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# Calculus I Exam 3 Study Guide Calculus I

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This 4 page Study Guide was uploaded by Rachael Chandler on Saturday April 2, 2016. The Study Guide belongs to Calculus I at The University of Cincinnati taught by Dr. Yu-Juan Jien in Winter 2016. Since its upload, it has received 20 views. For similar materials see Calculus I in Math at The University of Cincinnati.

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Date Created: 04/02/16

Calculus I Exam 3 Study Guide 4.5: Curve Sketching -Features of the graph of y = f(x) 1. Domain: D = the set of x for which y = f(x) is defined 2. Intercepts: x-intercepts (solve the equation when y = 0), y-intercepts (Solve the equation for x = 0) 3. Symmetry: a. If f(-x) = f(x) for all x in D, then f is an even function and the curve is symmetric about the y-axis (you can reflect it over the y-axis) b. If f(-x) = -f(x) for all x in D, then f is an odd function and the curve is symmetric about the origin (it can be reflected over the point (0, 0)) c. If f(x + p) = f(x) for all x in D where p is a positive constant, then f is a periodic function (it repeats after every period) and the curve of this function is translationally symmetric for every period (This usually occurs with trigonometric functions) 4. Asymptotes: a. Horizontal: y = L is a horizontal asymptote if lim ???? ???? = ???? ????→ ±∞ b. Vertical: x = a is a vertical asymptote if li− ???? ???? = ±∞ or lim ???? ???? + ±∞ ????→ ???? ????→ ???? c. Slant: y = mx + b if a slant asymptote if????→ ±∞???? ???? − ???????? + ???? = 0 **This is very uncommon and doesn’t really need to be checked 5. Interval of Increase/Decrease: Check the signs of f’ (See section 4.3) 6. Local Max/Min: Use previous test and check critical points (See sections 4.1, 4.3) 7. Concavity: Check signs of f” and look for inflection points (See section 4.3) 4.7: Optimization Problems **These are almost all application problems, and the best way to study them is to look at examples and try to do them. See my previous notes/homework problems for examples. 4.9: Antiderivatives -Antiderivative: A function, F, is called an antiderivative of f on an interval, I, if F’(x) = f(x) for all x in I. -If F is an antiderivative of f on an interval, I, then the most general antiderivative of f on I is F(x) + C where C is an arbitrary constant. **Adding this C is easy to forget, but it’s really important and you’ll lose points for missing it. Table of Antiderivatives: Particular Antiderivative Function Function Particular Antiderivative f(x) F(x) -csc (x) cot(x) C * f(x) C * F(x) f(x) + g(x) F(x) + G(x) sec(x) * tan(x) sec(x) n ????+1 x **when n ≠ -1 ???? -csc(x) * cot(x) csc(x) ???? + 1 -1 0 1 sin (x) x = 1 x √1 − ???? 2 x =1 1 ln|????| 1 -1 ???? − cos (x) e x e x √1 − ???? 2 1 tan (x) cos(x) sin(x) 2 sin(x) -cos(x) 1 + ???? 2 1 cot (x) sec (x) tan(x) − 1 + ???? 2 -Special Function Notations: - s(t): position function -v(t) = s’(t): velocity function -a(t) = v’(t) = s”(t): acceleration function 5.1: Areas and Distances Upper/Lower Estimates -An Upper (Over) Estimate is an estimate that is guaranteed to be larger than the actual area under the curve. -A lower (under) estimate is an estimate that is guaranteed to be smaller than the actual area under the curve. -Usually a midpoint sum is neither an upper nor lower estimate because it is impossible to tell whether it will be larger or smaller than the actual area. Left Hand, Right Hand, and Midpoint Sums -In the above chart the blue bars represent the rectangles that would make up a right hand sum (Rn), whereas the orange bars represent the rectangles that would make up the left hand sum (n ). Both are estimations of the area under the curve between x = 0 and x = 5. (n = the number of rectangles used.) -Another estimation of the area under the curve is the midpoint sumn(M ) in which the height at the midpoint between each of the points would be used. (For example, the height at 0.5, 1.5, 2.5, etc.) -For n ,nL , andnM the larger n will produce a more accurate area estimation. Therefore… ???? = l????→ ∞ ???? ???? = l????→ ∞ ???? ???? = l????→ ∞ ???? Equations for n ,nL , andnM ???? = ???? ???? ???? ∗ ∆???? ???? = ???? ???? ???? ∗ ∆???? ???? = ???? ???? ???? ∗ ∗ ∆???? ???? ????=1 ???? ???? ????=1 ????−1 ???? ????=1 ???? Combined (Example of Rn) ???? ???? = lim ???? ???? ???? ∆???? ????→ ∞ ????=1 Rnand Lnare Riemann Sums ???? 2 ???? ????+1 2????+1 ** ????=1 ???? = 6 5.2: The Definite Integrals The definite integral of f over [a, b] where a = lower limit, b = upper limit, and f(x) = the integrand, is ???? ???? ???? ???? ???? ???????? = ????=1???? ???? ???? ∆???? = ???? = ???????? ???? ???? If ???? ???? ≥ 0 over [a, b] then???? ???? ???? ???????? = ???? under the graph of f(x) over [a, b] ???? If ???? ???? < 0 over [a, b] then???? ???? ???? ???????? = −???? under the graph of f(x) over [a, b] Properties of Definite Integrals (Where c is a constant) ???? ???? ???? ???? ???? ???????? = − ???? ???? ???? ???????? ???? ???? ???? ???? ???????? = 0 ???? ???? ???? ???????? = ???? ???? − ???? ???? ???? ???? ????[???? ???? ± ???? ???? ] ???????? = ???????? ???? ???????? ± ???? ???? ???? ???????? ???? ???? ???? ???? ∗ ???? ???? ???????? = ???? ∗ ???? ???? ???? ???????? ???? ???? ???? ???? ???? ???? ???????? + ???? ???? ???? ???????? = ???? ???? ???? ???????? Properties of Comparison ???? ???? If ???? ???? ≤ ???? ???? on [a, b] then ???? ???? ???? ???????? ≤ ???? ???? ???? ???????? ???? If ???? ≤ ???? ???? ≤ ???? on [a, b] then ???? ???? − ???? ≤ ???? ???? ???? ???????? ≤ ???? ???? − ???? 5.3: The Fundamental Theorem of Calculus If f is continuous of [a, b], then… ???? o Part 1: Define ???? ???? = ???? ???? ???? ????????, where ???? ≤ ???? ≤ ????, g(x) is continuous on [a, b] and ???? ???? differentiable on (a, b) and ???? ???? = ???? ???? . Or, ???? ???? ???????? = ???? ???? ???????? ???? o Part 2: ???? ???? ???? ???????? = ???? ???? − ???? ???? where F is any antiderivative of f. (F’ = f) ???? 5.4: Indefinite Integrals and the Net Change Theorem Indefinite Integral of f(x): ???? ???? ???????? o Notice: There are no limits, and this integral is the most general antiderivative of f, which means the formula must include C! (Ex. f = F(x) + C) **Because section 5.5 has not been coveredyet in class, itis not on this study guide, but it WILL be on exam 3. For information on section 5.5, see my notes on it, which will be posted by class on Tuesday (and possibly earlier). For extra help on any of these sections, feel free to reference my older notes or our online text book. Another helpful way to study is to go to WebAssign, and click the button that says ‘View My Class Insights’. It will take you to a page where it will give you aperformancereport onyourhomework, andgivepracticeproblemson thequestions you did poorly on. The practice test is also a good resource. Good luck on exam 3!

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