Page 2 - DEHRADUN PUBLIC SCHOOL
P. 2
CALCULUS
Q.1 Evaluate the left hand and right hand limits of the function defined by
+1 x 2 if 0 x 1
f (x ) = at x = 1.
– if
2 x x 1
Q.2 Find the left hand and right had limits of the greatest integer function f(x) = {x} = greatest integer
less than or equal to x, at x = k, where k is an integer. Also, show that lim f (x ) does not exist.
x→ k
1 + x + 1 – x
Q.3 Evaluate lim
x→0 1 + x
x 2 – 4
Q.4 Evaluate : lim
x → 2 3x – 2 – x + 2
Q.5 Discuss the continuity of the function of given by f(x) = |x–1| + |x+2| at x = 1 and x = 2.
Q.6 Find the value of P so that f(x) is continuous
+
1 px – 1 – px
, if – x 0
f (x ) = x
2x +1 , if 0 x 1
x – 5
Q.7 Differentiate the following functions with respect to x by first principle :
2 + 3
x
(i) ax + (ii)
b
3 + 2
x
Q.8 If f(x) = mx +c and f(0) = f (0) = 1. What is f(2)?
′
Q.9 Differentiate the following functions with respect to x :
x
(i) e (ii) log7(log7 x)
e
n dy ny
2
y =
Q.10 If x + x + a 2 , then prove that =
dx x + a 2
2
dy
Q.11 If x + 2xy + y = 42, find .
3
2
dx
Q.12 Find the point on the curve y = 2x – 6x – 4 at which the tangent is parallel to the x - axis.
2
Q.13 Find a point on the curve y = (x – 3) , where the tangent is parallel to the line joining (4, 1)
2
and (3, 0).
Q.14 Find the equation of tangent line to y = 2x + 7 which is parallel to the line 4x – y + 3 = 0
2
Q.15 Find the equation of the tangents to the curve 3x – y = 8, which passes through the point
2
2
(4/3, 0).
1
Q.16 Find the equations of all lines of slope zero and that are tangent to the curve =
y
2 +
x 2 – x 3
Case-based question
Let f(x) be a differential function. Consider the curve y = f(x). Then derivative of the function
y = f(x) at point P(x1 ,y1) gives the slope of tangent at point P(x1 ,y1) .
On the basis of above information, answer the following questions.
x − 1 dy
(i) If =y , then find .
x + 2 dx
(ii) If y = 2x + x -1, then find the slope of tangent at x = 1.
4
6
1 dy k . f ( ' ) x
(iii) If y = , and = , then find the value of k .
f (x ) dx (xf ) 2
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