| Calculus I Assignment # 2 |
This assignment covers
Derivatives with power, product, quotient and chain rule
Implicit Differentiation
Higer Order Derivatives
QUESTIONS
A) Find the first derivative
| 1) |
2) |
3) |
| 4) |
5) f (x) = sin 2 (4x 3) | 6) |
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B) Use implicit differentiation to find the derivative
| 1) 5x 3 - 2x 2 y 2 + 4y 3 - 7 = 0 | 2) 3x 2 - x y 2 + y - 1 = 1 | 3) xy 2 = sin (x + 2y) |
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C)
1) Find equations of the tangent and normal lines to x 2 y - y 3 = 8 at P(-3, 1)
2) Find the first, second and third derivative of y = 5x 3 + 4x 1 / 2
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SOLUTIONS
A) Find the first derivative
| . 1) |
2)  |
3)  |
4)  |
| 5) f '(x) = 24x 2 sin (4x 3)cos (4x 3) |
6)  |
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B) Use implicit differentiation to find the derivative
| 1) 15x 2 - 4xy 2 - 4x 2 yy' + 12y 2 y' = 0 factor out y', transpose and divide to get |
2)  to simplify, multiply through by y 2. |
3) |
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C)
1) 2xy + x 2 y' -3y 2 y' = 0
we solve for y' and substitute x = -3, y = 1
The equation of the normal line with slope = -1 is y = -x -2.
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2)
| y' = 15x 2 + 2x - 1 / 2 | y'' = 30x - x - 3 / 2 | y''' = 30 + (3/2) x - 5 / 2 |
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