Calculus I (MAT 145)Dr. Day Wednesday Jan 28,2015

Questions about Limits or Continuity?

Tangent Lines and Derivatives (2.7 & 2.8)

Assignments

Wednesday, January 28, 2015

MAT 145

MAT 145Wednesday, January 28, 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 145Wednesday, January 28, 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 145Wednesday, January 28, 2015

MAT 145Wednesday, January 28, 2015

A

B

C

D

E

MAT 145Wednesday, January 28, 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 145Wednesday, January 28, 2015

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

MAT 145Wednesday, January 28, 2015

We call this slope calculation the

derivative of f at x = a.

MAT 145Wednesday, January 28, 2015

MAT 145Wednesday, January 28, 2015

MAT 145Wednesday, January 28, 2015

MAT 145Wednesday, January 28, 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 145Wednesday, January 28, 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 145Wednesday, January 28, 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 145Wednesday, January 28, 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 145Wednesday, January 28, 2015

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

x value where a derivative exists?

MAT 145Wednesday, January 28, 2015

MAT 145Wednesday, January 28, 2015

MAT 145Wednesday, January 28, 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 145Wednesday, January 28, 2015

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

MAT 145Wednesday, January 28, 2015

MAT 145

Assignments WebAssign : Sec 2.5: due tonight! Quiz #4: WebAssign

Available NOW! Due by midnight Tuesday, January 27.

Tangent return . . .

Wrapping It Up

Wednesday, January 28, 2015