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Math Assignment: Theorems & Curves

This document contains 12 problems related to calculus topics including: 1. Cauchy's Mean Value Theorem and Taylor's Theorem to approximate trigonometric functions. 2. Finding Taylor series expansions for specific functions around given points. 3. Approximating a function using a Taylor polynomial and estimating the error of approximations. 4. Sketching and tracing planar curves defined parametrically or implicitly.

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95 views2 pages

Math Assignment: Theorems & Curves

This document contains 12 problems related to calculus topics including: 1. Cauchy's Mean Value Theorem and Taylor's Theorem to approximate trigonometric functions. 2. Finding Taylor series expansions for specific functions around given points. 3. Approximating a function using a Taylor polynomial and estimating the error of approximations. 4. Sketching and tracing planar curves defined parametrically or implicitly.

Uploaded by

aryan richhariya
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Visvesvaraya National Institute of Technology, Nagpur

Department of Mathematics
Mathematics-I-MAL-101

Assignment-II
Topics: Cauchy’s Mean Value Theorem, Taylor’s Theorem and Trace the curves.

(1) Using Cauchy Mean value theorem, show that


x2
(a) 1 − < cos x for x ̸= 0.
2!
x3
(b) x − < sin x for x > 0.
3!
x2 x 4
(c) cos x < 1 − + for x ̸= 0.
2! 4!
x 3 x5
(d) sin x < x − + for x ≥ 0.
3! 5!
(2) Find the Taylor series for,
(a) f (x) = sin x, about x = 0
(b) f (x) = x4 + x2 + 1, about x = −2.
√
(3) Approximate the function f (x) = 3 x by a Taylor polynomial of degree 2 at the
point a = 8. How accurate is the approximation when 7 ≤ x ≤ 9?

x3 x5
(4) What is the maximum error possible in using the approximation sin x ≈ x− +
3! 5!
when −0.3 ≤ x ≤ 0.3? Use this approximation to find sin 12◦ correct to six deci-
mal places. For what values of x is this approximation accurate to within 0.00005?

(5) Show that the Taylor’s series generated by f (x) = ex at x = 0 converges to f (x)
for every real value of x.

(6) Show that the Maclaurin series for cos x converges to cos x for every value of x.
5
(7) Verify Taylor’s theorem for f (x) = (1 − x) 2 with Lagrange form of remainder upto
2 terms in the interval [0, 1].
√
−1 (x) < 1, for 0 ≤ x < 1.
ln(1+x)
(8) Show that 1−x 1+x
< sin

(9) Show that, for all x ≥ 0


x2 x2 x
1+x+ ≤e ≤1+x+ e .
x
2 2
(10) Sketch the following curves.
(i) y 2 (a + x) = x2 (3a − x)
2
(ii) f (x) = x2x
2 −1 .
2 1
(iii) f (x) = x 3 (6 − x) 3 .
(iv) x2 y 2 = a2 (y 2 − x2 )
3
(v) y = x2a+a2 .
(11) Trace the following curves,
(i) r2 = a2 sin 2θ
(ii) r = a cos 3θ
(iii) r = 2(1 − sin θ)
(iv) r = a sin 2θ
(v) r = a(1 + sin θ)

(12) Trace the following curves,


(i) x = a(θ − sin θ) ; y = a(1 − cos θ)
(ii) x = a cos3 (t) ; y = b sin3 (t)
3at 3at2
(iii) x = 1+t 3 ; y = 1+t3

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