Calculus Answers

Explain the relationship between distance, velocity, acceleration and time using calculus.

A plane takes off from an airport which is at sea level. The aircraft’s altitude in metres at a time of t minutes is modelled by the equation

h = 5400 ln(t + 1)

a) What is the height of the plane after five minutes?

b) What is the rate of climb of the plane after five minutes?

c) Using differentiation, show the curve y = h(t) has no stationary points, and hence comment on how realistic the model would be for large t.

A particle moves in a straight line and, at time t, has velocity v ms-1 with v = 4t – 12e-t (t ≥ 0)

a) Find an expression for the acceleration of the particle at time t.

b) When t=0, the particle is at the origin.Write an expression for the displacement of the particle at time t.

The distance, x mm, moved by a particle is related to the time in seconds. We have

x = 1.2 sin(2t)

Find the distance travelled, the velocity and the acceleration of the particle at a time of t = 0.3 seconds.

The velocity ,vms-1,of a fairground ride t seconds after it begins is given by v = (3e-t + 7)sin t

What is the acceleration of the ride 2 seconds after it begins?

What is the area enclosed by the graph of

y = x2 + x – 2

 below the x-axis?

find the area enclosed by the curve r²-4cosø

As a phase line, use a U-shaped curve. Let its downward-sloping segment intersect the 45 degree line at point L, and let its upward-sloping segment intersect the 45° line at point R. Answer the following five questions:

(a) Is this a case of multiple equilibria?

(b) If the initial value y0 lies to the left of L, what kind of time path will be obtained?

(c) What if the initial value lies between Land R?

(d) What if the initial value lies to the right of R?

(e) What can you conclude about the dynamic stability of equilibrium at L and at R, respectively? 

Consider the R²−R function f defined by f(x,y)= lnxy and the R−R² function r defined by r(t) = (t²,e^t). Determine the value of (f◦r)′(1) by using the General Chain Rule

1. find the sum, if possible:

     1+3+1/3+1/9+1/27+.........