(a) If a function is defined by
     f(x)={1+x, if x<=-1
     {x2, if x> -1,
     evaluate f(-3), f(-1) and f(0) and sketch the graph.

(b) Prove that lim_x→0 ∣x∣​ / x  does not exist.

(a) Sketch the curve y=x2 +1 with the guidelines of sketching.

(b) If z=xy2 + y3 , x= sint, y=cost, find dz/dt at t=0

(a) Estimate the area between the curve y=x² and  the lines x=0 and x=1, using rectangle method, with four sub intervals.

(b) 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.

(a) Define initial value problem. Solve:

y^H+y^‘ -6y =0, y(0)=1, y^‘(0)=0

(b) 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=2x² + 1.

State Rolle’s theorem and verify the theorem for f(x) = x2 – 9, x ε[-3,3]

Starting with x1= 1, find the third  approximate x3 to the root of the equation x³ – x – 5 = 0

Show the integral coverages

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)= x² + x³y² – y² + xy, at (1,2).

Evaluate
₀∫³  ₁∫² x²y dxdy