a.


b. Dry air is moving upward. If the ground temperature is 200 and the temperature at a height of 1km is 100 C, express the temperature T in 0C as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b)Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2km?

c. Find the equation of the tangent to the parabola y = x² + x + 1 at (0, 1)

a .A farmer has 2000 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area?

b.

If f(x) = x² – 1, g(x) = 2x + 1, find fog and gof and domain of fog.

Define continuity of a function at a point x = a. Show that the function f(x) = √(1 − x²) is continuous on the interval [−1, 1].

State Rolle’s theorem and verify the Rolle’s theorem for f(x) = x³ – x² – 6x + 2 in [0, 3]

Find the third approximation x₃ to the root of the equation f(x) = x³ – 2x – 7, setting x₁ = 2.

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” + y’ = 0, y(0) = 5, y(π/4) = 3

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) = x3 + 2x³y³ – 3y² + x + y, at (2,1)

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