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Calculus Problem

Calculus Problem. By: Ashley Kim Period 6. Problem. A curve is defined by x 2 y-3y 2 =48. a) Verify that dy/dx = 2xy/6y-x 2 b) Write an equation of the linearization of this curve at ( 5,3).

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Calculus Problem

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  1. Calculus Problem By: Ashley Kim Period 6

  2. Problem • A curve is defined by x2y-3y2=48. • a) Verify that dy/dx = 2xy/6y-x2 • b) Write an equation of the linearization of this curve at ( 5,3). • c) Using your equation from part a, estimate the y-coordinate of the point on the curve where x=4.93.

  3. d) Find equations of all horizontal tangent lines. • e) Find equations of all vertical tangent lines.

  4. First Problem • a) x2 y-3y2=48 • Step #1: Differentiate • x2 (dy/dx)+ 2xy -6y(dy/dx)=0 • Step #2 : Subtract each side by -2xy. • (x2 -6y) dy/dx =-2xy • Step #3 : Divide • dy/dx= 2xy/ 6y-x2

  5. Second Problem • b) Write equation of the linearization of this curve at (5,3) • Step #1: Find slope using (5,3) • dy/dx = 2(5)(3)/ 6(3)-52 • 30/ 18-25 = -30/7  Slope!! • Step #2: Use y-y1= m (x-x1). • y-3 = -30/7 ( x-5)

  6. Third Problem • c) Using your equation from part a, estimate the y-coordinate of the point on the curve where x= 4.93. • Step #1: Just plug in 4.93 into x in y-3 = -30/7 ( x-5). • y-3= -30/7( 4.93 -5) y-3 =0.3 • y= 3.3

  7. Fourth Problem • d) Find the equations of all horizontal tangent lines. • Step #1: Tangent Lines are horizontal when dy/dx = 2xy/6y-x2 =0. This is only possible when x=0 or y=0 since 2xy=0. • Step #2: For x=0 plug in 0 into the x. • 0y-3y2=48, so it has no real solutions.

  8. Step #3: For y=0, plug in 0 into y. • x2(0) -3(02) =48, that doesn’t make sense and impossible. • So there are no horizontal tangents.

  9. Fifth Problem • Find equations of all vertical tangents. • Step #1: For vertical tangent lines the slope is undefined. So it’s only possible when 6y-x2 =0. • Which is y= x2 /6. • Step #2: Substitute that into y into the original equation: x2y-3y2=48.

  10. x2 (x2 /6) -3 (x2 /6) =48. • (x4 /6)- ( x4/12)=48 • Step #3: Multiply each side by 12. • 2x4 - x4= 48. • Step #4: x4 =48 • x= + or - (√ 24) • x= 4.898979 or – 4.898979

  11. The End • Hope you had fun learning this valuable math problem!!

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