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Application of Derivatives

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Long Answer Type

421.

Find the equation of tangent to the curve x = sin 3t, y = cos 2t, at t = $\frac{π}{4}$

422.
Show that the rectangle of maximum area that can be inscribed in a circle is a square.

Let a rectangle ABCD be inscribed in a circle with radius r.

$Let ∠ DBC = θ In right ∆ BCD : \frac{BC}{BD} = cosθ \Rightarrow BC = BD cosθ = 2 r cosθ \frac{CD}{BD} = sinθ \Rightarrow CD = BD sinθ = 2 r sinθ$

Let A be the area of the rectangle ABCD.

$∴ A = BC X CD \Rightarrow A = 2 r cosθ x 2 r sinθ = 4 r^{2} sinθcosθ \Rightarrow A = 2 r^{2} \sin 2 θ . . . . . . . (\sin 2 θ = 2 sinθcosθ) ∴ \frac{dA}{dθ} = 2 x 2 r^{2} \cos 2 θ = 4 r^{2} \cos 2 θ Now, \frac{dA}{dθ} = 0 \Rightarrow 4 r^{2} \cos 2 θ = 0 \Rightarrow \cos 2 θ = 0 \Rightarrow \cos 2 θ = \cos \frac{π}{2} \Rightarrow θ = \frac{π}{4} \frac{d^{2} A}{d^{2} θ} = - 2 x 4 r^{2} \sin 2 θ = - 8 r^{2} \sin 2 θ ∴ {(\frac{d^{2} A}{d^{2} θ})}_{(θ = \frac{π}{4})} = - 8 r^{2} \sin (2 x \frac{π}{4}) = - 8 r^{2} x 1 = - 8 r^{2} < 0$

Therefore, by the second derivative test, $θ = \frac{π}{4}$ is the point of the local maxima of A.

So, the area of the rectangle ABCD is the maximum at $θ = \frac{π}{4}$

Now, $θ = \frac{π}{4}$

$\Rightarrow \frac{CD}{BC} = \tan \frac{π}{4} \Rightarrow \frac{CD}{BC} = 1 \Rightarrow CD = BC \Rightarrow Rectangle ABCD is a square .$

Hence, the rectangle of the maximum area that can be inscribed in a circle is a square.

423.

Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height h is $\frac{1}{3}$ h.

424.

Find the equation of the tangent to the curve $y = \sqrt{3 x - 2}$ which is parallel to the line 4x – 2y + 5 = 0

425.

Find the intervals in which the function f given by $f (x) = x^{3} + \frac{1}{x^{3}}, x \neq 0$ is

(i) increasing

(ii) decreasing.

426.

Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.

427.

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m³. If building of tank costs Rs. 70 per square metre for the base and Rs. 45 per square metre for sides, what is the cost of least expensive tank?

428.

Find the equations of the normals to the curve y = x³ + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

429.

Find the values of x for which f(x) = [x(x - 2)]2 is an increasing function. Also, find the points on the curve where the tangent is parallel to x-axis.

430.

Show that the right circular cylinder, open at the top, and of given surface area and maximum volume is such that its height is equal to the radius of the base.

Previous Year Papers

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Long Answer Type

422. Show that the rectangle of maximum area that can be inscribed in a circle is a square.

422.
Show that the rectangle of maximum area that can be inscribed in a circle is a square.