Find the volume of the solid obtained by revolving the curve
X = a cos3 θ , y = a sin 3 θ about the y-axis.
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Verify that f , defined by f (x) = x3 + 4x +1on [1,5], satisfies Lagrange’s mean value theorem.
Find the maxima and minima, if any, for the function f given by
f (x) = x4 - 14x2 + 24x + 6.
Also find the values of the function at those points.
Calculate the approximate value of 
x 3 dx by taking 6 subintervals of equal length and applying Simpson’s rule.
A jogger runs from her home to a point A, which is 6 km away. For there 6 km, she begins by running at a constant speed till she reaches a hilly portion 2 km from her home. Here her speed slows down while she runs up the hill, which is a 1-km run. Then she speeds up while running down the hill. The last 2 km of the run are again at
constant speed. Draw a graph to show the jogger’s speed as a function of the distance from her home. Also find the range of this function.
Differentiate tan -1 [sin x - cos x /sin x + cos x]
with respect to 1/2 x.
If y = emtan -1 x , show that (1 + x2 ) y n+1 (2nx - m) yn n(n -1)y n -1 = 0.
Give an example of a function which is one to one but not onto, with reason.