M 160 Exam 2 Study Guide
MATH 160 Study Guide for Sections 2.7, 3.1-3.7, 4.1, 4.3 Spring Semester, 2008
This Study Guide describes everything you are expected to know, understand, and be able to do from sections 2.7, 3.1-3.7, 4.1, and 4.3. Questions on Exam 2 covering this material will ask you to do one or more of the tasks described in this Study Guide. When you have mastered an item on the Study Guide, check it off. When every item is checked, you are ready for any question from these chapters that could appear on the second midterm exam!
Differentiation (Sections 2.7, 3.1 & 3.2)
• 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 the third Calculator Lab and 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 derived function 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 derived function y = f′(x).
Representative homework problems Page156; #27 – 30, 33, 34.
• Use the definition of continuity of a function at a specific point x = c to determine whether a function given by a specific formula or having given properties is or is not continuous at a given point x = c.
(Review of sec 2.6 and Exercises #35 – 40 on page 133.)
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.) 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: 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 + [pic] is differentiable at x = 1.
Page 158; #54 and #58, page 236 – 237, #65 – 67.
• Write a complete statement (as a theorem with hypotheses and conclusion) of the differentiation formulas for the sum, difference, and product of two or more differentiable functions and for the quotient of two differentiable functions.
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.
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.
Sec. 3.3 The Derivative as a Rate of Change
• 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 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.
Sec. 3.4 Derivatives of Trigonometric Functions
• 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 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.
4. Obtain the double angle formula for the cosine by differentiating the double angle formula for the sine.
• 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 formulas for the derivative 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 resulting function is differentiable, use the definition to find its derivative.
Representative homework problems: Page 189; #48. Page 237; #68. Page 242; #15 – 18.
Sec. 3.5 The Chain Rule
• Given two functions f and g (that may be the same function) form the composite functions fog and gof.
Given a function h write it as a non-trivial composition fog (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 compete 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: The Concept Quiz on the Chain Rule asked you to do this!
Sec 3.6 Implicit Differentiation
• Given an equation F(x, y) = c (c a constant), the graph of this equation, and a point (xo, yo) 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 (xo, yo).
(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 (xo, yo) (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 (xo, yo) 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 (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.
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.
Sec 3.7 Related Rates
• 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.
Sec. 4.1 Extreme Values of Functions
• Write complete definitions of the following words and phrases:
a) absolute maximum (aka global maximum) of a function (pg 244)
b) absolute minimum (aka global minimum) of a function (pg 244)
c) local maximum (aka relative maximum) (pg 247)
e) local minimum (aka relative minimum) (pg 247)
f) critical point (of a function) (pg 248).
Illustrate each of the above by sketching a graph.
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.)
Study Suggestion: Study Sec. 4.1.
Representative homework problems; Page 252; #1 – 10
• Write a complete statement of the Extreme Value Theorem (pg 246).
Illustrate the Extreme Value Theorem graphically.
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.
Study Suggestion: Study Theorem 1 in Sec. 4.1. Find examples among exercises 1 – 14, pg 252.
• Write a complete statement of the First Derivative Theorem for Local Extreme Values (Theorem 2, pg 247).
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).
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.
Representative homework problems: Page 252; #11 – 14.
• Describe a step-by-step procedure for finding the absolute extrema of a continuous function on a closed interval of finite length (summarized in the box top of page 249).
Use this procedure to find the absolute maximum and minimum values of a given function and the points where these extreme values are attained.
Representative homework problems: Page 253; #15 – 34 and #35 – 44.
• 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.
Representative homework problems: Pages 253 – 254; #53, 65, 68, 69, 70.
Sec. 4.3 Monotonic Functions and the First Derivative Test
• Write a complete definition of what it means for a function to be increasing and for a function to be decreasing on an interval (pg 263).
Explain how to use the derivative to determine the intervals on which a function is increasing and on which it is decreasing. Explain how to use the first derivative to determine whether a function has a local maximum, a local minimum, or neither at its critical points and the end points of its domain. Illustrate your explanations with graphs (pg 263).
Study Suggestion: Study Sec. 4.3.
Related homework problems: Page 267; #43 – 46.
• Given a function defined by an expression y = f(x), calculate and analyze its first derivatives (using algebra, not a calculator) to find the critical points of the function.
Knowing the critical points of a function, use the First Derivative Test for Monotonic Functions (page 263) to determine the intervals where the function is increasing and the intervals where the function is decreasing.
Use the First Derivative Test for Local Extrema (page 265) to determine whether the function has a local maximum, a local minimum, or neither at each critical point and, if applicable, at the end point of its domain.
Representative homework problems: Page 266; #1 –36; Page 267; #37, 38, 41, 42, 47, 48.
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