Rate Of Change Of Quantities

Step 1: Concept
Differentiate

f(x)

and equate the resulting expression to

x~cos~x

.
Step 2: Analysis

f'(x)%20%3D%20a~sin~x%20%2B%20(ax%2Bb)cos~x%20%2B%20c~cos~x%20-%20(cx%2Bd)sin~x

. Rearrange:

f'(x)%20%3D%20(a-cx-d)sin~x%20%2B%20(ax%2Bb%2Bc)cos~x

.
Step 3: Evaluation
Equate coefficients with

x~cos~x

: Coefficient of

x~cos~x%20%5CRightarrow%20a%3D1

. Coefficient of

x~sin~x%20%5CRightarrow%20c%3D0

. Constant for

cos~x%20%5CRightarrow%20b%2Bc%3D0%20%5CRightarrow%20b%3D0

. Constant for

sin~x%20%5CRightarrow%20a-d%3D0%20%5CRightarrow%20d%3D1

.
Step 4: Conclusion
Thus,

a%3D1%2C%20b%3D0%2C%20c%3D0%2C%20d%3D1

, which corresponds to

b%3Dc%3D0

and

a%3Dd%3D1

.
Final Answer: (a)

Concept: Volume of cylinder:

V%20%3D%20%5Cpi%20r%5E2%20h

.

Step 1: Differentiate with respect to time

t

%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cpi%20%5Cleft(%202r%20%5Cfrac%7Bdr%7D%7Bdt%7D%20%5Ccdot%20h%20%2B%20r%5E2%20%5Cfrac%7Bdh%7D%7Bdt%7D%20%5Cright)

Step 2: Substitute given values.

%5Cfrac%7Bdr%7D%7Bdt%7D%20%3D%202%2C%5Cquad%20%5Cfrac%7Bdh%7D%7Bdt%7D%20%3D%20-3%2C%5Cquad%20r%20%3D%203%2C%5Cquad%20h%20%3D%205

%5Cfrac%7BdV%7D%7Bdt%7D%20%3D%20%5Cpi%20%5Cleft(%202%20%5Ctimes%203%20%5Ctimes%202%20%5Ctimes%205%20%2B%20(3)%5E2%20%5Ctimes%20(-3)%20%5Cright)

%3D%20%5Cpi%20%5Cleft(%202%20%5Ctimes%203%20%5Ctimes%202%20%5Ctimes%205%20%3D%2060%2C%5Cquad%209%20%5Ctimes%20(-3)%20%3D%20-27%20%5Cright)

%3D%20%5Cpi%20(60%20-%2027)%20%3D%2033%5Cpi

Chapter Questions
2 MCQs