Chapter 3 Differentiation and Chapter 4 Applications of Derivatives Sec 4.1 – 4.2
Chapter 3 Differentiation
Section
2.1
3.1 &
3.2
Concepts
Rates of Change and Tangents to Curves
Representative
homework problems
See Ch 2 Study Guide
Tangents and the Derivative at a Point
State the definition of derivative of a function y = f(x) at a specific point x = c (as a limit of difference
quotients).
o Illustrate and explain each part of the definition graphically (using secant lines, tangent lines, etc.).
o Explain why the definition requires taking a limit and what the limit means in this setting.
o Comment: Section 2.1 introduces the problem of finding instantaneous rate of change and slope of a tangent line.
These two problems lead to the definition of the derivative. Review section 2.1.
§
The definition of the derivative of a function at a point is on page 103.
§
The formula for the slope of a curve on page 102 also defines the derivative at a specific point.
§
A good graphical interpretation of this definition is found in Figure 3.1 in the margin on page 102. The
derivative of a function at a point is also the subject of the third calculator lab.
Written Homework
3.2: The Derivative as a Function
Explain how you can tell from its graph whether a function is differentiable at a point. Given the graph of a
function, indicate (or list) the points (x-values) where the function is differentiable and where it is not
differentiable.
Estimate the derivative at points where the function is differentiable from the graph.
Pg 108; #1 – 4.
Pgs 116 – 117; #37 – 48.
The derivative of a function y = f(x) is another function f′(x) called the derivative (or the derived function).
o Explain in terms of the graph of y = f(x) what the derivative f′(x) tells you.
o Explain in terms of physical quantities (e.g. time and position, altitude and air pressure, or something else) what
the derived function f′(x) tells you.
o Given the graph of a function y = f(x), sketch the graph of the derivative f′(x).
Pg 115; #27 – 30, 31
Pg 116; 33, 34.
Pg 179, #89, 90.
Use a graph to explain why a function that is not continuous at a point x = c must also not be differentiable at
the point x = c. (Therefore, in order for a function to be differentiable at a point, it must be continuous there.)
o Give examples, by graphs and/or equations, of functions that are continuous but not differentiable at a point.
o Use the definition of a function being differentiable at a point x = c and the definition of continuity at a point to
show that a function that has a derivative at a point x = c must also be continuous at that point (see Theorem 1,
pg 111).
Use the definition of the derivative of a function at a specific point x = c to determine whether a function given
by a specific formula (perhaps defined piecewise) is or is not differentiable at a given point x = c.
o Explain how to see from the definition that the function is or is not differentiable at x = c. (The language of onesided derivatives can be useful.) Confirm your conclusion by examining the graph of the function.
o Use the definition of the derivative to determine:
(a) whether the function g(x) = x|x| is differentiable at x = 0;
(b) whether the function h(x) = 2 + | x −1| is differentiable at x = 1.
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Pg 117; #43 – 48.
Pg 109; #35, 36.
Pg 117; #41, 42, 58.
Pg 178; #65 – 67.
3.3: Differentiation Rules
MATH 160 Study Guide: Chptr. 3 Differentiation & Chptr. 4 Applications of Derivatives Sec 4.1 – 4.2
Write complete statements (as a theorem with hypotheses and conclusion) of the differentiation formulas
for the:
o -sum,
o -difference, and
o -product of two or more differentiable functions, and
o -for the quotient of two differentiable functions.
o Give examples to show that if the hypotheses of the theorem (the “if” part) are not satisfied the
conclusion might or might not be true.
Use the differentiation formulas for sums, products, and quotients to find the value of the derivative of a
combination of functions at a given point (x-value) from information about the values of the functions and
their first derivatives at that point.
Pg 125; #41, 42.
Pg 178; #55, 56.
Use the differentiation formulas to calculate first and second derivatives of functions defined by
expressions that involve constant multiples, sums, differences, products, and/or quotients of power and
root functions.
Pg 125; #1 – 40.
Pg 177; #1 – 4, 6, 9, 10.
3.4: The Derivative as a Rate of Change
Find an equation for the line tangent to the graph of a differentiable function at a given point. Find the
points on the graph of a differentiable function where the tangent line has specified slope.
3.5: Derivatives of Trigonometric Functions
If possible, give 2 different
examples of a pair of functions
f(x) and g(x) with the
properties that f(x) is
differentiable at x = 0, g(x) is
not differentiable at x = 0, and
p(x) = f(x) g(x) is differentiable
at x = 0.
o If it is impossible to give 2
different pairs of examples,
explain how you know.
o Create four more questions
by replacing f(x) g(x) by (i)
f(x) +g(x) and by (ii) f(x)\g(x).
o
o
Given an expression for a function that describes the motion of a body along a straight-line path, find:
§
the displacement over a time interval
§
the distance travelled over a time interval, and
§
the average velocity of the body over a time interval.
§
Explain the difference between displacement and distance travelled.
Given a function that describes the motion of a body along a straight-line path, find functions that
describe the velocity and acceleration of the moving body.
§
Then find the position, the velocity and speed (not the same as velocity), and the acceleration of the
body at a given time.
§
Use these results to answer questions about the position and motion of the body.
Pg 108; #23 – 26.
Pg 125; #43 – 46.
Pg 126; #49 – 52.
Pgs 178 – 179; #69 – 73.
Pgs 133 – 134; #1 – 10.
Pg 137; #33 – 36.
Pg 183; #10 – 12, 14.
From the graph of a function that describes the motion of a body along a straight line path, make
reasonable estimates of the position and velocity of the body at a given time and determine whether the
velocity of the body is increasing or decreasing at that time. Explain the basis for your estimates.
Pgs 135 – 136; #17, 18, 20, 21,
22.
Given a function y = f(x) that models the a physical situation (e.g. volume as a function of time, volume
as a function of radius) find the (instantaneous) rate of change of the dependent variable with respect to
the independent variable.
o _ Use this result to analyze and answer questions about how the dependent variable changes as the
independent variable changes.
Pgs 108 – 109; #29 – 34.
Pgs 136 – 137; #25, 27, 28,
30- 30.
Prerequisite facts from trigonometry:
o Sketch the graphs of the six basic trig functions without a calculator.
o Express the sine, cosine, and tangent of angles in terms of the lengths of the sides of a right triangle.
o Express the sine, cosine, and tangent of angles in terms the coordinates of a point on the unit circle.
o Know the identities that allow one to represent all six trig functions in terms of the sine and cosine, the
Pythagorean identity, and the double angle formula for the sine.
o Obtain the double angle formula for the cosine by differentiating the double angle formula for the sine.
Review Sec 1.3
Trigonometric Functions
Use the graphs of the sine and cosine functions to explain how one might conjecture the formulas for the
derivatives of the sine and cosine functions.
o Compute the derivatives of the tangent, cotangent, secant and cosecant using the quotient rule and the
derivatives of sine and cosine.
Calculate derivatives of functions formed from trigonometric functions and power functions (with positive
or negative rational exponents) by adding, subtracting, multiplying, and dividing.
o Find equations for tangent lines to graphs of these functions at given points.
o Find the points on the graph of a differentiable function where the tangent line has specified slope.
Illustrate/verify graphically that your tangent lines are correct (using a calculator).
Pg 141; #1 – 34.
Pgs148 – 149; select problems
w/ trig functions #9 – 58.
Pgs 177 – 178; select
problems w/ trig functions #1 –
40.
Suggested practice for finding
tangent lines:
Pgs 141 – 142; #35 – 45.
Pgs 178 – 179; #69 – 79.
3.7: Implicit Differentiation
3.6: The Chain Rule
MATH 160 Study Guide: Chptr. 3 Differentiation & Chptr. 4 Applications of Derivatives Sec 4.1 – 4.2
Given a function, use the mathematical definitions to determine whether the function is continuous and
whether it is differentiable at specified points.
o Given a function that may be defined piecewise by expressions involving parameters, determine whether the
parameters may be assigned values so as to make the function be continuous and/or differentiable at
specified points.
o If the function has a removable discontinuity at a point x = c, determine what value should be assigned to f(c)
so as to make the function be continuous at the point x = c. Determine whether the resulting function is
differentiable. If it is differentiable, use the definition (of the derivative as a limit) to find its derivative.
Pg 126; #57, 58.
Pg 142; #57, 58.
Pg 178; #65 – 68.
Pg 183; #15 – 18.
Given two functions f and g (that may be the same function) form the composite functions fog and gof.
o Given a function h, write it as a non-trivial composition such as fog (perhaps in more than one way).
(Non-trivial means neither f nor g is the identity function y = x.)
Pg 148; #9 – 18
Write each function explicitly as
a composition.
Write a complete statement of the Chain Rule (as a theorem with hypothesis and conclusion).
Use the Chain Rule, perhaps in combination with other differentiation formulas, to calculate derivatives of
functions formed by adding, subtracting, multiplying, dividing, and composing constants, power functions
(with positive or negative rational exponents), and trigonometric functions.
Given information about the values of two functions f and g and their derivatives at certain points, find the
value of the derivative of functions formed from f, g, and/or additional specific functions by using algebraic
operations and/or composition. (Functions may not be given explicitly, but information about function values
and values of the derivative at specific points is given or can be found from graphs of the functions.)
Given an equation F(x, y) = c (c a constant), the graph of this equation, and a point (xo, yo) on the graph,
See discussion and examples on pages 151 – 152.
Find the derivative of the function y = f(x) defined in (a) by implicit differentiation; and
Find the equation for the tangent line at a given point.
Explain why the formula for the derivative of the function y = f(x) obtained by implicit differentiation gives the
slope of the tangent line at each point on the graph of F(x, y) = c (not just at the point (xo, yo) used to
define the function y = f(x).)
o
o
o
o
Given an equation F(x, y) = c (c a constant) that can be solved explicitly for y and given a point (xo, yo)
on the graph of F(x, y) = c ,
(a) find an explicit expression for a differentiable function y = f(x) that is defined implicitly by F(x, y) = c
and whose graph includes the point (xo, yo) ;
(b) compute the derivative of the function y = f(x) explicitly (using the expression found in (a));
(c) compute the derivative of the function y = f(x) by implicit differentiation; and
(d) demonstrate that the results of (b) and (c) are the same.
4.1 Extreme Values of Functions
Calculate first and second derivatives by implicit differentiation.
Write complete definitions of the following words and phrases:
o Identify and label all absolute extrema, relative extrema, and critical points.
o Given a sketch of the graph of a function, identify and label all of the above. (Notice that whether a function
has absolute and local extrema depends on the domain of the function as well as the equation/formula for the
function.)
Pgs 148-149; select from #1 –
64.
Pgs 177 – 178 Practice
Exercises; select from #7 – 40.
Pg 149; #71 – 74.
Pg 178; #55, 56.
Study recommendation: Some
problems in the written
homework asked you to do this!
Pg 155-156; #29 – 38 and 41
– 44 (perhaps with extended
instructions)
Study Suggestion: Study
Examples 1 & 2 on
151 – 152.
Pg 155; #1 – 18, 19 – 26.
Pgs 190 - 191; #1 – 10.
Pg 192; #77 – 80.
Write the complete statement of the Extreme Value Theorem (as a theorem with hypotheses and conclusion).
o Illustrate the Extreme Value Theorem graphically.
o Give graphical examples to show that when any of the hypotheses of the Extreme Value Theorem are not
satisfied, the function may or may not have an absolute maximum or minimum.
Pg 192; #15 – 20.
Pg 243; #1 – 3, #5 – 8.
Pg 245; # 1 – 3.
Study Suggestion: Study
Theorem 1 in Sec. 4.1.
Write a complete statement of the First Derivative Theorem for Local Extreme Values (Theorem 2, pg 188).
o Use a graph to explain in terms of secant and tangent lines how to see that this theorem is plausible
(short of a formal proof).
o Give examples to show that a function might or might not to have a local extremum at a point where its
derivative is 0.
Study Suggestion: Study
Theorem 2 and the discussion
of Finding Extrema in Sec.
4.1.
Explain the connection between critical points and local extrema of a function.
o Find the critical points, domain endpoints, and absolute and local extrema of a given function (without using a
calculator).
Pg 191; #11 – 14.
Pg 192; 49 – 66
Sec. 4.2 The Mean Value Theorem
MATH 160 Study Guide: Chptr. 3 Differentiation & Chptr. 4 Applications of Derivatives Sec 4.1 – 4.2
Describe a step-by-step procedure for finding the absolute extrema of a continuous function on a closed
interval of finite length (page 189).
o Use this procedure to find the absolute maximum and minimum values of a given function and the points
where these extreme values are attained.
Pg 191; select from #15 – 40.
Give examples to show that a function can fail to have a local extremum at a critical point and at an end
point of its domain.
Pg 192; #67 – 68.
Write the complete statement of the Mean Value Theorem (as a theorem with hypotheses and conclusion).
o Explain what the Mean Value Theorem says graphically by sketching the graph of a function and the relevant
secant and tangent lines, and then explaining how these lines are related, and
o Explain what the Mean Value Theorem says physically by interpreting the hypotheses and the conclusion of
the theorem in terms of an object moving along a straight-line path.
Pg 197; #1 – 6 (think
graphically).
Pg 198; #49 – 53 (think
physically).
Pg 199; #60 (think about h ( x)
= f(x) – g ( x)), #61 – 63, #67.
Study Suggestion:
Pgs 193 – 195; Graphical
interpretation
Pg 195; Physical interpretation
Explain the connection between Rolle’s Theorem and the Mean Value Theorem.
Given the graph of a function over a closed interval, determine whether the Mean Value Theorem is
applicable or not. In both cases, EITHER locate the point or points that satisfy the conclusion of the Mean
Value Theorem (and explain how to see that the conclusion is satisfied at these points) OR explain how to
see that there are no such points.
Given a function defined on a given closed interval [a, b] by an expression y = f ( x ), determine whether the
Mean Value Theorem is applicable or not. In both cases, use calculus (and algebra) to EITHER locate the point
or points that satisfy the conclusion of the Mean Value Theorem OR to explain how to see that there are no such
points.
Apply the fact that two functions that have the same derivative must differ by a constant, your knowledge
of the basic differentiation formulas, and your skill with algebra and trigonometry to:
o find all functions that have a given polynomial, power function, or simple trigonometric function as first
derivative,
o show that two seemingly different functions differ only by a constant.
Study Suggestion: This is just
the graphical interpretation of
the Mean Value Theorem in
the other direction. Read Sec.
4.2.
Pg 197; #7 – 13 (If there is a
point c that satisfies the
conclusion of the Mean Value
Theorem, find it).
For Mathophiles:
Pg 199; #55, 56, 64.
Pg 198; #27 – 40.
Pg 199: #57.
Pg 243; #17, 18.