Math 160 Study Guide for Chapter 3 Differentiation

This Study Guide describes everything you are expected to know, understand, and be able to do from Chapter 3.
Questions on Exam 2 covering this material will ask you to do one or more of the tasks described in this Study
Guide.

__ • State the definition of derivative of a function y = f(x) at a specific point x = c (as a limit).
Illustrate and explain each part of the definition graphically (using secant lines, tangent lines, etc.).
Explain why the definition requires taking a limit and what the limit means in this setting.
Comment: Definition of the derivative of a function is on page 147. Formula for the slope of a curve on page
137 also defines the derivative at a specific point. Best graphical interpretation of the definition is on pages
135-136. This idea is the subject of a Concept Quiz.

__ • Explain how to tell from its graph whether a function is differentiable or not. Given the graph of a function,
indicate 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.
Representative homework problems: Page 140; #1 – 4; Pages 156 – 157; 35 – 44 .

__ • The derivative of a function y = f(x) is another function y = f ′ (x). Explain in terms of the graph of y = f(x)
what the derivative y = f ′ (x) tells you.
Explain in terms of physical quantities (e.g. time and position, altitude and air pressure, or something else)
what the function y = f ′ (x) tells you.
Given the graph of a function y = f(x), sketch the graph of the derivative y = f ′ (x).
Representative homework problems Page156; #27 – 30, 33, 34.

__ • 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 154).
Give examples, by graphs and/or equations, of functions that are continuous but not differentiable at a point.
Representative homework problems: Page 157 – 158; #39 – 44.

__ • 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.
Explain how you see from the definition that the function is or is not differentiable at x = c. (The language of
one-sided derivatives can be useful.) Confirm your conclusion by examining the graph of the function.
Representative homework problems: Page 158; #54 and #58. Page 236, #65 – 67.

__ • Write the differentiation formulas for the sum, difference, and product of two or more differentiable functions
and for the quotient of two differentiable functions.
Use these differentiation formulas 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.
Representative homework problems: Page 169; #39, 40. Page 236; #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.
Representative homework problems: Page 169; #1 – 38. Page 235; #1 – 4, 6, 9, 10, 31.

__ • Find an equation in point-slope form 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.
Representative homework problems: Page 140; #23 – 26. Page 169; #41 – 44.

__ • Given an expression for a function that describes the motion of a body along a straight-line path,
(a) find the displacement and (b) find the average velocity of the body over a time interval.
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.
Representative homework problems: Page 179; #1 – 16. Page 183; #31 – 34.

__ • 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.
Representative homework problems: Page 181 – 182; #17, 18, 20 – 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
Page 2
independent variable. Use this result to analyze and answer questions about how the dependent variable
changes as the independent variable changes.
Representative homework problems: Page 182 – 183; #25 – 30.

___• Prerequisite facts from trigonometry:
1. Sketch the graphs of the six basic trig functions without a calculator.
2. Express the sine, cosine, and tangent of angles in terms of the lengths of the sides of a right triangle.
3. Know how to represent all six trig functions in terms of sine and cosine and the Pythagorean identity(ies).

___• 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. Derive the formulas for the derivatives of the tangent, cotangent,
secant and cosecant from the derivatives of the sine and cosine.

___• Calculate derivatives of functions formed from trigonometric functions and power functions (with positive or
negative rational exponents) by adding, subtracting, multiplying, dividing, and forming compositions.
Find equations for tangent lines to graphs of these functions (in point-slope form, of course).
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).

Representative homework problems:
Page 188; #1 – 26.
Page 201; select problems involving trig functions from #9 – 48
Pages 235 – 236; select problems involving trig functions from #11 – 40.

Suggested problems for practicing finding tangent lines:
Page 188; #27 – 38
Page 237; #69 – 72.
No exam questions will involve normal lines.

__ • Given a function that may be defined piecewise and may involve trigonometric functions, use the definitions
to determine whether the function is continuous and whether it is differentiable at specified points.
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.

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. If the function is differentiable, use the definition
to find its derivative.
Representative homework problems: Page 189; #48. Page 237; #68. Page 242; #15 – 18.

___• Given two functions f and g (that may be the same function) form the composite functions f o g and g o f.
Given a function h write it as a non-trivial composition f o g (perhaps in more than one way).
(Non-trivial means neither f nor g is the identity function y = x.)
Representative homework problems: Page 201 #1 – 18.

___• 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

Representative homework problems:
Page 201 – 202; select from #1 – 58, 63, 64.
Page 211; #1 – 18.
Pages 235 – 236 Practice Exercises; select from #7 – 40.

___• 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 are given or can be found from graphs of the functions.)
Representative homework problems: Page 202; #59, 60. Page 236; #55, 56.
Study recommendation: A Concept Quiz on the Chain Rule asked you to do this!

___• Solve related rate problems in which time ( t ) is the independent variable, there are two or three dependent
variables, and an equation connecting the dependent variables can be found by using at most two of the
following relations:
a) Pythagorean Theorem,
b) similar triangles,
c) area formulas for simple plane figures (e.g. circle, triangle, rectangle, parallelogram, trapezoid),
d) volume formulas for familiar solids (e.g. sphere, cylinder, cone, rectangular parallelepiped)
e) surface area formulas for a sphere or cube, and
f) definition of the sine, cosine, or tangent function in terms of a right triangle.

Representative homework problems: Pages 218 – 219; #10, 11 – 18, 20, 21 – 23, 30, 31, 33, 34, 35.
NOTE: You are required to know the Pythagorean Theorem; properties of similar triangles; formulas for
areas of simple plane figures; formulas for volumes for spheres, cylinders, cones, and rectangular
parallelepipeds (boxes); and the definitions of the trigonometric functions in terms of right triangles.

Section 3.6 Implicit Differentiation (This section will probably not be tested but may be assessed in a quiz.)

___• Given an equation F(x, y) = c (c a constant), the graph of this equation, and a point (x_o, y_o) on the graph,
a) identify (e.g. by encircling or shading) a portion of the graph that defines a function y = f(x) whose
derivative gives the slope of the line tangent to the graph of F(x, y) = c at the point (x_o, y_o).
(See discussion and examples on pp 205 – 206.)
b) find the derivative of the function y = f (x) defined in (a) by implicit differentiation; and
c) find the equation for the tangent line at (x_o, y_o) (in point-slope form, of course!).
d) Explain why the formula for the derivative of the function y = f (x) obtained by implicit differentiation the
slope of the tangent line at each point on the graph F(x, y) = c (not just at the point (x_o, y_o) used to get
the function y = f (x).)

Representative homework problems:
Page 211; #47 – 56 and 59 – 62 (perhaps with extended instructions)
No exam questions will involve the normal line.

___• Given an equation F(x, y) = c (c a constant) that can be solved explicitly for y and given a point (x_o, y_o) 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 (x_o, y_o) ;
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.
Study Suggestion: Study Examples 1 & 2 on page 206.

___• Calculate first and second derivatives by implicit differentiation.
Representative homework problems:
Page 211; #19 – 36
Page 211; #37 – 44.