Questions and Answers
What is the radius of curvature for the curve $x^3 + y^3 = 3ay$ at the point $(\frac{3a}{2}, \frac{3a}{2})$?
For the cardioid $r = a(1 + \cos \theta)$, what is the relationship of the tangents at $\theta = \frac{3\pi}{2}$ and $\theta = \pi$?
What type of curve is defined by the equation $r^2 \sec^2 \theta = a^2$?
Which of the following expressions represents the polar form of the radius of curvature?
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At what point does the curve $y^2 = a^2(a - x)$ meet the x-axis?
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Which of the following is true regarding the angle between the curves $r = a\cos \theta$ and $r = 2$?
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What is the primary challenge in deriving the radius of curvature in polar form?
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In the derivation of the radius of curvature for a curve, which factor is not considered important?
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Which equation represents a circle centered at the point (a, b) with radius k?
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What is the locus of the center of curvature for a given curve?
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At any point P(x, y) on the curve y = f(x), how does the center of curvature relate to the function?
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Given u = log(x² + y² + z²), what is the partial derivative of u with respect to z?
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If u = x * y * log(x), what is ∂u/∂y?
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If f(x, y) has dy as a function of x, which of the following expressions is correct?
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Which of the following is NOT a form of a quadratic equation?
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For the function u = x² + y² + z², which of the following expressions represents its partial derivative with respect to x?
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What is the equation of the parabola given in the content?
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At which point should the Center of curvature be found for the curve $xy^2 + x^2y = 2$?
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Which condition must be shown for the evolute of the parabola $y^2 = 4ax$?
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What geometric shape is the center of curvature at the origin for the curve $x + y = ax^2 + by^2 + ex^3$?
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For the function $u = e^{ax+by} f(ax - by)$, what identity must be proven involving partial derivatives?
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What relationship must be shown for the equation $z(x+y) = x^2 + y^2$ with respect to partial derivatives?
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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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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.