7. Sketch the graph of the following function by using information from its firs

Question

7. Sketch the graph of the following function by using information from its first and second derivative: y = 27x - x^3 Calculate dy/dx and obtain two values of X for which dy/dx = 0 Determine which value of x is MIN and which One is MAX with d^2y/dx^2 9. In studying water waves, the vertical displacement y(in ft) of a wave was determined to be Y = (15 sin(2t) + 0.3 cos(t), where t is the time (in s). Find the velocity and acceleration for T = 0.4s Find the derivative of the following function : y = 4x+3/cos pi x

Explanation / Answer

dy/dx = 27 - 3x^2 = 0

-3(-9 + x^2) = 0

So, x^2 - 9 = 0

(x - 3)(x + 3) = 0

So, x = -3 and 3

d^2y/dx^2 = -6x

When x = -3, d^2y/dx^2 = -6(-3) = 18 --> positive
So, x = -3 is a MIN

When x = 3, d^2y/dx^2 = -6(3) = -18 --> negative
So, x = 3 is a MAX

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Length = 8 - 2x , Width = 8 - 2x , height = x

Volume = (8 - 2x)(8 - 2x)(x)

V = 4x^3 -32x^2 + 64x

dV/dx = 12x^2 - 64x + 64 = 0

3x^2 - 16x + 16 = 0

3x^2 - 12x - 4x + 16 = 0

(3x - 4)(x - 4) = 0

x = 4 or 4/3

d^2V/dx^2 = 24x - 64

When x = 4/3, second derivative is 24(4/3) - 64 ---> 32 - 64 --> -32 --> negative

So, x = 4/3 is the maximum case

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y = 0.5sin(2t) + 0.3cos(t)

v = y' = cos(2t) - 0.3sin(t)
v = cos(2*0.4) - 0.3sin(0.4)
v = 0.57988

a = y'' = -2sin(2t) - 0.3cos(t)
a = -2sin(2*0.4) - 0.3cos(0.4)
a = -1.711

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