Examine the continuity of the function
f(x) = x3 + 2x – 1 at x = 1
Find which of the functions is continuous or discontinuous at the indicated points:
at x = 2
at x = 4
at x = 0
Check continuity at x =a
f(x) = |x| + |x – 1| at x = 1
Find the value of k so that the function f is continuous at the indicated point:
Prove that the function f defined by
remains discontinuous at x=0, regardless the choice of k.
Find the values of a and b such that the function f defined by
is a continuous function at x = 4.
Given the function f(x) = Find the point of discontinuity of the composite function y = (f(x)).
Find all points of discontinuity of the function where
Show that the function f(x) = |sin x + cos x| is continuous at x = .
Examine the differentiability of f, where f is defined by
Show that f(x) = |x – 5| is continuous but not differentiable at x = 5.
A function satisfies the equation f(x + y) = f(x) f(y) for all x, y . Suppose that the function is differentiable at x = 0 and f’(0) = 2. Prove that f’(x) = 2f(x).
Differentiate each of the following w.r.t. x
log [log (log x5)]
sinn (ax2 + bx + c)
sinx2 + sin2x + sin2 (x2)
(sin x)cos x
sinmx . cosnx
(x + 1)2 (x + 2)3 (x + 3)4
and
Find of each of the functions expressed in parametric form in
If prove that
If x = asin 2t (1 + cos2t) and y = bcos2t, show that
if x = 3sint – sin3t, y = 3cost – cos 3t, find
Differentiate w.r.t. sinx.
Differentiate w.r.t. tan-1x when
Find when x and y are connected by the relation given
sec (x + y) = xy
tan-1 (x2 + y2) = a
(x2 + y2)2 = xy
If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, then show that
If yx = ey–x, prove that
If , show that
If x sin (a + y) + sin a cos (a + y) = 0, prove that
If y = tan-1x, find in terms of y alone.
Verify the Rolle’s theorem for each of the functions
f(x) = x (x – 1)2 in [0, 1].
f(x) = log (x2 + 2) – log3 in [– 1, 1].
f(x) = x(x + 3)e-x/2 in [–3, 0].
Discuss the applicability of Rolle’s theorem on the function given by
Find the points on the curve y = (cosx – 1) in [0, ], where the tangent is parallel to x-axis.
Using Rolle’s theorem, find the point on the curve y = x(x – 4), where the tangent is parallel to x-axis.
Verify mean value theorem for each of the functions given
f(x) = x3 – 2x2 – x + 3 in [0, 1]
f(x) = sinx – sin2x in
Find a point on the curve y = (x – 3)2,where the tangent is parallel to the chord joining the points (3, 0) and (4, 1).
Using mean value theorem, prove that there is point on the curve y = 2x2 – 5x + 3 between the points A(1, 0) and B(2, 1), where tangent is parallel to the chord A. Also, find that point.
Find the values of p and q so that
Is differentiable at x = 1.
If x = sint and y = sin pt, prove that
Find
If f(x) = 2x and g(x) = then which of the following can be a discontinuous function.
The function f(x) = is
The set of points where the function f given by f(x) = |2x – 1| sinx differentiable is
The function f(x) = cot x is discontinuous on the set
The function f(x) = e|x| is
If where , then the value of the function f at x = 0, so that the function is continuous at x = 0, is
If is continuous at then
Let f(x) = |sinx|. Then
If then is equal to
The derivative of cos-1(2x2 – 1) w.r.t. cos-1 x is
If x = t2, y = t3, then is
The value of c in Rolle’s theorem for the function f(x) = x3 – 3x in the interval
For the function the value of c for mean value theorem is
Fill in the blanks in each of the
An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is _______.
Derivative of x2 w.r.t. x3 is _______.
If f(x) = |cos x|, then f’ ______.
If f(x) = |cosx – sinx|, then f’ ____.
For the curve is _______.
State True or False for the statements
Rolle’s theorem is applicable for the function f(x) = |x – 1| in [0, 2].
If f is continuous on its domain D, then |f| is also continuous on D.
The composition of two continuous functions is a continuous function.
Trigonometric and inverse–trigonometric functions are differentiable in their respective domain.
If f .g is continuous at x = a, then f and g are separately continuous at x = a.
