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Madhava's Contributions to Calculus and Infinite Series

Madhava's Contributions to Calculus and Infinite Series

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Questions

Question 1 Multiple Choice (Single Answer)

Who is considered to be the founder of the Kerala school of astronomy and mathematics?

  1. Madhava of Sangamagrama
  2. Aryabhata
  3. Bhaskara II
  4. Brahmagupta
Question 2 Multiple Choice (Single Answer)

What is the name of the series that Madhava used to approximate the sine function?

  1. Madhava series
  2. Taylor series
  3. Fourier series
  4. Maclaurin series
Question 3 Multiple Choice (Single Answer)

What is the general formula for the Madhava series?

  1. $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
  2. $$\sin x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$
  3. $$\sin x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$
  4. $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
Question 4 Multiple Choice (Single Answer)

What is the name of the series that Madhava used to approximate the cosine function?

  1. Madhava series
  2. Taylor series
  3. Fourier series
  4. Maclaurin series
Question 5 Multiple Choice (Single Answer)

What is the general formula for the Madhava series for the cosine function?

  1. $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$
  2. $$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{4} - \frac{x^6}{6} + \cdots$$
  3. $$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{4} - \frac{x^6}{6} + \cdots$$
  4. $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$
Question 6 Multiple Choice (Single Answer)

What is the name of the series that Madhava used to approximate the arctangent function?

  1. Madhava series
  2. Taylor series
  3. Fourier series
  4. Maclaurin series
Question 7 Multiple Choice (Single Answer)

What is the general formula for the Madhava series for the arctangent function?

  1. $$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$
  2. $$\arctan x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
  3. $$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$
  4. $$\arctan x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
Question 8 Multiple Choice (Single Answer)

What is the name of the series that Madhava used to approximate the pi?

  1. Madhava series
  2. Taylor series
  3. Fourier series
  4. Maclaurin series
Question 9 Multiple Choice (Single Answer)

What is the general formula for the Madhava series for pi?

  1. $$\pi = 4 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  2. $$\pi = 4 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  3. $$\pi = 4 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  4. $$\pi = 4 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
Question 10 Multiple Choice (Single Answer)

What is the name of the series that Madhava used to approximate the area of a circle?

  1. Madhava series
  2. Taylor series
  3. Fourier series
  4. Maclaurin series
Question 11 Multiple Choice (Single Answer)

What is the general formula for the Madhava series for the area of a circle?

  1. $$A = \pi r^2 = \frac{4r^2}{3} \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  2. $$A = \pi r^2 = \frac{4r^2}{3} \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  3. $$A = \pi r^2 = \frac{4r^2}{3} \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  4. $$A = \pi r^2 = \frac{4r^2}{3} \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
Question 12 Multiple Choice (Single Answer)

What is the name of the series that Madhava used to approximate the volume of a sphere?

  1. Madhava series
  2. Taylor series
  3. Fourier series
  4. Maclaurin series
Question 13 Multiple Choice (Single Answer)

What is the general formula for the Madhava series for the volume of a sphere?

  1. $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  2. $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  3. $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
  4. $$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$$
Question 14 Multiple Choice (Single Answer)

What is the name of the series that Madhava used to approximate the circumference of an ellipse?

  1. Madhava series
  2. Taylor series
  3. Fourier series
  4. Maclaurin series
Question 15 Multiple Choice (Single Answer)

What is the general formula for the Madhava series for the circumference of an ellipse?

  1. $$C = 4aE(e) = 4a \left(1 - \frac{e^2}{2} + \frac{e^4}{2\cdot4} - \frac{e^6}{2\cdot4\cdot6} + \cdots\right)$$
  2. $$C = 4aE(e) = 4a \left(1 - \frac{e^2}{2} + \frac{e^4}{2\cdot4} - \frac{e^6}{2\cdot4\cdot6} + \cdots\right)$$
  3. $$C = 4aE(e) = 4a \left(1 - \frac{e^2}{2} + \frac{e^4}{2\cdot4} - \frac{e^6}{2\cdot4\cdot6} + \cdots\right)$$
  4. $$C = 4aE(e) = 4a \left(1 - \frac{e^2}{2} + \frac{e^4}{2\cdot4} - \frac{e^6}{2\cdot4\cdot6} + \cdots\right)$$
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