FDWKCalcSM_ch6

# FDWKCalcSM_ch6 - 264 Section 6.1 Chapter 6 Differential...

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264 Section 6.1 Chapter 6 Differential Equations and Mathematical Modeling Section 6.1 Slope Fields and Euler’s Method (pp. 321–330) Exploration 1 Seeing the Slopes 1. Since dy dx = 0 represents a line with a slope of 0, we should expect to see intervals with no change in y . We see this at odd multiples of π /. 2 2. Since y is the dependent variable, I t will have no effect on the value of dy dx x = cos . 3. The graph of dy dx will look the same at all values of y . 4. When x x === 01 ,c o s . This can be seen on the graph near the origin. At that point, the change in y and change in x are the same. 5. When x x == = o s. 1 This can be seen in the graph at x = . At this point, the change in y is negative of the change in x . 6. This is true because each point on the graph has a negative of itself. Quick Review 6.1 1. Yes. d dx ee xx = 2. d dx 44 4 = 3. No. d xe x () 22 2 =+ 4. Yes. d ex e 2 = 5. No. d e 52 += 6. d dx x 2 1 2 1 2 7. Yes. d x sec sec tan = 8. No. d dx −− =− 12 9. yx xC + + 34 2 2 3(1) 4 (1) 5 2 C C 10. yxx C =−+ 23 sin cos 4 2 sin(0) 3 cos(0) 7 C C 11. ye xC x + 2 sec 50 3 20 + = eC C sec( ) 12. yx x + tan ln(2 1) 1 C + = tan (1) ln(2(1) 1) 3 4 1 C C Section 6.1 Exercises 1. x x dx ∫∫ (s e c ) 5 42 =− + 5 tan 2. x x e x (sec tan ) e C x + sec 3. x e x + (sin ) 8 3 e x C x + + + cos 2 4 4. dy x x dx x x C + 11 1 2 ln 5. x x C + + 55 1 1 5 2 1 ln tan 6. x x x x C = + 1 1 1 2 2 1 sin 7. t t dt t C + ( c o s ) s i n 3 33 8. te t = cos sin t sin 9. x x = (sec ( )( )) 25 4 5 tan 5 10. u udu = 4 3 (sin ) cos (sin ) uC 4 11. x C + sin cos 23 0 5 + = cos( ) , CC + 35 cos 12. e e x C = + cos sin 32 0 1 21 0 + = + C x x sin( ) , sin 13. du x x dx x x x C + = + + 735 5 62 7 3 11 1 5 4 54 73 =−++ = =−+− ux x x ,

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Section 6.1 265 14. dA x x x dx x x x x C =+ + = + + + () 10 5 2 4 4 9 4 1 042 61 1 1 4 1 1 41 1 1 =+−+ + = + + , CC Ax x x x 15. dy xx dx x x x c =−−+ =+++ −− 13 12 12 24 31 1 1 2 1 1 1 12 1 =++ + = =++− , yx x x 1 0 x > 16. dy x x dx x x c =− + 5 3 2 5 23 2 sec tan / 75 0 0 7 57 32 += + t a n () () , tan / / yx x 17. t dt t C tt = + + + 1 1 22 2 2 1 ln tan 30 2 2 2 10 1 + tan ( ) , tan yt C t 18. dx t t dt t t t C + 11 66 2 1 ln 01 1 6 1 7 6 1 1 + + = , xt t t 7 0 t > 19. dv t t e t dt t e t C + + + 46 4 3 2 sec tan sec 54 0 3 0 0 02 + sec( ) ( ) , eC C Vt e t t t + 43 2 2 sec π < < 2 20. ds t t dt t t C =−= + 01 1 0 =−+ = , st t 21. d ftd t t d t x a x == s in( ) 2 1 x sin( ) 5 2 1 22. du dx d dx f t dt t dt x a x + cos 2 0 ut d t x 0 cos 23. Fx d dx f t dt e dt t x a x 1 2 cos e d t t x cos 9 2 24. Gs d t td t x a x tan 3 0 t x 3 0 4 25. Graph (b). (sin 0) (sin1) (sin ( 2 2 = 0 2 >0 > )) 26. Graph (c).
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## FDWKCalcSM_ch6 - 264 Section 6.1 Chapter 6 Differential...

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