Calculus: Angle Between Curves Quiz

Questions and Answers

What is the radius of curvature for the curve $x^3 + y^3 = 3ay$ at the point $(\frac{3a}{2}, \frac{3a}{2})$?

$\frac{3}{2a}$

$\frac{3a^2}{2}$ (correct)

$\frac{a^2}{3}$

$3a$

For the cardioid $r = a(1 + \cos \theta)$, what is the relationship of the tangents at $\theta = \frac{3\pi}{2}$ and $\theta = \pi$?

There is no defined relationship between the tangents.

Both tangents are parallel to the initial line.

Both tangents are perpendicular to the initial line.

The tangent at $\theta = \frac{3\pi}{2}$ is parallel, and at $\theta = \pi$ is perpendicular. (correct)

What type of curve is defined by the equation $r^2 \sec^2 \theta = a^2$?

Hyperbolic

Parabolic

Elliptic

Cardioid (correct)

Which of the following expressions represents the polar form of the radius of curvature?

$\rho = \frac{r^3}{(\frac{d^2 r}{d\theta^2} + r)^{3/2}}$ Signup and view all the answers

At what point does the curve $y^2 = a^2(a - x)$ meet the x-axis?

(a, 0) Signup and view all the answers

Which of the following is true regarding the angle between the curves $r = a\cos \theta$ and $r = 2$?

The angle is determined by the points of intersection. Signup and view all the answers

What is the primary challenge in deriving the radius of curvature in polar form?

Complexity of the differentiation process Signup and view all the answers

In the derivation of the radius of curvature for a curve, which factor is not considered important?

Length of the curve Signup and view all the answers

Which equation represents a circle centered at the point (a, b) with radius k?

(x - a)² + (y - b)² = k² Signup and view all the answers

What is the locus of the center of curvature for a given curve?

An evolute Signup and view all the answers

At any point P(x, y) on the curve y = f(x), how does the center of curvature relate to the function?

y̅ = y + (y² / (1 + y²)) Signup and view all the answers

Given u = log(x² + y² + z²), what is the partial derivative of u with respect to z?

2z/(x² + y² + z²) Signup and view all the answers

If u = x * y * log(x), what is ∂u/∂y?

x log(x) Signup and view all the answers

If f(x, y) has dy as a function of x, which of the following expressions is correct?

du = ∂u + ∂y dy Signup and view all the answers

Which of the following is NOT a form of a quadratic equation?

(x - a)² + (y - b)² = k Signup and view all the answers

For the function u = x² + y² + z², which of the following expressions represents its partial derivative with respect to x?

2x Signup and view all the answers

What is the equation of the parabola given in the content?

$x^2 = 4ay$ Signup and view all the answers

At which point should the Center of curvature be found for the curve $xy^2 + x^2y = 2$?

(1,1) Signup and view all the answers

Which condition must be shown for the evolute of the parabola $y^2 = 4ax$?

$27aY^2 = 4(X - 2a)^3$ Signup and view all the answers

What geometric shape is the center of curvature at the origin for the curve $x + y = ax^2 + by^2 + ex^3$?

Circle Signup and view all the answers

For the function $u = e^{ax+by} f(ax - by)$, what identity must be proven involving partial derivatives?

$b rac{ ext{d} u}{ ext{d} x} + a rac{ ext{d} u}{ ext{d} y} = 2ab u$ Signup and view all the answers

What relationship must be shown for the equation $z(x+y) = x^2 + y^2$ with respect to partial derivatives?

$rac{ ext{d} z}{ ext{d} z^2} = 4[1 - rac{ ext{d} x}{ ext{d} y}]$ Signup and view all the answers

What formula must be verified for the second derivatives when $u = f(r)$ where $r^2 = x^2 + y^2$?

$rac{ ext{d}^2 u}{ ext{d} x^2} + rac{ ext{d}^2 u}{ ext{d} y^2} = f''(r) + f'(r) rac{1}{r}$ Signup and view all the answers

Study Notes

Curves and Angles

Find the angle between the curves represented by polar equations ( r = a \cos \theta ) and ( r = 2 ).
Determine the angle between ( r^n \cos n\theta = a^n ) and ( r^n \sin n\theta = b^n ).
Calculate the angle between ( r = a\theta ) and ( r = \frac{a}{1+\theta} \frac{1}{1+\theta^2} ).

Cardioids and Tangent Lines

Show that tangents to the cardioid defined by ( r = a(1 + \cos \theta) ) at angles ( \theta = \frac{3\pi}{2} ) and ( \theta = \frac{\pi}{3} ):
- The tangent at ( \frac{3\pi}{2} ) is parallel to the initial line.
- The tangent at ( \frac{\pi}{3} ) is perpendicular to the initial line.

Radius of Curvature

Derive the radius of curvature in Cartesian coordinates.
Find radius of curvature for the curve ( x^3 + y^3 = 3axy ) at the point ( \left( \frac{3a}{2}, \frac{3a}{2} \right) ).
Calculate radius of curvature for ( y^2 = a^2 (a - x) ) where it meets the x-axis.
Derive the radius of curvature in polar coordinates.

Taylor and Maclaurin Series

Obtain Taylor series expansion of ( \log \cos x ) about ( x = 3 ) up to the 4th degree.
Derive Taylor series expansion of ( \tan x ) at ( x = 4 ) up to the 4th degree.
Obtain Maclaurin series expansion for ( y = \log(1 + \sin x) ) up to third degree terms.
Expand ( e^{\sin x} ) using Maclaurin series up to the term containing ( x^4 ).

Curvature and Evolutes

Locus of the center of curvature is termed as the evolute.
At any point ( P(x, y) ) on the curve ( y = f(x) ), the center of curvature coordinate is given by ( \bar{y} = y \pm \frac{y^2}{1 + y^2} ).

Engineering Mathematics Principles

Apply concepts of partial differentiation to find results related to functions involving multiple variables.
Prove that for a specified function ( u ), the relationship ( b \frac{\partial u}{\partial x} + a \frac{\partial u}{\partial y} = 2ab u \frac{\partial u}{\partial u} ) holds true.
Show relationships and derive equations related to ( z(x+y) = x^2 + y^2 ).

Centre of Curvature

Find the coordinates of the center of curvature at any point of the parabola defined by ( x^2 = 4ay ).
Calculate the center of curvature for the curve ( xy^2 + x^2y = 2 ) at the point (1, 1).
Show that the circle of curvature at origin for ( x + y = a x^2 + b y^2 + ex^3 ) follows the equation ( (a + b)(x^2 + y^2) = 2(x + y) ) and establish the evolute of the parabola ( y^2 = 4ax ).

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Description

Test your knowledge on finding angles between various polar curves such as 𝑟 = 𝑎𝑐𝑜𝑠𝜃 and 𝑟 = 2, and more complex forms. This quiz covers fundamental concepts in calculus related to polar coordinates and tangents. Prepare to apply your understanding of derivatives and polar equations.