Question 1:
If functions f and g are such that
f(x) = g(x) + k
where k is a constant, then
(A) f '(x) = g '(x) + k
(B) f '(x) = g '(x)
(C) None of the above
Answer :
(B). The derivative of a sum of two functions is equal to the sum of the derivatives of the two functions and also the derivative of constant is equal to zero.
Question 2:
If f(x) = g(u) and u = u(x) then
(A) f '(x) = g '(u)
(B) f '(x) = g '(u) . u '(x)
(C) f '(x) = u '(x)
(D) None of the above
Answer :
(B). The derivative of the composition of two functions is given by the chain rule.
Question 3:
lim [e^{ x} 1] / x as x approaches 0
is equal to
(A) 1
(B) 0
(C) is of the form 0 / 0 and cannot be calculated.
Answer :
(A). The definition of the derivative at x = a is given by
f '(a) = lim [f(x)  f(a)] / (x  a) as x approaches a.
The given limit is the derivative of e^{ x} at x = 0.
Question 4:
True or False. The derivative of [g(x)] ^{ 2} is equal to [g '(x)]^{ 2}.
Answer :
False. The derivative of [g(x)] ^{ 2} is equal to 2 g '(x) . g(x)].
Question 5:
True or False. The derivative of f(x) . g(x) is equal to f '(x) g(x) + f(x).g '(x).
Answer:
True.
Question 6:
If f(x) is a differentiable function such that f '(0) = 2, f '(2) = 3 and f '(5) = 7 then the limit
lim [f(x)  f(4)] / (x  4) as x approaches 4.
is equal to
(A) 2
(B) 3
(C) 7
(D) None of the above
Answer :
(D). The given limit is equal to f '(4).
Question 7:
If f(x) and g(x) are differentiable functions such that
f '(x) = 3 x and g '(x) = 2 x^{ 2}
then the limit
lim [(f(x) + g(x))  (f(1) + g(1))] / (x  1) as x approaches 1.
is equal to
(A) 5
(B) 0
(C) 20
(D) None of the above
Answer :
(A). The given limit is the definition of the derivative of f(x) + g(x) at x = 1. The derivative of the sum is equal to the sum of the derivatives. Hence the given limit is equal to f '(1) + g '(1) = 5.
Question 8:
Below is the graph of function f. This graph has a maximum point at B.
If xA, xB and xC are the x coordinates of points A, B and C respectively and f ' is the first derivative of f, then
(A) f '(xA) > 0 , f '(xB) > 0 and f '(xC) > 0
(B) f '(xA) > 0 , f '(xB) = 0 and f '(xC) > 0
(C) f '(xA) > 0 , f '(xB) = 0 and f '(xC) < 0
(D) f '(xA) < 0 , f '(xB) = 0 and f '(xC) > 0
Answer :
(C). f is increasing (f '(x) > 0) at point A, decreasing (f '(x) < 0) at C and has a maximum (f '(x) = 0) at B.
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