Differentiation Short Questions (Question 22 - 25)

Question 22:
Given that  $y=\frac{3}{4}{x}^{2}$ , find the approximate change in x which will cause y to decrease from 48 to 47.7.

Solution:

Question 23:
Given that $y=15x+\frac{24}{{x}^{3}}$ ,

(a) Find the value of $\frac{dy}{dx}$ when x = 2,
(b) Express in terms of k, the approximate change in y when x changes from 2 to
2 + k, where k is a small change.

Solution:
(a)

(b)

Question 24:
If the radius of a circle increases from 4 cm to 4.01 cm, find the approximate increase in the area.

Solution:

Example 25:
Given that y =3t + 5t2 and x = 5t 1.
(a) Find $\frac{dy}{dx}$  in terms of x,
(b) If x increases from 5 to 5.01, find the small increase in t.

Solution:
$\begin{array}{l}y=3t+5{t}^{2}\\ \frac{dy}{dt}=3+10t\\ \\ x=5t-1\\ \frac{dx}{dt}=5\end{array}$
(a)
$\begin{array}{l}\frac{dy}{dx}=\frac{dy}{dt}×\frac{dt}{dx}\\ \frac{dy}{dx}=\left(3+10t\right)×\frac{1}{5}\\ \frac{dy}{dx}=\frac{3+10\left(\frac{x+1}{5}\right)}{5}←\overline{)\begin{array}{l}x=5t-1\\ t=\frac{x+1}{5}\end{array}}\\ \frac{dy}{dx}=\frac{3+2x+2}{5}\\ \frac{dy}{dx}=\frac{5+2x}{5}\end{array}$
(b)