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Questions and Answers
What is the radius of curvature for the curve $x^3 + y^3 = 3ay$ at the point $(\frac{3a}{2}, \frac{3a}{2})$?
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 $\theta = \pi$?
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 $r^2 \sec^2 \theta = a^2$?
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?
Which of the following expressions represents the polar form of the radius of curvature?
At what point does the curve $y^2 = a^2(a - x)$ meet the x-axis?
At what point does the curve $y^2 = a^2(a - x)$ meet the x-axis?
Which of the following is true regarding the angle between the curves $r = a\cos \theta$ and $r = 2$?
Which of the following is true regarding the angle between the curves $r = a\cos \theta$ and $r = 2$?
What is the primary challenge in deriving the radius of curvature in polar form?
What is the primary challenge in deriving the radius of curvature in polar form?
In the derivation of the radius of curvature for a curve, which factor is not considered important?
In the derivation of the radius of curvature for a curve, which factor is not considered important?
Which equation represents a circle centered at the point (a, b) with radius k?
Which equation represents a circle centered at the point (a, b) with radius k?
What is the locus of the center of curvature for a given curve?
What is the locus of the center of curvature for a given curve?
At any point P(x, y) on the curve y = f(x), how does the center of curvature relate to the function?
At any point P(x, y) on the curve y = f(x), how does the center of curvature relate to the function?
Given u = log(x² + y² + z²), what is the partial derivative of u with respect to z?
Given u = log(x² + y² + z²), what is the partial derivative of u with respect to z?
If u = x * y * log(x), what is ∂u/∂y?
If u = x * y * log(x), what is ∂u/∂y?
If f(x, y) has dy as a function of x, which of the following expressions is correct?
If f(x, y) has dy as a function of x, which of the following expressions is correct?
Which of the following is NOT a form of a quadratic equation?
Which of the following is NOT a form of a quadratic equation?
For the function u = x² + y² + z², which of the following expressions represents its partial derivative with respect to x?
For the function u = x² + y² + z², which of the following expressions represents its partial derivative with respect to x?
What is the equation of the parabola given in the content?
What is the equation of the parabola given in the content?
At which point should the Center of curvature be found for the curve $xy^2 + x^2y = 2$?
At which point should the Center of curvature be found for the curve $xy^2 + x^2y = 2$?
Which condition must be shown for the evolute of the parabola $y^2 = 4ax$?
Which condition must be shown for the evolute of the parabola $y^2 = 4ax$?
What geometric shape is the center of curvature at the origin for the curve $x + y = ax^2 + by^2 + ex^3$?
What geometric shape is the center of curvature at the origin for the curve $x + y = ax^2 + by^2 + ex^3$?
For the function $u = e^{ax+by} f(ax - by)$, what identity must be proven involving partial derivatives?
For the function $u = e^{ax+by} f(ax - by)$, what identity must be proven involving partial derivatives?
What relationship must be shown for the equation $z(x+y) = x^2 + y^2$ with respect to partial derivatives?
What relationship must be shown for the equation $z(x+y) = x^2 + y^2$ with respect to partial derivatives?
What formula must be verified for the second derivatives when $u = f(r)$ where $r^2 = x^2 + y^2$?
What formula must be verified for the second derivatives when $u = f(r)$ where $r^2 = x^2 + y^2$?
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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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