MorrisonCalc1Fall2024MT1reviewsheet.pdf

MATH1206: Calculus, I First Midterm Review Sheet–October 8, 2024 Professor Ian Morrison Below are the instructions that will appear on the actual midterm, to be held in your class period on Thursday, October 8. Instructions: Answer all questions. Points for each are shown: there...

MATH1206: Calculus, I First Midterm Review Sheet–October 8, 2024 Professor Ian Morrison Below are the instructions that will appear on the actual midterm, to be held in your class period on Thursday, October 8. Instructions: Answer all questions. Points for each are shown: there are a total of 66 possible points but the test will be graded out of 60. Please read the questions carefully and make sure that you answer the question I have asked. Explain your reasoning clearly. Unexplained answers — even correct ones — will not receive full credit. No calculators, cellphones, tablets or laptops are allowed in the exam room. You may use a single 3” × 5” index card as an aid. The test will contain questions covering the following topics although not necessarily in the order given below. There will not be any questions that deal only with the pre-calculus review topics covered in Chapter 1 of Stewart, but these skills will be needed answering in questions on other topics. Section(s) of Stewart covering each topic are indicated in parentheses. Topics marked with a star (*) will be the most important. Practical Section: 94% of credit 1. Values of sin, cos and tan at common angles between 0 and π2. 2. * Determining existence and values of finite and infinite limits algebraically using flowchart dis- cussed in class or, more precisely, using the algorithmic procedure outlined and used in class to reduce to direct substitution when possible. 3. Recognizing existence or non-existence of limits as x → ∞ and finding their values when they exist. 4. * Recognizing continuity and differentiability properties, or their failure, either from formulas or from graphs. Also, vice-versa, drawing graphs that exhibit specified limit, continuity and derivative properties. 5. Applying (but not proving) the important theorems about limits and continuous functions: the Squeeze Theorem for limits and the Intermediate Value Theorem for continuous functions. 6. * Finding derivatives from the limit definition. 7. * Computation of derivatives by differentiation rules and implicit differentiation, and the use of these derivatives to give the equations of tangent lines. 8. Estimation derivatives and average rates of change from formulas, from graphs and from numerical data. 9. * Graphing f ′ given the graph of f or f given the graph of f ′ and sketching these graphs using verbal and numerical descriptions of f. Most of the questions involve calculations but there will be one word problem. 1 MATH1206: Calculus, I First Midterm Review Sheet–October 8, 2024 Professor Ian Morrison Theory Section: 16*% of credit One of parts 1.–3. below and one of parts 4.–6 below.: you will have some choice of which parts you discuss. Parts 1.–3. are the main steps in the proof of the Product Rule. Parts 4.–6. are the main steps in computing limh→0 sin(h) h. 1. Draw a rectangle that shows f (x), g (x), f (x +h) and g (x +h) and explain indicate on it an area equal to f (x + h)g (x + h) − f (x)g (x). 2. Explain how to obtain the formula ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ f (x + h)g (x + h) − f (x)g (x) = f (x + h) − f (x) · g (x) + f (x) · g (x + h) − g (x) + f (x + h) − f (x) · g (x + h) − g (x) from the picture below. f (x + h) g (x + h) − g (x) g (x + h) g (x) f (x) f (x + h) − f (x) 3. Use the formula ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ f (x + h)g (x + h) − f (x)g (x) = f (x + h) − f (x) · g (x) + f (x) · g (x + h) − g (x) + f (x + h) − f (x) · g (x + h) − g (x) to verify the Product Rule. Please explain carefully what you operations you are performing at each step and how you know the values of any limits you take. 4. Fill in the missing coordinates of the points A, B , and D in the diagram below. Be sure to explain carefully how you know the value of each coordinate. ¡ ¢ ¡ ¢ A= , B = 1, ¡ ¢ O = (0, 0) h D= ,0 E = (1, 0) x2 + y 2 = 1 sin(h) h sin(h) 5. Explain how to obtain the inequalities ≤ ≤ from the diagram, justifying any 2 2 2 cos(h) formulas for areas that you use. sin(h) 6. Use these inequalities to show that lim = 1, being sure to explain each of your steps and to h→0 h justify your use of any Theorems you choose to invoke. 2