Calculus_205th_20Edition_20-_20James_20Stewart_Part25

# Calculus_205th_20Edition_20-_20James_20Stewart_Part25 - 600...

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22. Use Euler’s method with step size to estimate , where is the solution of the initial-value problem , . Use Euler’s method with step size to estimate , where is the solution of the initial-value problem , . 24. (a) Use Euler’s method with step size to estimate , where is the solution of the initial-value problem ,. (b) Repeat part (a) with step size . ; 25. (a) Program a calculator or computer to use Euler’s method to compute , where is the solution of the initial-value problem (i) (ii) (iii) (iv) (b) Verify that is the exact solution of the differ- ential equation. (c) Find the errors in using Euler’s method to compute with the step sizes in part (a). What happens to the error when the step size is divided by 10? 26. (a) Program your computer algebra system, using Euler’s method with step size 0.01, to calculate , where is the solution of the initial-value problem (b) Check your work by using the CAS to draw the solution curve. 27. The ﬁgure shows a circuit containing an electromotive force, a capacitor with a capacitance of farads (F), and a resistor with a resistance of ohms ( ). The voltage drop across the capaci- tor is , where is the charge (in coulombs), so in this case Kirchhoff’s Law gives But , so we have Suppose the resistance is , the capacitance is F, and a battery gives a constant voltage of 60 V. (a) Draw a direction ﬁeld for this differential equation. (b) What is the limiting value of the charge? C E R 0.05 ± 5 R dQ dt ² 1 C Q ± E ± t ² I ± dQ ³ dt RI ² Q C ± E ± t ² Q Q ³ C ± R C y ± 0 ² ± 1 y ³ ± x 3 ´ y 3 y y ± 2 ² CAS y ± 1 ² y ± 2 ² e ´ x 3 h ± 0.001 h ± 0.01 h ± 0.1 h ± 1 y ± 0 ² ± 3 dy dx ² 3 x 2 y ± 6 x 2 y ± x ² y ± 1 ² 0.1 y ± 1 ² ± 0 y ³ ± x ´ xy y ± x ² y ± 1.4 ² 0.2 y ± 0 ² ± 1 y ³ ± y ² y ± x ² y ± 0.5 ² 0.1 23. y ± 0 ² ± 0 y ³ ± 1 ´ y ± x ² y ± 1 ² 0.2 Make a rough sketch of a direction ﬁeld for the autonomous differential equation , where the graph of is as shown. How does the limiting behavior of solutions depend on the value of ? (a) Use Euler’s method with each of the following step sizes to estimate the value of , where is the solution of the initial-value problem . (i) (ii) (iii) (b) We know that the exact solution of the initial-value problem in part (a) is . Draw, as accurately as you can, the graph of , together with the Euler approximations using the step sizes in part (a). (Your sketches should resemble Figures 12, 13, and 14.) Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates. (c) The error in Euler’s method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler’s method to estimate the true value of , namely . What happens to the error each time the step size is halved? 20. A direction ﬁeld for a differential equation is shown. Draw, with a ruler, the graphs of the Euler approximations to the solution curve that passes through the origin. Use step sizes and . Will the Euler estimates be underestimates or overestimates? Explain.

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Calculus_205th_20Edition_20-_20James_20Stewart_Part25 - 600...

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