Chapter 3



Q101.BC.NOTES: Chapter 2.3, 2.4, 3.1, 3.2

Define [pic]: Definition of [pic]:

PART 1

IS f CONTINUOUS AT x = a?

Definition of a function continuous at x = a:

Theorem:

1. Let [pic] Prove that f is or is not continuous at [pic] Support graphically.

2. Let [pic]. Prove that g is or is not continuous at [pic]Support graphically.

3. Let [pic]. Prove that p is or is not continuous at [pic].

PART 2

Average versus Instantaneous Rate of Change

Discovering the derivative at x = a: Slopes of secants and tangents to a curve

DEF: A function f is differentiable at x = a if exists.

If the limit does not exists then we say that the function is not differentiable at x = a.

DEF (alt): A function f is differentiable at x = a if exists.

If the limit does not exists then we say that the function is not differentiable at x = a.

DEF: The derivative of the function [pic]at the point [pic]is [pic], provided it exists.

DEF (Alt): The derivative of the function [pic]at the point [pic]is [pic], provided it exists.

1. Derivative of f at x = a / Equation of tangent lines.

Consider the function [pic]where[pic]

a. Find the slope of the tangent line to the graph of [pic] at [pic], i.e. the derivative of f at x = 1. (Precede your answer with Lagrange notation).

b. Write an equation of the tangent line [pic]to the graph of [pic]at [pic]

2. Instantaneous rate of change versus average rate of change

Let [pic]. Find the average rate of change from [pic]hrs to [pic]hrs.

Find the instantaneous rate of change at [pic]hrs.

3. Instantaneous rate of change versus average rate of change

Let [pic]. Find the average rate of change from [pic]hrs to [pic]hrs.

Find the instantaneous rate of change at [pic]hrs.

4. Estimating the derivative with average on a small neighborhood

The coordinates f of a moving body for various values of t are given.

t (sec) |0 |0.5 |1.0 |1.5 |2.0 |2.5 |3.0 |3.5 |4.0 | |f (ft) |-12 |-15 |-16 |-15 |-12 |-7 |0 |9 |20 | |

Assuming a smooth curve represents the motion of the body, estimate the velocity at [pic] and [pic]

[pic]

5. The derivative at a breaking point x = a.

Let [pic] Find [pic], if it exists.

PART 3

Is the function differentiable at x = a?

DEF: A function f is differentiable at x = a if [pic]exists.

If the limit does not exists then we say that the function is not differentiable at x = a.

DEF (alt): A function f is differentiable at x = a if [pic]exists.

If the limit does not exists then we say that the function is not differentiable at x = a.

Relating Continuity to Differentiability at x = a.

Theorem:

Notation for a one-sided derivative:

Right-hand derivative at x = a: [pic] , provided it exists

Left-hand derivative at x = b: [pic], provided it exists

Alternatively… [pic] and [pic] provided they exists.

Classification for a function that fails to be differentiable at x = a

A function fails the be differentiable at x = a if the following exist at x = a: a discontinuity, corner, cusp, or a vertical line tangent.

Corner

Cusp

Vertical line tangent

EX1. Prove that g is or is not differentiable at x = 0. [pic] Classify.

EX2: Prove that f is or is not differentiable at x = 1. [pic] Classify.

EX3: Prove that b is or is not differentiable at x = 1. [pic] Classify.

DEF: Differentiable at an endpoint

A function is differentiable at a left endpoint x = a if the respective one-sided derivative exists.

A function is differentiable at a right endpoint x = b if the respective one-sided derivative exists.

Part 4

What is happening as x varies?

DEF: A continuous function is a function that is continuous at each point in its domain.

THM: Polynomial functions, unless restricted or broken, are continuous for all values of x.

DEF: The derivative of the function [pic]with respect to the variable [pic]is the function [pic]whose value at [pic] is [pic], provided it exists. (No alternate definition)

DEF: Differentiable on an interval

A function is differentiable on an open interval (a, b) if [pic]exists for every x in that interval.

A function is differentiable on a closed interval [a, b] if f is differentiable on the open interval (a, b) and if the following limits exist:

[pic] and [pic].

THM: Polynomial functions, unless restricted or broken, are differentiable for all values of x.

1. Find the function [pic] and interpret it.

If [pic], find the function [pic] and interpret it.

2. Graph [pic]from [pic]

The graph of [pic]shown here is made of line segments joined end to end. Graph the function’s derivative.

[pic] [pic]

3. Graph [pic] from [pic]

Sketch a possible graph of a continuous function f that has domain [-3, 3], where [pic]and the equation of [pic]is shown below.

[pic]

[pic]

Deeper Level Thinking Problems

1. Let f be a function that is differentiable throughout its domain and that has the following properties.

(i) [pic]for all real numbers x, y, and [pic] in the domain of f.

(ii) [pic] (iii) [pic]

a) Show that [pic]

b) Use the definition of the derivative to show that [pic]. Indicate where properties (i), (ii), and (iii) are used.

2. Suppose that a function f is differentiable at [pic], [pic], and [pic]. Find [pic].

3. Suppose that f is a differentiable function with the property that [pic] and [pic]. Find [pic]and [pic].

4. Suppose that f has the property [pic] for all values of x and y and that [pic]. Show that f is differentiable and [pic]

LESSON 1 – HW:

1. Prove that [pic]is or is not continuous at [pic].

2. Prove that [pic] is or is not continuous at [pic].

3. Prove that [pic]is or is not continuous at [pic].

4. Prove that [pic] is or is not continuous at [pic].

5. A. Prove that [pic]is not continuous at [pic].

B. Extend g(x), making it a piecewise function that is continuous at[pic].

6. A. Prove that [pic]is not continuous at [pic].

B. Extend d(x), making it a piecewise function that is continuous at[pic].

7. Prove that [pic] is or is not continuous at [pic]

Hint: See the Sandwich Theorem of Limits.

8. If [pic]and [pic], find an equation of (a) the tangent line, and (b) the normal line to the graph of [pic]at the point where [pic].

9. Consider the function [pic]where[pic]

a. Find the average rate of change in f on the interval [-2, 4]

b. Find the slope of the tangent line to the graph of [pic] at [pic], i.e. the derivative of f at x = 1.

c. Write an equation of the tangent line [pic]to the graph of [pic]at [pic]

10. Consider the function [pic]where[pic]

a. Find the average rate of change in f on the interval [-2, 4]

b. Find the slope of the tangent line to the graph of [pic] at [pic], i.e. the derivative of f at x = 0.

c. Write an equation of the tangent line [pic]to the graph of [pic]at [pic]

11. Consider the function [pic].

a. Find [pic]using the standard definition of the derivative at x = a.

b. Find [pic]using the alternate definition of the derivative at x = a.

12. Use the table below to estimate a) [pic] and b) [pic]

t |0.00 |0.56 |0.92 |1.19 |1.30 |1.39 |1.57 |1.74 |1.98 |2.18 |2.41 |2.64 |3.24 | |f(t) |1577 |1512 |1448 |1384 |1319 |1255 |1191 |1126 |1062 |998 |933 |869 |805 | |

LESSON 2 – HW:

1. Prove that [pic]is or is not differentiable at [pic].

Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].

2. Prove that [pic]is or is not differentiable at [pic].

Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].

3. Prove that [pic]is or is not differentiable at [pic].

Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].

4. Prove that [pic]is or is not differentiable at [pic].

Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].

5. Prove that [pic]is or is not differentiable at [pic].

Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].

6. Prove that [pic]is or is not differentiable at [pic].

Classify [pic]as either smooth a corner, a cusp, a vertical line tangent, or discontinuous at [pic].

7. Prove that [pic]is or is not differentiable at [pic].

Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].

8. Find the unique values of a and b that will make g both continuous and differentiable.

[pic]

9. The graph of the function [pic]shown here is made of line segments joined end to end.

Graph [pic]and state its domain.

[pic] [pic]

10. Sketch the graph of a continuous function with domain [-2,2],[pic], and [pic].

[pic]

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