Indian Institute of Space Science and Technology
Thiruvananthapuram
MA111 - Calculus
Assignment∗
1. Let f(x) = tan π
4+x1/x ,for x6= 0 and kfor x= 0.For what values of k, f (x) is
continuous at x= 0.
2. Verify the applicability of Rolle’s theorem for the function
f(x) =
x2+ 1,0≤x≤1
3−x1< x ≤2.
3. Let fbe a differentiable function on IR and |f(x)−f(y)| ≤ (x−y)2. Then show that fis
constant.
4. For what values of a, b, c, d does the function f(x) = ax3+bx2+cx +d, x ∈Rhave a local
minimum at -1, a point of inflection at 1 and satisfy f(−1) = 10, f(1) = −6?
5. Using applications of Mean Value Theorem, prove that
x2
2(1 + x)< x −ln(1 + x)<x2
2, x > 0.
6. Find the first three nonzero terms of the Maclaurin series of the function
f(x) = (sin x) ln(1 + x) and the values of x for which the series converges absolutely.
7. Let fbe the function defined by f(x, y) = 3x2y−y3
x2+y2for (x, y)6= 0 and f(0,0) = 0.[5]
(a) Verify the continuity of fat (0,0).
(b) Find the value of fy(x, 0) for any x6= 0.
(c) Verify the continuity of fyat (0,0).
8. Show that the function f(x, y) =
(x2+y2) cos 1
px2+y2!,(x, y)6= (0,0)
0,(x, y) = (0,0)
is differentiable at (0,0) but fx, fyare not continuous at (0,0).
9. Find the global maximum and minimum values of the function f(x, y) = 2 +2x+ 4y−x2−y2
on the triangular region in the first quadrant bounded by the lines x= 0, y = 0,and
y= 9 −x.
∗Submission deadline: 20/02/2022
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