Question

19-22: (a) Find the differential \(dy\) and (b) evaluate \(dy\) for the given values of x and \(dx\).

20. \(y = \cos \pi x,{\rm{ }}x = \frac{1}{3},{\rm{ }}dx = - 0.02\)

Short answer

a) The differential of the function is \(dy = - \pi \sin \pi xdx\).

b) The value of \(dy\) is 0.054.

Step-by-step solution

Definition of differentials

The equation establishes the differential \(dy\)with respect to \(dx\) as shown below:

\(dy = f'\left( x \right)dx\)

Where \(dx\) represent the independent variableand \(dy\) represent the dependent variable.

Determine the differential of the function

a)

The function is \(y = f\left( x \right) = \cos \pi x\).

Obtain the differential of the function as shown below:

\(\begin{aligned}dy &= f'\left( x \right)dx\\dy &= - \sin \pi x \cdot \pi dx\\dy &= - \pi \sin \pi xdx\end{aligned}\)

Thus, the differential of the function is \(dy = - \pi \sin \pi xdx\).

Evaluate the value of \(dy\)

b)

Substitute\(x = \frac{1}{3}\)and\(dx = - 0.02\)to obtain\(dy\)as shown below:

\(\begin{aligned}dy &= - \pi \sin \frac{\pi }{3}\left( { - 0.02} \right)\\ &= \pi \frac{{\sqrt 3 }}{2}\left( { - 0.02} \right)\\ &= 0.01\pi \sqrt 3 \\ \approx 0.054\end{aligned}\)

Thus, the value of \(dy\) is 0.054.