calculus tutorial

Indian Institute of Technology Guwahati Department of Mathematics MA 101 – Mathematics-I Tutorial Sheet-4 Calculus Date: 02-Nov-2012 Topics Covered: • Limit of a function • Continuity, Intermedia Value Theorem • Differentiability - Definition and Properties 1. If f : R → R be a function such that lim x→c f (x)= L then for p q ∈ Q, show that lim x→c (f (x)) p q = L p q . 2. Let f : (1, 2) → R be such that -16 - sin 2 (x - 2) <f (x) < x 2 |4x - 8| x - 2 for x ∈ (1, 2). Show that lim x→2 f (x) exists. Find the limit. 3. Let f : (0, ∞) → R be such that f (x)= 0 if x is irrational 1 q if x = p q and gcd(p, q)=1. Show that f is continuous at each positive irrational and discontinuous at each positive rational. 4. Let n ∈ N and 0 <a ∈ R. Show that there is unique 0 <b ∈ R such that b n = a. 5. Let S ⊂ R be a compact set and f : S → R be a continuous function. Then f attains its bounds (maximum and minimum) in S . i.e. there exists r, s ∈ S such that f (r) = inf {f (x) | x ∈ S } f (s) = sup{f (x) | x ∈ S }. 6. Show that the equation 17x 7 - 19x 5 - 1 = 0 has a solution p which satisfies -1 <p< 0. 7. Does there exists a continuous function f : [1, 10] → R such that f (1) = 0,f (10) = 11 and f ([1, 10]) ⊂ [-1, 0] ∪ [1, 11]. 8. Let f :[a, b] → R be continuous and one-to-one. If f (a) <f (b) then show that f is strictly increasing. 9. Let f,g : R → R be such that g ◦ f is defined. Let d = f (c). Suppose f is differentiable at c and g is differentiable at f (c)= d, then g ◦ f is differentiable at c. (g ◦ f ) 0 (c)= g 0 (f (c)) · f 0 (c). 10. If f :(a, b) → R is differentiable at c ∈ (a, b) then show that lim h→0 + f (c + h) - f (c - h) 2h exists and is equal to f 0 (c). Is the converse true? 11. Let f :[a, b] → R be differentiable. Show that f has the Intermediate Value Property on [a, b], that is, f assumes all values between f (a) and f (b).

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Transcript of calculus tutorial

Indian Institute of Technology Guwahati

Department of Mathematics

MA 101 Mathematics-I Tutorial Sheet-4Calculus Date: 02-Nov-2012

Topics Covered:

Limit of a function Continuity, Intermedia Value Theorem

Differentiability - Definition and Properties

1. If f : R R be a function such that limxc

f(x) = L then for pq Q, show that lim

xc(f(x))

pq = L

pq .

2. Let f : (1, 2) R be such that 16 sin2(x 2) < f(x) < x2|4x 8|x 2 for x (1, 2). Show

that limx2

f(x) exists. Find the limit.

3. Let f : (0,) R be such that f(x) ={

0 if x is irrational1q

if x = pq

and gcd(p, q) = 1.Show that f is

continuous at each positive irrational and discontinuous at each positive rational.

4. Let n N and 0 < a R. Show that there is unique 0 < b R such that bn = a.5. Let S R be a compact set and f : S R be a continuous function. Then f attains its

bounds (maximum and minimum) in S. i.e. there exists r, s S such that

f(r) = inf{f(x) | x S}f(s) = sup{f(x) | x S}.

6. Show that the equation 17x7 19x5 1 = 0 has a solution p which satisfies 1 < p < 0.7. Does there exists a continuous function f : [1, 10] R such that f(1) = 0, f(10) = 11 and

f([1, 10]) [1, 0] [1, 11].8. Let f : [a, b] R be continuous and one-to-one. If f(a) < f(b) then show that f is strictly

increasing.

9. Let f, g : R R be such that g f is defined. Let d = f(c). Suppose f is differentiable at cand g is differentiable at f(c) = d, then g f is differentiable at c.

(g f)(c) = g(f(c)) f (c).

10. If f : (a, b) R is differentiable at c (a, b) then show that limh0+

f(c + h) f(c h)2h

exists

and is equal to f (c). Is the converse true?

11. Let f : [a, b] R be differentiable. Show that f has the Intermediate Value Property on[a, b], that is, f assumes all values between f(a) and f(b).