Mathematics I old question papers

If a function is defined by

f(x)={1+x, if x<=-1

{x^2, if x> -1,

evaluate f(-3), f(-1) and f(0) and sketch the graph.

Sketch the curve y=x^2 +1 with the guidelines of sketching.

Estimate the area between the curve y=x^2 and the lines x=0 and x=1, using rectangle method, with four sub intervals.

Define initial value problem. Solve:

y^H+y^' -6y =0, y(0)=1, y^'(0)=0

Prove that ${\lim }_{x\to 0}\frac{|x|}{x}$ the does not exist.

If z=xy^2 + y^3 , x= sint, y=cost, find dz/dt at t=0

A particle moves a line so that its velocity v at time t is

(1) Find the displacement of the particle during the fine period 1 ≤ t ≤ 4

(2) Find the distance travelled during this time period.

Find the Taylor's series expansion for cosx at x=0.

Dry air is moving upward. If the ground temperature is 20^° and the temperature at a height of 2km is 10^° c, express the temperature T in ^° c as a function of the height h(in km), assuming that a linear model is appropriate. (b) Draw the graph of the function and find the slope. Hence, give the meaning of slope. (c) What is the temperature at a height of 2km?

Find the equation of the tangent at (1,3) to the curve y=^2x^2 + 1.

State Rolle's theorem and verify the theorem for f(x) = x^2 – 9, x ε[-3,3]

Starting with x_1= 1, find the third approximate x_3 to the root of the equation x³ – x – 5 = 0

Show the integral coverages

∫_0^³ dx/x-1

Use Trapezoidal rule to approximate the integral _1 ∫^² dx/x, with n=5.

Find the derivative of (r(t)) = t^2i – te^(-t)j + sin(2t)k and find the unit tangent vector at t = 0

What is sequence? Is the sequence

$a_n=\frac{n}{\sqrt{5+n}}$

convergent?

Find the angle between the vectors a = (2, 2, -1) and b = (1, 3, 2)

Find the partial derivative f_xx and f_yy of f(x,y)= x^2 + x^3y^2 – y^2 + xy, at (1,2).

Evaluate

_0∫^3 _1∫^2 x^2y dxdy