Calculus Answers

The extension, y, of a material with an applied force, F, is given by y = e^F x 1 x 10^-3. (e is to the power of F x 1 x 10; 10 is to the power of -3 in that equation)

A) calculate work done if the force increases from 100N to 500N using:
i) an analytical integration technique?
ii) a numerical integration technique?

( work done is given by the area under the curve )

B) compare the two answers?
C) on a spreadsheet increase the number of values used for your numerical method?
D) analyse any affect the size of numerical step has on the result?

Questions: A) find the indefinite integral of the function y = 3t^2 + 2e^3t + 1/t + 2cos3t?
B) calculate the definite integral: (this is an integral operator, with a 1 at the bottom and 2 at the top of the integral operator) 1int2 3t^2 + 2e^3t + 1/t + 2cos3t dt ?

**MUST SHOW GRAPHICAL METHOD FOR CALCULATING GRADIENT**

Question: The equation for the instantaneous voltage across a discharging capacitor is given by v= Voe^-t/t, where Vo is the initial voltage and t is the time constant of the circuit..

A) draw a graph that includes a graphical method for calculating gradient of voltage against time for Vo = 12V and t= 2s, between t = 0s and t= 10s?

B) calculate the gradient at t = 2s and t= 4s? (must be done on a numbered graph)

C) differentiate v= 12e^-t/2 and calculate the value of dv/dt at t=2s and t=4s?

D) compare results for part b and c?

E) calculate the second derivative of the instantaneous voltage (d^2 v / dt^2)?

**THERE MUST BE A GRAPHICAL METHOD FOR CALCULATING THE GRADIENT**

Question:
The equation for a distance, s (m), travelled in time t (s) by an object starting with an initial velocity u (ms^-1) and uniform acceleration a (ms^-2) is:

S= ut + 1 / 2 at^2

A) plot a graph showing a graphical method for calculating gradient of distance (s) vs time (t) for the first 10 seconds of motion if u = 10ms^-1 and a= 5ms^-2?

B) determine the gradient of the graph at t= 2s and t= 6s?

C) differentiate the equation to find the functions for:
i) velocity (v= ds/dt)
ii) acceleration (a= dv/dt = d^2 s / dt^2)

D) use your result from part c to calculate the velocity at t= 2s and t= 6s?

E) compare results from parts b and d?

MUST be answered in graphical format if possible!
Question:
A company is required to fence off the area around a robot arm to comply with health and safety law. They have 750m of fencing available...

The task is to find the maximum area they can fence off?

The answer also need to be presented in a graphical way if possible?

Question: MUST be answered in analytical format if possible?
You plan to make a simple, open topped box from a piece of sheet metal by cutting a square - of equal size - from each corner and fold up the sides.
If L (length) is 200mm and W (width) is 150mm, calculate:

A) the value of x (x is the value of each corner) which will give the maximum volume?

B) the maximum volume of the box?

C) comment of the value obtained in part b?

You are driving on a country road when the road curves gently to the right. You are
able to drive 40km/h around the curve. Which statement is correct? Defend your
choice and explain why the others are incorrect.
a) You are driving at a velocity of 40 km/h.
b) Your speed is constantly changing as you go around the curve.
c) Your velocity is constantly changing as you go around the curve.

Because of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 4 miles to make a turn and climb to an altitude of 4.2 miles. The model for the path of the plane during this maneuver is
r(t) = 10 cos 10π t i + 10 sin 10 π t j + ( 4+ 4t )k , 0≤ t≤ 1/20
where t is the time in hours and r is the position vector.
i. Determine the speed of the plane
ii. Calculate αT and αN

(a) Find the arc length of the parametrized curve
x= et cos t y= et z= et sin t for 0≤ t ≤2π

(a) Find expressions for a plant leaf epidermal cell’s absorption and consumption of a nutrient (as for the spherical cell in Chapter 1). Assume the absorption rate per unit surface area is k1 and the consumption rate per unit volume is k2.
(b) For what values of r is absorption faster than consumption? You answer should give a range of r values defined in terms of the other parameters given in the problem (h, k1, k2).