Math 131 E Lab 4: DERIVATIVE - MEASUREMENT OF CHANGE
Purpose
The purpose of today's lab is to demonstrate graphically:
1) that the derivative is the limit of a difference quotient,
2) that the derivative measures the instantaneous rate of change of a function and
3) that the derivative measures the slope of the tangent line to a function.
Applications
For the following functions:
i. sketch the graph;
ii. complete the tables;
iii. guess the value of f ¢(c);
iv.draw at least four different secant lines through (c,f(c));
v. sketch the tangent line if possible otherwise explain;
Use different colors for the graph, secant lines, & the tangent line.
vi. answer the questions -
Is f differentiable at c? ______If so, what is f¢(c)? ______
Is f continuous at c? ______
Group 1 f(x) = sin x c = 1
Group 2 c = 3 c = -1
9. Based on your experimentation, can you establish some criteria so that you can look at a graph and determine where the derivative does and does not exist? Perhaps further experimentation would help.
10. What does your experimentation tell you about the relationship between differentiability and continuity?
Assessment
1. What is the best measurement, if it exists, of how a function is changing at a point?
2. What is the relationship between the difference quotient and the derivative?
3. What is the relationship between secant lines and the tangent line to a function at a point?
4. What is the relationship between f ¢(c) and the line tangent to f at c?
5. What can you say about f ¢(c) when f has a point discontinuity atc?
6. What can you say about f ¢(c) when f has a jump discontinuity at c?
7. What can you say about f ¢(c) if f has a point(jump) discontinuity at a where a < c < b?
8. If the tangent line is vertical at c, what do you know about f ¢(c)?
Part 2: THE DERIVATIVE AS A FUNCTION
Purpose
The purpose of this lab is to recognize a derivative as a function and to explore the relationship between a function and its derivative.
Consider the graph in #8 of page 83 of your text. Sketch tangent lines to approximate the following.
Carefully sketch tangent lines on f when x = -2, -1, 0, 1, 2, 3, 3.5. With a ruler, estimate the slope of each tangent line to determine the following:
f ¢(-2) = ______f¢ ¢(-1) = ______f ¢(0) = ______f ¢(1) = ______
f ¢(2) = ______f ¢(3) = ______f ¢(3.5) = ______
Use the above information and the graph of f(x) to sketch f¢ ¢(x).
Assessment
1. Given that f ¢(-4) = 5 and f ¢(3) = -8,
is f steeper when x = -4 or when x = 3? ______
The slope of the line tangent to f when x = -4 is ______.
2. Describe the relationship between f ¢and the steepness of f.
3. Describe the behavior of f when f ¢ is close to zero.
4. What, if anything, does the sign of f ¢(x) tell you about f(x)?
5. What, if anything, does the sign of f(x) tell you about f ¢(x)?
6. What, if anything, does the sign of f ²(x) tell you about f¢(x)?
7. For each of i-iv, sketch a function, f, on [a, b] which has the indicated properties:
i) f(x) > 0 and f¢(x) > 0 ii) f(x) < 0 and f¢(x) > 0
iii) f(x) > 0 and f¢(x) < 0 iv) f(x) < 0 and f¢(x) < 0