Friday, September 18, 2015MAT 145. Friday, September 18, 2015MAT 145.

Calculus I (MAT 145)Dr. Day Friday Sept 18, 2015

The Tangent-Line Slope Function: Also Known as The Derivative (2.7 and 2.8)

Finish the small-group secant-line and tangent-line slope tasks: Graphs 1 through 8.

Complete the contextual problems on the gold sheet: 8 scenarios

WA 2.7 due tonight; 2.8 due Monday night.

Friday, September 18, 2015

MAT 145

MAT 145Friday, September 18, 2015

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


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


Friday, September 18, 2015MAT 145. Friday, September 18, 2015MAT 145.

Documents

Transcript of Friday, September 18, 2015MAT 145. Friday, September 18, 2015MAT 145.