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40 Find the stationary points of...

This question uses the function f(x) = -2x^3 - 9x^2 + 24x + 40

Find the stationary points of this function

B. Using the strategy to apply the first derivative test, classify the left hand stationary point found in the first question

Then using second derivative test classify the right hand stationary point found in first question

C. Find the y coordinate of each of the stationary points on the graph of the function f(x), and evaluate f(0)

D. Draw a graph of the function f(x)

2a. Simplify the answers where appropriate

Write down the derivative of each of the functions:

F(x) = x^4 and g(x) = sin(5x)

Hence by using product rule, differentiate the function

K(x) = x^4sin(5x)

Write down the derivative of each of the functions f(t) = e^6t and g(t) = t^3 + 8

Hence by using quotient rule, differentiate the function

K(t) = e^6t/t^3 + 8 (t > -2)

Write down derivative of the function f(x) = 6cos(1/3x)

Hence using the composite rule differentiate the function k(x) = ln (6cos(1/3x)) (-3/2 delta < x 0)

h(u) = cos^2 (6u)

B. Evaluate indefinite integral with 8 at top and 6 at bottom x(4 + 7x^2)dx

C. Write down a definite integral that will give the value of the area under the curve y=x^2sin(3/4x) between x = 1/3 delta and x = delta

( do not evaluate the integral by hand)

Use mathcad to find the area described in previous question giving answer to 4 decimal places and give print out of the working

4. A rocket is modelled by a particle that moves along a vertical line. From launch the rocket rises until it's motor cuts out after 17 seconds. At this time it has reached a height of 580 metres above launch pad and attained an upward velocity of 120ms-1. from this time on the rocket has a constant upward acceleration of -10ms-2 ( due to gravity effect)

Choose the s axis ( for position of particle that represents the rocket) to point upwards with origin at launch pad. Take t = 0 to be the time when rocket motor cuts

What is the max height above launch pad reached by rocket?

How long from launch does rocket take to reach this max height?

After how long from launch does rocket crash onto launch pad?

Give your answers in seconds to 1 decimal place

5. Solve the initial value problem dy/dx = sin(4x)/3+cos(4x). Y = 5 when x = 0

Show where c is an arbitrary constant in this equation

Indefinate integral 2e^2x - 5/(e^2x - 5x)^2/3 dx=3(e^2x - 5x)^1/3 + c

Hence find it's implicit form, the general solution of the differential equation

Dy/dx = 2y^1/2(2e^2x -5)/3(e^2x -5x)^2/3 (y>0)

Find corresponding particular solution (in implicit form) that satisfies the initial condition y=1 when x = 0

Find the explicit form of this particular solution

What is the value of y given by this particular solution when x =2? Give your answer to 4 sig figs

Find the stationary points of this function

B. Using the strategy to apply the first derivative test, classify the left hand stationary point found in the first question

Then using second derivative test classify the right hand stationary point found in first question

C. Find the y coordinate of each of the stationary points on the graph of the function f(x), and evaluate f(0)

D. Draw a graph of the function f(x)

2a. Simplify the answers where appropriate

Write down the derivative of each of the functions:

F(x) = x^4 and g(x) = sin(5x)

Hence by using product rule, differentiate the function

K(x) = x^4sin(5x)

Write down the derivative of each of the functions f(t) = e^6t and g(t) = t^3 + 8

Hence by using quotient rule, differentiate the function

K(t) = e^6t/t^3 + 8 (t > -2)

Write down derivative of the function f(x) = 6cos(1/3x)

Hence using the composite rule differentiate the function k(x) = ln (6cos(1/3x)) (-3/2 delta < x 0)

h(u) = cos^2 (6u)

B. Evaluate indefinite integral with 8 at top and 6 at bottom x(4 + 7x^2)dx

C. Write down a definite integral that will give the value of the area under the curve y=x^2sin(3/4x) between x = 1/3 delta and x = delta

( do not evaluate the integral by hand)

Use mathcad to find the area described in previous question giving answer to 4 decimal places and give print out of the working

4. A rocket is modelled by a particle that moves along a vertical line. From launch the rocket rises until it's motor cuts out after 17 seconds. At this time it has reached a height of 580 metres above launch pad and attained an upward velocity of 120ms-1. from this time on the rocket has a constant upward acceleration of -10ms-2 ( due to gravity effect)

Choose the s axis ( for position of particle that represents the rocket) to point upwards with origin at launch pad. Take t = 0 to be the time when rocket motor cuts

What is the max height above launch pad reached by rocket?

How long from launch does rocket take to reach this max height?

After how long from launch does rocket crash onto launch pad?

Give your answers in seconds to 1 decimal place

5. Solve the initial value problem dy/dx = sin(4x)/3+cos(4x). Y = 5 when x = 0

Show where c is an arbitrary constant in this equation

Indefinate integral 2e^2x - 5/(e^2x - 5x)^2/3 dx=3(e^2x - 5x)^1/3 + c

Hence find it's implicit form, the general solution of the differential equation

Dy/dx = 2y^1/2(2e^2x -5)/3(e^2x -5x)^2/3 (y>0)

Find corresponding particular solution (in implicit form) that satisfies the initial condition y=1 when x = 0

Find the explicit form of this particular solution

What is the value of y given by this particular solution when x =2? Give your answer to 4 sig figs

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