Find the length of the curve x a 2 3 + y b 2 3 = 1 .

Given that, the equation of the curve is x a 2 3 + y b 2 3 = 1 Now, the length of the curve is L = ∫ x = a x = b 1 + d y d x 2 d x Therefore, L = 4 × length of curve from x = 0 to x = a , so L = 4 × ∫ 0 a 1 + d y d x 2 d x Now, y b 2 3 = 1 - x a 2 3 ⇒ y b = 1 - x a 2 3 3 2 ⇒ y = b 1 - x a 2 3 3 2 Now, differentiate y with respect to x , then d y d x = b × 3 2 1 - x a 2 3 1 2 d d x 1 - x a 2 3 ⇒ d y d x = b × 3 2 1 - x a 2 3 1 2 2 3 x a - 1 3 - 1 a ⇒ d y d x = - b a x a - 1 3 1 - x a 2 3 1 2 ⇒ d y d x 2 = b 2 a 2 x a - 2 3 1 - x a 2 3 ⇒ d y d x 2 = b 2 a 2 x a - 2 3 - 1 Therefore, the length of the curve is L = 4 × ∫ 0 a 1 + b 2 a 2 x a - 2 3 - 1 d x Lets assume a = b , then L = 4 × ∫ 0 a 1 + x a - 2 3 - 1 d x ⇒ L = 4 × ∫ 0 a x a - 1 3 d x ⇒ L = 4 × 3 a 2 x a 2 3 0 a ⇒ L = 4 × 3 a 2 a a 2 3 - 0 ⇒ L = 4 × 3 a 2 = 6 a Hence, the required length of the curve is 6 a units.

Find the asymptotes of the following curve: f x = x 2 - 6 x - 1 x + 3

Given f x = x 2 - 6 x - 1 x + 3 For Horizontal asymptote: lim x → + ∞ x 2 - 6 x - 1 x + 3 = lim x → + ∞ x - 6 x - 1 x 1 + 3 x = ∞ lim x → - ∞ x 2 - 6 x - 1 x + 3 = lim x → - ∞ x - 6 x - 1 x 1 + 3 x = - ∞ So no horizontal asymptotes. For Vertical asymptote: x + 3 = 0 ⇒ x = - 3 lim x → - 3 + x 2 - 6 x - 1 x + 3 = ∞ lim x → - 3 - x 2 - 6 x - 1 x + 3 = - ∞ So x = - 3 is a vertical asymptote. For Slant asymptote: The polynomial in the numerator is a higher degree than the polynomial in the denominator. To find the slant asymptote you must divide the numerator by the denominator using either long division or synthetic division. x + 3 x 2 - 6 x - 1 x - 9 x 2 + 3 x - 9 x - 1 - 9 x - 27 26 By performing synthetic division we get f x = x - 9 + 26 x + 3 So y = x - 9 is a slant asymptote. So, given curve has a vertical asymptote x = - 3 and slant asymptote y = x - 9 .

Find the asymptotes of the following curve: f x = x 2 x + 1

Given curve is f x = x 2 x + 1 For Horizontal asymptote: lim x → + ∞ x 2 x + 1 = ∞ lim x → - ∞ x 2 x + 1 = - ∞ So no horizontal asymptotes. For Vertical asymptotes: Finding where the curve is undefined, that is when x + 1 = 0 ⇒ x = - 1 lim x → - 1 + x 2 x + 1 = ∞ lim x → - 1 - x 2 x + 1 = - ∞ So x = - 1 is a vertical asymptote. For Slant asymptotes: Since the polynomial in numerator has a higher degree than polynomial in the denominator, we have a slant asymptote. Slant asymptote occurs for a curve if lim x → ∞ f x - y = 0 , where f x is the curve and y is asymptote. ⇒ lim x → ∞ x 2 x + 1 - y = 0 ⇒ lim x → ∞ x 2 - 1 + 1 x + 1 - y = 0 ⇒ lim x → ∞ x + 1 x - 1 x + 1 + 1 x + 1 - y = 0 ⇒ lim x → ∞ x - 1 - y + lim x → ∞ 1 x + 1 = 0 ⇒ lim x → ∞ x - 1 - y = 0 This happens if, y = x - 1 So, slant asymptote is y = x - 1 . So, given curve has a vertical asymptote x = - 1 and slant asymptote y = x - 1 .

Find the asymptotes of the following curve: f x = 3 x x 2 + 2

Given curve is f x = 3 x x 2 + 2 . For Vertical asymptote: Solve x 2 + 2 = 0 ⇒ x 2 + 2 = 0 x 2 = - 2 so, curve is always defined So there are no vertical asymptotes. For Horizontal asymptote: lim x → + ∞ 3 x x 2 + 2 = lim x → + ∞ 3 1 + 2 x 2 = ± 3 lim x → - ∞ 3 x x 2 + 2 = lim x → - ∞ 3 1 + 2 x 2 = ± 3 So y = 3 and y = - 3 are horizontal asymptotes. For Slant asymptote: The Polynomial in numerator does not have a higher degree than the polynomial in the denominator. So no slant asymptotes.

MEDIUM

Earn 100

Show that the graph of the function $f (x) = x^{3}$ is symmetric with respect to the Origin

Important Questions on Applications of Differential Calculus

MEDIUM

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

The number of real solutions of

x^{7} + 5 x^{3} + 3 x + 1 = 0

is equal to _____.

EASY

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

The curve $y^{2} (x - 2) = x^{2} (1 + x)$ has

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Trace the curve $y = x^{3}$ .

MEDIUM

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Let

f : R - \{2, 6\} \to R

be real valued function defined as

f (x) = \frac{x + 2 x + 1}{x^{2} - 8 x + 12}

. Then range of

f

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Find the length of the curve ${(\frac{x}{a})}^{\frac{2}{3}} + {(\frac{y}{b})}^{\frac{2}{3}} = 1$ .

MEDIUM

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Show that the graph of the function

f (x) = 2 x^{3} - 4 x

is symmetric with respect to the Origin

MEDIUM

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

which of the following curve are symmetrical about x-axis or y-axis or origin $5 x^{2} + 7 y - 6 = 0$ .

MEDIUM

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Show that the graph of the function

f (x) = x^{3} + x

is symmetric with respect to the Origin

MEDIUM

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Show that the graph of the function

f (x) = x^{5}

is symmetric with respect to the Origin

MEDIUM

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Represent the union of two sets by Venn diagram for each of the following.

$X = {x | x$ is a prime number between $80$ and $100}$

$Y = {y | y$ is an odd number between $90$ and $100}$

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Find the asymptotes of the following curve:

f (x) = \frac{x^{2} - 6 x - 1}{x + 3}

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

which of the following curve are symmetrical about x-axis or y-axis or origin $y^{2} = x + 9$ .

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Which of the following options is not the asymptote of the curve

3 x^{3} + 2 x^{2} y - 7 x y^{2} + 2 y^{3} - 14 x y + 7 y^{2} + 4 x + 5 y = 0

MEDIUM

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Show that the graph of the function

f (x) = x^{5} + x^{3} + x

is symmetric with respect to the Origin

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Find the asymptotes of the following curve:

f (x) = \frac{x^{2}}{x + 1}

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

which of the following curve are symmetrical about x-axis or y-axis or origin $x y = 1$ .

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

Find the asymptotes of the following curve:

f (x) = \frac{3 x}{\sqrt{x^{2} + 2}}

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

which of the following curve are symmetrical about x-axis or y-axis or origin $x^{2} + y^{2} = 25$ .

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

If a curve is symmetrical about each of the co-ordinate axes then prove that is is also symmetrical about the origin. What can you say about the converse?

HARD

Mathematics>Differential Calculus>Applications of Differential Calculus>Symmetry and Asymptotes then Sketching of Curve

which of the following curve are symmetrical about x-axis or y-axis or origin $x^{2} = 3 y^{2} + 7$ .

Show that the graph of the function fx=x3 is symmetric with respect to the Origin

Important Questions on Applications of Differential Calculus

Learn and Prepare for any exam you want !

Show that the graph of the function $f (x) = x^{3}$ is symmetric with respect to the Origin