Monday, September 7, 2015MAT 145. Monday, September 7, 2015MAT 145

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


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!








The line y = L is called a

horizontal asymptoteof the curve y = f(x) if either:






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



Here is a graph of a function f. At which values is f discontinuous? Why?



Removable DiscontinuityInfiniteDiscontinuity

JumpDiscontinuity

Removable Discontinuity



The following types of functions are continuous at every number in their domains.PolynomialsRational functionsRoot functionsTrigonometric functions Inverse trigonometric functionsExponential functions Logarithmic functions


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).”




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!


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!



A

B

C

D

E


Calculate the slope at x = −2.

Calculate the slope at x = −1.

Calculate the slope at x = 0.

Calculate the slope at x = a.


Calculate the slope of f(x) = x2 at x = a.


We call this slope calculation the

derivative of f at x = a.





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.


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?



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?



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?


Can we create a derivative function f that will be true for any

x value where a derivative exists?




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.


Calculate the derivative function, f ’(x), for f(x) = x2. Use the limit definition of the derivative.


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






Documents

Monday, September 7, 2015MAT 145. Monday, September 7, 2015MAT 145