Name: Vectors and Their Operations
Uploaded: 2019-07-19
Duration: 10 min 16 s
Description: Do you know vector is a term that refers colloquially to some quantities? There are physical quantities which are vectors. Let's understand them in detail.

Question

Draw a graph from the following data. Draw tangents at $x = 2, 4, 6$ and $8 .$ Find the slopes of these tangents. Verify that the curve drawn is $y = 2 x^{2}$ and the slope of tangent is $\tan θ = \frac{d y}{d x} = 4 x$ .

$\begin{matrix} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ y & 2 & 8 & 18 & 32 & 50 & 72 & 98 & 128 & 162 & 200 \end{matrix}$

Accepted Answer

The above diagram shows the curve plotted for $y = 2 x^{2}$ with the tangent making an angle $8 .$ with the positive $x$ -axis with slope $y = 2 x^{2}$ .

The first tangent at $(2, 4)$ makes an angle $θ_{1}$ with the positive $x$ -axis at $(1, 0),$ the second at $(4, 32)$ makes $θ_{2}$ at $(2, 0),$ the third at $(6, 72)$ makes $θ_{3}$ at $(3, 0)$ and the fourth at $(8, 128)$ makes an angle $θ_{4}$ at the point $(4, 0) .$

Clearly, the slopes of these tangents taken in order are given by

$m_{1} = \tan θ_{1} = \frac{8}{2 - 1} = 8$

$8 .$

$m_{3} = \tan θ_{3} = \frac{72}{6 - 3} = 24$

$m_{4} = \tan θ_{4} = \frac{128}{8 - 4} = 32$

Putting the given abscissa in the equation $y = 2 x^{2},$ we get

$y_{1} = 2 {(1)}^{2} = 2, y_{2} = 2 {(2)}^{2} = 8, y_{3} = 2 {(3)}^{2} = 18, y_{4} = 2 {(4)}^{2} = 32,$ and so on. Since, the given abscissa, on putting in the given equation of the curve, yield the given respective ordinates, the curve drawn is certainly $y = 2 x^{2} .$

Also, $\frac{d y}{d x} = \frac{d (2 x^{2})}{d x} = 4 x$

$\frac{d y}{d x}$ at the point $(2, 8)$ is $4 x = 4 \times 2 = 8 = m_{1} = \tan θ_{1}$ .

Similarly,

$\frac{d y}{d x}$ at the point $(4, 32)$ is $4 x = 4 \times 4 = 16 = m_{2} = \tan θ_{2}$ .

$\frac{d y}{d x}$ at the point $(6, 72)$ is $4 x = 4 \times 6 = 24 = m_{3} = \tan θ_{3}$ .

$\frac{d y}{d x}$ at the point $(8, 128)$ is $4 x = 4 \times 8 = 32 = m_{4} = \tan θ_{4}$ .

Hence, the slope of the tangent is given by the formula $\frac{d y}{d x} = \tan θ$ .

Important Questions on Physics and Mathematics

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Draw a graph from the following data. Draw tangents at x=2,4,6 and 8. Find the slopes of these tangents. Verify that the curve drawn is y=2x2 and the slope of tangent is tanθ=dydx=4x. x12345678910y281832507298128162200

Important Questions on Physics and Mathematics

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