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Exercise 5.2 - Chapter 5 Continuity & Differentiability class 12 ncert solutions Maths - SaraNextGen [2024]


Question 1:

Differentiate the functions with respect to x.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6853/Chapter%205_html_m4d909628.gif

Answer:

Let f(x)=sinx2+5, ux=x2+5, and v(t)=sint

 

Then, vou=vux=vx2+5=tanx2+5=f(x)

Thus, f is a composite of two functions.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6853/Chapter%205_html_m76e301f4.gif

Alternate method

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6853/Chapter%205_html_7e76c191.gif

Question 2:

Differentiate the functions with respect to x.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6856/Chapter%205_html_m73e47a83.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6856/Chapter%205_html_m26cdca88.gif

Thus, is a composite function of two functions.

Put t = u (x) = sin x

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6856/Chapter%205_html_4ceae8bd.gif

By chain rule,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6856/Chapter%205_html_m119d5d9e.gif

Alternate method

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6856/Chapter%205_html_664cfa69.gif

Question 3:

Differentiate the functions with respect to x.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6858/Chapter%205_html_me9e6b3b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6858/Chapter%205_html_m4ff198ec.gif

Thus, is a composite function of two functions, u and v.

Put t = u (x) = ax + b

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6858/Chapter%205_html_7838a27b.gif

Hence, by chain rule, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6858/Chapter%205_html_e4e4928.gif

Alternate method

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6858/Chapter%205_html_5e6626a8.gif

Question 4:

Differentiate the functions with respect to x.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6860/Chapter%205_html_m39f8d27b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6860/Chapter%205_html_714e4461.gif

Thus, is a composite function of three functions, u, v, and w.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6860/Chapter%205_html_m27e732b4.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6860/Chapter%205_html_6bc2354c.gif

Hence, by chain rule, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6860/Chapter%205_html_21a687a3.gif

Alternate method

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6860/Chapter%205_html_2ba7a7d4.gif

Question 5:

Differentiate the functions with respect to x.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6862/Chapter%205_html_m74aa0a8b.gif

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6862/Chapter%205_html_85ef1b7.gif , where g (x) = sin (ax + b) and

h (x) = cos (cx d)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6862/Chapter%205_html_55c8a329.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6862/Chapter%205_html_2129c4ad.gif

∴ is a composite function of two functions, u and v.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6862/Chapter%205_html_56dd424a.gif

Therefore, by chain rule, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6862/Chapter%205_html_m5cf33a7.gif

h is a composite function of two functions, p and q.

Put y = p (x) = cx d

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6862/Chapter%205_html_71b91de8.gif

Therefore, by chain rule, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6862/Chapter%205_html_7b73ba25.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6862/Chapter%205_html_695dd5a1.gif

Question 6:

Differentiate the functions with respect to x.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6865/Chapter%205_html_5333257e.gif

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6865/Chapter%205_html_5333257e.gif .

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6865/Chapter%205_html_2f2abb3d.gif

Question 7:

Differentiate the functions with respect to x.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6866/Chapter%205_html_728931d9.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6866/Chapter%205_html_231350d7.gif

Question 8:

Differentiate the functions with respect to x.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6869/Chapter%205_html_m69c04209.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6869/Chapter%205_html_38a98fe8.gif

Clearly, is a composite function of two functions, and v, such that

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6869/Chapter%205_html_m18ee6d2d.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6869/Chapter%205_html_7147c9af.gif

By using chain rule, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6869/Chapter%205_html_me24073e.gif

Alternate method

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6869/Chapter%205_html_61c17f97.gif

Question 9:

Prove that the function given by

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6873/Chapter%205_html_m1a89e7ec.gif  is notdifferentiable at x = 1.

Answer:

The given function ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6873/Chapter%205_html_m1a89e7ec.gif

It is known that a function f is differentiable at a point x = c in its domain if both

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6873/Chapter%205_html_m246fba49.gif are finite and equal.

To check the differentiability of the given function at x = 1,

consider the left hand limit of f at x = 1

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6873/Chapter%205_html_m169cbd00.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6873/Chapter%205_html_m274af491.gif

Since the left and right hand limits of f at x = 1 are not equal, f is not differentiable at x = 1

Question 10:

Prove that the greatest integer function defined byhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6875/Chapter%205_html_2ba7aba7.gif is not

differentiable at x = 1 and x = 2.

Answer:

The given function f ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6875/Chapter%205_html_2ba7aba7.gif

It is known that a function f is differentiable at a point x = c in its domain if both

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6875/Chapter%205_html_m246fba49.gif are finite and equal.

To check the differentiability of the given function at x = 1, consider the left hand limit of f at x = 1

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6875/Chapter%205_html_5dc0d8f0.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6875/Chapter%205_html_7bb8909e.gif

Since the left and right hand limits of f at x = 1 are not equal, f is not differentiable at

x = 1

To check the differentiability of the given function at x = 2, consider the left hand limit

of f at x = 2

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6875/Chapter%205_html_6d34caac.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6875/Chapter%205_html_14d07761.gif

Since the left and right hand limits of f at x = 2 are not equal, f is not differentiable at x = 2

Also Read : Exercise-5.3-Chapter-5-Continuity-&-Differentiability-class-12-ncert-solutions-Maths

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