# Math 1 CALCULUS I STUDY GUIDE AND REVIEW FOR EXAM I ...

Math 1 CALCULUS I STUDY GUIDE AND REVIEW FOR EXAM I ...

Math 1 CALCULUS I STUDY GUIDE AND REVIEW FOR EXAM I ...

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<strong>Math</strong> 1 <strong>CALCULUS</strong> I<strong>STUDY</strong> <strong>GUIDE</strong> <strong>AND</strong> <strong>REVIEW</strong> <strong>FOR</strong> <strong>EXAM</strong> IExam I will be given Friday, September 11. It will cover Chapter 2, all sections, andChapter 3, sections 1 and 2. You may use a scientific calculator during the exam;graphing calculators are not allowed. Cell phones and other electronic devices may notbe used during the exam.To prepare for the exam, you should review the concepts and skills described below andwork the problems on the Review Exercises handout. It is important that you understand,and are able to apply, the concepts, definitions and theorems discussed in the text and inclass.CHAPTER TWOSection 1. Calculate average and instantaneous rates of change from a table of values.For example, from a table of function values, calculate slopes of secant lines over severalintervals and estimate the slope of the tangent line as the secant line approaches thetangent. Or, calculate average velocity over several intervals and use the results toestimate instantaneous velocity at a point.Section 2. Given the graph of a function f, estimate lim f ( x)or lim f ( x)or lim f ( x)x→a+x→a−x→aDetermine whether limits do or do not exist and justify your conclusions in terms ofdefinitions 1, 2 and 3. Discuss the existence of vertical asymptotes in terms ofdefinitions 4, 5 and 6. Use limit notation correctly and appropriately.Section 3. Use limit laws, properties of limits and algebraic manipulations to calculatelimits. Find limits for piece-wise defined functions. Use limit notation correctly andappropriately.Section 4. Use the precise definition of limit (the δ, ε definition) to prove a limit.Section 5. Use the definition of continuity to discuss why a function is continuous ordiscontinuous at a point and classify its discontinuities as infinite, removable or jumpdiscontinuities. Determine if a piece-wise function is discontinuous at a point and sketchits graph. Know the results of Theorems 4, 5, 7, 8 and 9. Use the Intermediate ValueTheorem to show that an equation has a root in a specified interval..CHAPTER THREE( + ) − ( )f a h f aSection 1. Use the limit of the difference quotient limto find theh→0hinstantaneous rate of change in a function f at x = a. In particular, find the slope of theline tangent to the graph of f at x = a and write the equation of the tangent line.

Study Guide and Review Exercises for Exam I Page 2 of 5Section 2. Use the definition of the derivativef ( a+ h) − f ( a)f′ ( x)= limh→0hto find either f ′( a)or f ′( x). State the domain of the function f and its derivative f ′ .Interpret the derivative as the slope of a tangent line or as a rate of change (e.g., velocity).Given the graph of a function, determine all points at which the function is notdifferentiable and justify your answer in terms of the definition of differentiability.Given the graph of a function, sketch a graph that represents the behavior of thederivative. Use differentiation notation correctly and appropriately.<strong>REVIEW</strong> EXERCISES <strong>FOR</strong> <strong>EXAM</strong> IUse the graph of the function f ( x ) to solve problems 1 - 4 below.1. Find the limit, if it exists. If the limit does not exist, explain why not. Justify youranswer by referring to appropriate limit definitions, theorems, laws, or properties.Where appropriate, indicate a limit of +∞ or −∞ .(a) lim f ( x)−x→0(c) lim f ( x)x→0(b) lim f ( x)+x→0(d) lim f ( x)−x→2<strong>Math</strong> 1 Calculus IT. Henson

Study Guide and Review Exercises for Exam I Page 3 of 5(e) lim f ( x)+x→2(g) lim f ( x)−x→4(f) lim f ( x)x→2(h) lim f ( x)+x→4(i) lim f ( x)x→42. Identify all vertical asymptotes of f ( x ) , if any. Justify your answer usingappropriate definitions or theorems.3. State all values of x at which the function f ( x ) is discontinuous. Identify the typeof discontinuity and justify your answer using the definition of continuity.4. State all values of x at which the function f ( x ) is not differentiable. Give a reasonfor your answer.5. Given thatx→3( x)= lim g( x)= − 2 lim h( x)= − 8 h( x)lim f 5x→3−x→3lim = 3find the following limits, if they exist. If the limit does not exist, explain why not byciting appropriate limit definitions, theorems, laws, or properties.+x→3(a) lim f ( x)+x→3(c) f ( x)limx → 3⎡g( x) ⎤2⎣⎦( )(b) lim −4f ( x) i g( x)x→3(d) f ( x)+ 4lim+x→3g( x) − h( x)(e) lim g( x) i h( x)−x→36. Find the limit, if it exists. SHOW ALL STEPS of your work/thinking. If the limithas a numerical value, give that value. If it can be determined that the limit isinfinite ( +∞ or −∞ ), indicate that. If the limit does not exist, write DNE and give areason for your answer.(a)2x − 3x+7limx→122x + x−7(b)2x + 4x−5limx→122x − 7 x+5<strong>Math</strong> 1 Calculus IT. Henson

Study Guide and Review Exercises for Exam I Page 4 of 5(c)⎛lim ⎜⎝1 6++ −x→−32x 3 x 9⎞⎟⎠(Hint: add the fractions)(d)⎛lim ⎜⎝+x→4xx −4−x⎞⎟⎠lim 2 − 3 = 1.7. Using the δ, ε definition of the limit, prove ( x )x→28. Given ( )2⎧x+ 4 if x ≤ −1⎪f x = ⎨x if -1 ≤ x

Study Guide and Review Exercises for Exam I Page 5 of 5′ for12. On the same pair of axes, sketch a graph that represents the behavior of f ( x)the function f ( x ) sketched below.13. Find the equation of the line that is parallel to y = −4x− 5 and tangent to the curve( )2f x = x − 2x.14. A particle moves along a straight line. Its distance from the origin is given by thes = f t = t − t , where s is in meters and t is in seconds. Find the velocityfunction ( )3 2of the particle after 1 second. When is the particle at rest?<strong>Math</strong> 1 Calculus IT. Henson