Reference no: EM133967883

Question 1

Differentiate the following functions with respect to x.

Use * to show variables multiplying trigonometric functions such as y*sin(x) to represent ysin(x).

Use brackets to denote arguments of sinusoidal terms such as cos(4x) to represent cos(4x) as opposed to cos4x.

e(2x) is entered as e(2x) not as e2x which would give e²x.

(a)

Use the quotient rule to differentiate:

y=8x²+3x/7x+6y

dy/dx=____

(b)

Use the chain rule to differentiate:

y=3cos(x⁵-1)

dy/dx =

(c)

Select an appropriate rule to differentiate:

y=(3x³+5e^-3x)cos(6x)

dy/dx=____

Question 2

The angular displacement, θ radians, of the spoke of a wheel is given by the expression:

θ=1.4t³-t² where t is the time in seconds.

(a)

The angular velocity after 4 seconds (ω₄)

ω₄=____ rad/sec

(b)

The angular acceleration after 2 seconds (α₂)

α₂=

(c)

The time when the angular acceleration is zero (t₀) in seconds.

t₀ =

Round your answer to 2 decimal places.

Question 3

From a rectangular sheet measuring 250 mm by 200 mm, equal squares of side xxx are cut from each of the four corners. The remaining flaps are then folded upwards to form an open box.

(a) Write an expression for the volume (V) of the box in terms of x. Fully expand all brackets.

V=____ mm³V

(b) Find the value of xxx that gives the maximum volume and give your answer to 2 decimal places.

x_max =____ mm

Calculate the volume to the nearest mm³

(c)

₃∫⁶ (5t³/(√t⁴+3)) .dt

= ___

Round your answer to 2 decimal places.

Question 4

Calculate the following:

(a) Find the area bounded by the curve: y=8/x between x = 3 and x = 6. Give your answer to 3 decimal places.

(b) Find the area enclosed by the curve: y = cos(x) the x-axis and the lines x=0.4Π and x=1.4Π. Give your answer to 2 decimal places.

Hint: Sketching the curve of y=cos(x) between x=0 and x = 2π may help with this part. Get dependable, budget-friendly assignment help-starting today!

Question 5

If the instantaneous rate of change of a population (P) is given by:

95 t² - 304 t^3/2

(measured in individuals per year) and the initial population is 46000, then evaluate/calculate the following.

Use fractions where applicable such as (5/3)t to represent 5/3 t as opposed to 1.67t.

(a) What is the population after t years?

(b) What is the population after 30 years? Round up your answer to whole people.

P₃₀ =

Question 6

Use integration by parts to solve the following integral:

∫4xcos(4x) dx

Use fractions and (capital) C for the constant of integration where applicable.

Use * to indicate multiplication between functions of x and trigonometric functions such as x*cos(6x) for xcos(6x) instead of xcos(6x).