(a) If  and then find f0g and its domain and range.

(b) A rectangular storage container with an open top has a volume of 20m3. The length of its base is twice its width. Material for the base costs Rs.10 per square meter material for the sides costs Rs 4 per square meter. Express the cost of materials as a function of the width of the base.

(a) Using rectangular, estimate the area under the parabola y = x² from 0 to 1.

(b) A particle moves along a line so that its velocity v at time t is
v = t² – t + 6

a. Find the displacement of the particle during the time period 1 ≤ t ≤ 4.
b. Find the distance travelled during this time period.

(a) Find the area of the region bounded by y = x² and y = 2x – x² 

(b) Using trapezoidal rule, approximate ₁∫² 1/x dx with n = 5

(a) Solve y’ = x²/y², y(0) = 2

(b) Solve the initial value problem: y” + y’ – 6y = 0,  y(0) = 1, y'(0) = 0

Recent studies indicates that the average surface temperature of the earth has been rising rapidly. Some scientists have modeled the temperature by the linear function T = 0.03t + 8.50, where T is temperature in degree centigrade and t represents years since 1900.

What do the slope and T-intercept represent?
Use the equation to predict the average global surface temperature in 2100

Find the equation of tangent at (1, 2) to the curve y = 2x²

State Rolle’s theorem and verify the Rolle’s theorem for f(x) = x² – 3x + 2 in [0, 3]

Use Newton’s method to find ⁶√2 correct five decimal places.

Find the derivatives of r(t) = (1 + t²)i  – te^-tj + sin 2tk and find the unit tangent vector at t=0.

Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x².

Solve: y’ + 2xy – 1 = 0

What is sequence? Is the sequence

convergent?

Find a vector perpendicular to the plane that passes through the points:p(1, 4, 6), Q(-2, 5, -1) and R(1. -1, 1)

Find the partial derivative of f(x, y) = x² + 2x³y² – 3y² + x + y at (1. 2)

Find the local maximum and minimum values, saddle points of f(x,y) = x⁴ + y⁴ – 4xy + 1