Name: Derivative of Functions in Parametric Form
Uploaded: 2019-07-19
Duration: 8 min 37 s
Description: Learn everything you'll need to know about derivatives of functions in parametric forms with this quick and easy lesson that'll cover the concept comprehensi...

Question

A curve has equation

y = 3 \cos 2 x + 4 \sin 2 x + 1

for

0 ⩽ x ⩽ π,

Find the exact value of the

x

-coordinate of the stationary point of the curve, giving your answer correct to

3

significant figures.

Accepted Answer

Given equation of the curve is $y = 3 \cos 2 x + 4 \sin 2 x + 1$ for $0 \leq x \leq π$ .

We need to find the $3$ -coordinate of the stationary point of the curve.

We know that the stationary points are the points where the gradient of the curve is zero.

So,

$\frac{d y}{d x} = \frac{d}{d x} (3 \cos 2 x + 4 \sin 2 x + 1)$

$= 3 \frac{d}{d x} (\cos 2 x) + 4 \frac{d}{d x} (\sin 2 x) + \frac{d}{d x} (1)$

$= 3 \times (- \sin 2 x) \times \frac{d}{d x} (2 x) + 4 \cos 2 x \times \frac{d}{d x} (2 x) + 0$

$= - 3 \sin 2 x \times 2 + 4 \cos 2 x \times 2$

$= - 6 \sin 2 x + 8 \cos 2 x . . . . (i)$

Now, for the stationary points, we put

$\frac{d y}{d x} = 0$

$\Rightarrow - 6 \sin 2 x + 8 \cos 2 x = 0$

$\Rightarrow 8 \cos 2 x = 6 \sin 2 x$

$\Rightarrow \tan 2 x = \frac{4}{3}$

$\Rightarrow 2 x = \tan^{- 1} (\frac{4}{3})$

$\Rightarrow x = 0.464 + n \frac{π}{2}, n \in Z$

But we need the values of $x$ such that $0 \leq x \leq π$ .

Hence, $x = 0.464, 0.464 + \frac{π}{2}$

$\Rightarrow x = 0.464, 2.03$ .

A curve has equation $y = 3 \cos 2 x + 4 \sin 2 x + 1$ for $0 ⩽ x ⩽ π,$ Find the exact value of the $x$ -coordinate of the stationary point of the curve, giving your answer correct to $3$ significant figures.

Important Questions on Differentiation

Learn and Prepare for any exam you want !

A curve has equation y=3cos2x+4sin2x+1 for 0⩽x⩽π, Find the exact value of the x -coordinate of the stationary point of the curve, giving your answer correct to 3 significant figures.

Important Questions on Differentiation

Learn and Prepare for any exam you want !

A curve has equation $y = 3 \cos 2 x + 4 \sin 2 x + 1$ for $0 ⩽ x ⩽ π,$ Find the exact value of the $x$ -coordinate of the stationary point of the curve, giving your answer correct to $3$ significant figures.