Week 3 of m171
(§ 3, Mr. Lane)
6–10 February 2017

Math Learning Center (in Math 011) has tutors:
10am–4pm Monday–Thursday
and 10am–noon Friday

- Monday (6 February 2017) [d09]
WeBWorK:
RQ-2-2 closes at noon,
day-07 closes at 11 pm
- Study section 2.2 (
*The Derivative at a Point*). - I recommend explorations guided by interactive webpage
**The Derivative at a Point**, it is item 7 in the list at - Work on these items at end of §2.2: 3, 9, 12; 13, 15, 19, 25, 39; 51, 52, 56
- pre-class Learning Objectives:
- Write an expression that computes average rate-of-change for function g on interval [a,b].
- Sketch graphs of three
**different**functions such that- each graph goes through the point (2,3)
- the line y = 5 − x is tangent to each graph at the point (2,3)

- Suppose V(t) is a balloon's volume (cubic meters) at time t (seconds after it starts to be inflated). Interpret V(2) and V′(2) [remember to mention suitable units].

- Learning Objectives for class or further study
- Use the limit definition to compute the derivative of a quadratic polynomial p(x) at some specific point.
- Interpret the results of a derivative calculation in everyday terms.
- Write an equation for the line tangent to graph of g(x) = 2/x at the point where x=1/2.

- Also work on a few more items at end of §2.1: 7, 12–13; 18, 20; 32, 33

- Study section 2.2 (
- Tuesday [d10]
WeBWorK:
RQ-2-3 closes at noon,
day-08 closes at 11 pm
- Study section 2.3 (
*The Derivative Function*).- First paragraph summarizes the objective of §2.3
- Examples 1 & 2 present a graphical interpretation of that objective.
- Example 3 presents a tabular interpretation.
- Examples 4 & 5 produce derivative formulas.

- Work on these items at end of §2.3: 1, 3, 4, 11, 17; 28, 32, 40, 47, 49; 52, 53
- pre-class Learning Objectives:
- Use the limit definition to find a formula for the derivative of
g(x) = 3 − 4 x − 5 x
^{2} - Sketch a graph over the interval [0,10] which is smooth but not strictly increasing on some subinterval. Find where its corresponding derivative function will not have positive values.

- Use the limit definition to find a formula for the derivative of
g(x) = 3 − 4 x − 5 x
- Explore ways to analyze a function by considering its derivative
approximated (or evaluated) at several (or many) places.
- www.geogebra.org/m/KHauu78A (Derivative Builder)
- webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_as_a_function.html
Each red dot has coordinates (a,f′(a));
enabling
*Trace*lets several be seen at once - Choose a value for H which is close to zero, then plot the expression (f(x+H) − f(x))/H

- Learning Objectives for class or further study:
- Sketch graphs of three different functions such that each has the same derivative function: g′(x) = 1/2 for all values of x.
- Given a smooth graph of function g, sketch the graph of its derivative function.
- Investigate how shape of graph for function g affects shape of graph for its derivative function g′

- Also work on a few more items at end of §2.2: 5, 10, 11; 18, 21, 32; 57, 58
- Anticipate a quiz.

- Study section 2.3 (
- Wednesday [d11]
WeBWorK:
RQ-2-4 closes at noon,
day-09 closes at 11 pm
- Study section 2.4 (
*Interpretations of the Derivative*). - Work on these items at end of §2.4: 1, 4, 5, 7; 13, 15, 19, 20, 23, 29; 35, 37, 39
- pre-class Learning Objectives:
- Suppose function g is used in a context where units are known for its input (t, hours after an oil spill) and its output (v, gallons of oil pollution). Identify suitable units for input and output of its derivative function g′.
- Suppose the only information about a function is given in a table. Construct a corresponding table of estimates for derivative values.

- Learning Objectives for class or further study:
- Suppose the values of g(a) and g′(a) are known. Also suppose b is some number close to a. Compute a reasonable estimate for g(b).
- Suppose you are given a graph of a derivative function, g′(x). Sketch a plausible graph of a corresponding original function, g(x).
- Estimate the derivative of a function at a point in several ways: forward, backward, and centered difference quotients.

- Study section 2.4 (
- Friday (10 Feb 2017) [d12]
WeBWorK:
RQ-2-5 closes at noon,
day-10 closes Saturday at 11 pm
- Write-up solutions of these four problems —
show your work and interpret your results. Hand-in your written
homework at the
**start**of Friday's class.**section**2.1 2.2 2.3 2.4 **problems**24 36 48 24 - Study section 2.5 (
*The Second Derivative*). - Work on these items at end of §2.5:
1–4,
**5**, 7;**15**, 20, 21,**25**, 28; 33, 34, 37, 39, 41 - pre-class Learning Objectives:
- If g(t) is position of an object at time t, then interpret g′′(t) in terms of the object's motion. Hint: after interpreting g′(t), consider how to interpret changes in the values of g′(t).
- Interpret second derivative in terms of concavity for graph.

- Learning Objectives for class or further study:
- In the neighborhood of an inflection point, how is/are the tangent line(s) related to the function's graph?
- Suppose f and g are functions such that f′(x) = 1 and g′′(x) = 1 [for all x]. In what way(s) could graphs of f and g have different shapes?

- Also work on a few more items at end of §2.4: 3, 9; 14, 27; 34, 42

- Write-up solutions of these four problems —
show your work and interpret your results. Hand-in your written
homework at the

© Richard B Lane
Last modified: 3 February 2017, Friday 07:32