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Problem Solving Approach in Calculus-Part-I
Dhrubajyoti Mandal
Email: dhurbajyoti@nitsikkim
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in

Abstract
This is a collection of a few problems of College Level Calculus
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I request you to go through the note once
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1
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Which of the following is true ?
a) f is one-one in the interval [-1,1]
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c) f is NOT one-one in the interval [-4,0]
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Answer: First notice that f is a continuous function
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Now let us think when this kind of
situation happens
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Then there exist a neighbourhood of x = a say (a − δ, a + δ) where on both sides of x = a the value of the function
decreases (or increases) continuously if x = a is a local minima (or maxima)
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So f must not be one-one in
(a − δ, a + δ), in fact f must not be one-one in any interval containing (a − δ, a + δ)
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So first let us find out the local maxima and minima of f
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Now, x = 0 and x = 3 are interior points of
[−1, 1] and [2, 4] respectively
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Therefore options (a) and (b) are incorrect
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So [−4, 0] contains no local maxima or minima as its interior
point
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Lastly, [0, 4] contains the local minima x = 3 as its interior point
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Therefore the option (d) is the correct answer
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Question Suppose {an } be a sequence of positive real numbers
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n→∞

an+1
an
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Now
lim

n→∞

an+1
= l < 1 =⇒ ln < 1, ∀n
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Therefore
n→∞

lim an = 0
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Question: Let S is the sum of a convergent series
convergent ? If convergent then find the sum in terms of S,
Answer: As

∞
P

∞
P

an
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Then is

n=1
a1 , a2
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Again

n=1
N
X

tn = 3

n=1

Therefore

tn

n=1

an converges therefore the sequence N-th partial sum

n=1

∞
P

N
X

an − (2a1 + a2 )
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So

∞
X

tn = 3S − 2a1 − a2

n=1

4
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Which of the following statements are true?
a) There exist a continuous function f : [a b] → (a b) such that f is one-one
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c) There exist a continuous function f : (a b) → [a b] such that f is one-one
d) There exist a continuous function f : (a b) → [a b] such that f is onto

Answer: If [a b] and [c d] are two different closed and bounded intervals then there always exist a map f : [a b] →
[c d] defined by
f (x) = c1 x + c2
where c1 , c2 are functions of a, b, c, d and can be determined by imposing conditions like f (a) = c, f (b) = d
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Then it can be easily seen that this map is one-one (Can you see it!

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