1. (Vectors) The vectors a, b and c are given by a = (−2,3,1), b = (0,2,4) and c = (3,−1,−2).

(i) Sketch the vectors a, b and c, all on the same set of xyz-axes.

(ii) A force of 12 newtons acts in the direction of c. Find the force vector F. (Hint: start with a unit vector.)

(iii) Determine which of the vectors a, b and c are perpendicular to each other, if any.

(iv) Find the angle θ, in radians, between a and b, both in exact form and to four decimal places.

(v) Use the cross-product a×b to find the area of the triangle that has a and b as two of its sides.

(vi) Find the volume of the tetrahedron that has a, b and c as three of its edges.

2. (CriticalPointsandCurveSketching)Considerthefunctionf(x)=1x+sinx,for0≤x≤2π.

(a) Use the techniques of calculus to find the coordinates of the critical points of f(x), both in exact form and to four decimal places. Give both the x- and y-coordinates.

(b) Use the First Derivative Test (sign diagram for f′) to classify the critical points.

(c) Use the Second Derivative Test to classify the critical points.

(d) What are the global (i.e. absolute) maximum and minimum values of f(x) for 0 ≤ x ≤ 2π? Give both the x- and y-coordinates.

(e) Sketch the graph of y = f(x).

3. (Mathematical Modelling: Related Rates) Water is dripping slowly through the bottom of a small conical cup that is 4 cm across and 6 cm deep. The cup loses half a cubic centimetre of water each minute. How fast is the water level dropping at the moment when the water is 3 cm deep? Give your answer both in exact form and to four decimal places.

In your solution to this mathematical modelling question, clearly label the three steps of (1) Modelling, (2) Solving and (3) Interpreting.

SOLUTION

2.

We suppose, the tension in the left cable is T_{1} and tension in right cable is T_{2}.

Resolving the forces horizontally,

Resolving the forces vertically,

Therefore,

6. (a) Given,

(i) The function is continuous everywhere on [2,4]

Also,

(ii) The derivative exists every where on [2, 4]. Therefore, the function is differentiable on [2,4]

Therefore, both the conditions of Mean value theorem are satisfied.

Therefore, there must exist at least one point

Such that,

(b) Since,

(c) Given,

Using Fundamental theorem of Calculus, if

Then,

f(x) is the anti derivative of

Therefore,

Therefore,

Therefore,

LF33

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