate the slope at x = 0.
Calculate the slope at x = a.
MAT 145Monday, September 7, 2015
Calculate the slope of f(x) = x2 at x = a.
MAT 145Monday, September 7, 2015
We call this slope calculation the
derivative of f at x = a.
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
The value f ’(a) is called:• the derivative of f at x =
a,• the instantaneous rate
of change of f at x = a, • the slope of f at x = a,
and• the slope of the
tangent line to f at x = a.
MAT 145Monday, September 7, 2015
The derivative in action!
S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100?
MAT 145Monday, September 7, 2015
The derivative in action!
C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5?
MAT 145Monday, September 7, 2015
The derivative in action!
V(r) represents the volume of a sphere, where r is the radius of the sphere in cm. What is the meaning of V ’(3)=36π?
MAT 145Monday, September 7, 2015
Can we create a derivative function f that will be true for any
x value where a derivative exists?
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to h. As h approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true derivative of f at x is the limit of the value of the slope function as the secant lines get closer and closer to being a tangent line.
MAT 145Monday, September 7, 2015
Calculate the derivative function, f ’(x), for f(x) = x2. Use the limit definition of the derivative.
MAT 145Monday, September 7, 2015
MAT 145
Assignments WebAssign : Chapter 2 (limits) due throughout this
week WebAssign Quiz #4 : Open now, due Tuesday at
midnight, two submissions
Wrapping It Up
Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
MAT 145Monday, September 7, 2015
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