Calculus 1120, Fall 2012 Dan Barbasch October 25, 2012 Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 1 / 1
Summary of material for Prelim II Integration, chapter 8 Techniques of Integration 8.1-8.5 1 Numerical Integration 8.6 2 Improper Integrals 8.7 3 review problems are on page 507 Differential equations, 7.2, most of 9.1-9.4 Graphic methods, Slope fields 9.1, parts of 9.4 draw slope fields and 1 some solutions for problems on page 540 Separable and Linear Equations 7.2, 9.3 page 433 1-8, 9-22, page 526 2 1-21 Applications, parts of 9.1-9.4. page 434 23-46, page 540 13-18 3 Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 2 / 1
Integration Make sure you know the standard form of writing P ( x ) Q ( x ) as a partial fraction The standard form is a combination of x 1 ( ± a 2 + x 2 ) n , ( ± a 2 + x 2 ) n x 1 ( a 2 − x 2 ) n , ( a 2 − x 2 ) n In the first column substitute u = ± a 2 + x 2 , u = a 2 − x 2 . In the second column do a trigonometric substitution. The outcome is a trigonometric integral which you can do by double angle formulas, recursion formulas and so on. Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 3 / 1
Numerical Integration Make sure you review the left/right endoint approximations, and the over/under estimates. Formulas for error estimates will be given. Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 4 / 1
Improper Integrals You must divide the integral into a sum so that each integral has one singularity. Analyze the function at the singularity. ln x << x a , ( any a > 0) x a << b x , ( any a > 0 , b > 1) 1 b − x << x a , ( any b > 1 , a > 0) 1 1 x a << ( any a > 0) . ln x as x → ∞ . Remember that you must find a function g ( x ) to compare with, but also compute an integral to carry out the test. Often g ( x ) is 1 / x p where you know the answer. Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 5 / 1
Improper Integrals � ∞ 1 An exception might be x (ln x ) p dx . 2 � ln A � A 1 1 x (ln x ) p dx = u p du . 2 ln 2 � A � ln A � B 1 1 du lim x (ln x ) p dx = lim u p du = lim u p . A →∞ A →∞ B →∞ 2 ln 2 ln 2 So � ∞ � ∞ 1 1 x (ln x ) p dx = u p du . 2 ln 2 � ∞ dx Contrast this with (ln x ) p . In this case 2 1 1 0 ≤ x ap ≤ (ln x ) p and so the integral diverges for any p . Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 5 / 1
Differential Equations Problem 16, page 534 Carbon monoxide pollution. An executive conference room of a corporation contains 4500 ft 3 of air initially free of carbon monoxide. Starting at time t = 0, cigarette smoke containing 4% carbon monoxide is blown into the room at the rate of 0 . 3 ft 3 / min . A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of 0 . 3 ft 3 / min . Find the time when the concentration of carbon monoxide in the room reaches 0 . 01%. Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 6 / 1
Differential Equations Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 6 / 1
Differential Equations Autonomous Equation y ′ = F ( y ) . Equilibrium Solution y = c where F ( c ) = 0 . Problem: Consider the differential equation y ′ = 2 xy 2 . (i) Draw the direction field. Mark clearly where the slopes are negative, positive and zero. (ii) Find the equilibrium solution if there is one. Draw several solutions, in particular the ones with initial condition y (0) = 1 and y (0) = − 1 . (iii) What is the long term behaviour of the solutions ( i.e. what is lim x →∞ y ( x ))? (iv) Solve the differential equation. Compute the exact value of y (0 . 3) for the solution satisfying the initial condition y (0) = 1 . Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 6 / 1
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