Calculus Answers

A car has an initial velocity of 20 m s⁻¹

For the first 4 seconds of its motion it accelerates at 2.5 m·s^{−2 For} the next T seconds it travels at a constant velocity of V m s⁻¹. The car then decelerates to rest

Sketch the velocity-time graph for the whole journey of the car. 

Find V.

The total time for the journey is 40 seconds.

iii) If the total distance travelled by the car is 1090m, find T.

let f:[0,1]->R be a function defined by f(x)= x^m (1-x)^n, where m,n belong to N. Find the values of m and n such that the Rolle's theorem holds for the function f.

Consider the R − R 2 function r defined by r (t) = ￾ t, t2
; t ∈ [−3, 3] . (a) Determine the vector derivative r 0 (1) by using Definition 6.1.1(b) Sketch the curve r together with the vector r 0 (1), in order to illustrate the geometric meaning of the vector derivative. Note: The curve r is the image of r, so it consists of all points (x, y) = (t, t2 ); t ∈ [−3, 3]

Show that x2i +2xyj - 4xzk is solenoidal

Use a triple Integral to find the volume of the pyramid P whose base is the square with vertices (1,0,0), (0,1,0), (-1,0,0), and (0,-1,0) and whose top vertex is (0,0,1).

Use a triple Integral to determine the volume of the tetrahedron with vertices (0,0,0), (3,0,0), (0,4,0) and (0,0,5).

Compute the volume of the box with opposite corners at (0,0,0) and (2,2,3).

A. Find the centroid of each of the following systems of masses.

1. Equal masses of 100 kg at (0, 3), (2, 2) and (2, -1).

2. Masses of 200 lb, 500 lb and 1000 lb at (3, 2), (4, 0) and (1, 5).

B. Find the centroid of each of the areas bounded by the following curves.

1. y = 6 - 2x ; x = 0 ; y = 0

2. Y = X³, Y = 4X

A. Find the volume generated by revolving above the x-axis the areas bounded by the following curves.

1. y= 6x-x² ; y = 0

2. y = x² ; y = 2x

B. Find the volume generated by revolving above the y-axis the areas bounded by the following curves.

1. y = x² ; y = x

2. y = 12 - x² ; y = x ; x = 0

C. Find the volume of the solid generated by revolving above the indicated axis the area bounded by the giving curves.

1. y² = 4x ; x = 4 ; about x = 4

2. y² = 5-x ; x = 0 ; about x = 5

I. Find the area enclosed by each of the following.

1. r = 4 Sin θ

2. r = 2 - cos θ

II. Find the common area enclosed by the following pairs of curves.

1. r = 3 Cos θ

2. r = 1 + Cos θ

III. Find the area which is inside of the curve r = 5 Sin θ and the outside of the curve r = 2 + Sin θ.