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Harshdeep K.

Mathematics Tutor

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Pre-Calculus

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Question:

Does the infinite geometric series converge or diverge? $$1+\frac{5}{2}+\frac{25}{4}+\frac{125}{8}+ ...$$ If the series converges, find its sum.

Harshdeep K.

Answer:

To determine if the Geometric Progression series converges or diverges, find the common ratio r. Find the common ratio r. Take the ratio of the first two terms. $$r=\frac{\frac{5}{2}}{1}$$ Since the series is geometric, the ratio between any two consecutive terms is $$\frac{5}{2}$$ . Determine if the series converges or diverges. Take the absolute value of r. $$|r|=\frac{5}{2}\geq1$$ Since |r|≥1, the series diverges. Its sum does not exist

Basic Math

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Question:

The length and breadth of a rectangular courtyard is $$75\ m$$ and $$32\ m$$. Find the cost of leveling it at the rate of $$$3$$ per $$m^2$$. Also, find the distance covered by a boy to take 4 rounds of the courtyard.

Harshdeep K.

Answer:

Length of the courtyard = $$75\ m$$ Breadth of the courtyard = $$32\ m $$ Perimeter of the courtyard = $$2 \times (75 + 32) m$$ = $$2 \times 107\ m$$ = $$214\ m$$ Distance covered by the boy in taking 4 rounds = 4 × perimeter of courtyard = $$4 \times 214$$ = $$856\ m$$ We know that area of the courtyard = length × breadth = $$75 × 32\ m^2$$ = $$2400\ m^2$$ For $$1\ m^2$$, the cost of levelling = $$$3$$ For $$2400\ m^2$$, the cost of levelling = $$$3 × 2400$$ = $$$7200$$

Calculus

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Question:

If f(x) and g(x) are differentiable functions such that $$f '(x) = 7x^2\ and \ g '(x) = 5 x ^3$$ then the limit $$\lim_{x\to2}\frac{(f(x) + g(x)) - (f(2) + g(2)) }{ (x - 2) }$$ is equal to

Harshdeep K.

Answer:

Solution: $$\lim_{x\to1}\frac{(f(x) + g(x)) - (f(2) + g(2)) }{ (x - 2) }$$ The given limit is the definition of the derivative of $$f(x) + g(x)\ at\ x = 2$$. The derivative of the sum is equal to the sum of the derivatives. Hence the given limit is equal to$$ f '(2) + g '(2) = (7*(2)^2) + (5*(2)^3)=28+40=68$$

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