MATH 160 Study Guide for Sections 2.7, 3.1-3.7, 4.1, 4.3 Spring Semester, 2008MATH 160 Study Guide for Sections 2.7, 3.1-3.7, 4.1, 4.3 Spring Semester, 2008This 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 + x 1 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.Page 1 February 14, 2008____• 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;