Questions on binomial expansion exponential functions and logarithmic function :
Here we are going to see some practice questions on finding binomial expansion, expansion of exponential function and logarithmic functions.
(1) Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
(i) 1/(5 + x) Solution
(ii) 2/(3 + 4x)2 Solution
(iii) (5 + x2)2/3 Solution
(iv) (x + 2)-2/3 Solution
(2) Find 3√1001 approximately (two decimal places).
(3) Prove that 3√(x3 + 6) − 3√(x3 + 3) is approximately equal to 1/x2 when x is sufficiently large. Solution
(4) Prove that √(1−x)/(1+x) is approximately equal to 1 − x + x2 when x is very small. Solution
(5) Write the first 6 terms of the exponential series
(i) e5x Solution
(ii) e−2x Solution
(iii) e(1/2)x Solution
(6) Write the first 4 terms of the logarithmic series
(i) log (1 + 4x) Solution
(ii) log(1 − 2x) Solution
(iii) log [(1+3x)/(1−3x)] Solution
(iv) log [(1-2x)/(1+2x)] Solution
(7) If y = x + x2/2 + x3/3 + x4/4 + · · · , then show that x = y − y2/2! + y3/3! − y4/4! + · · · . Solution
(8) If p − q is small compared to either p or q, then show that
(9) Find the coefficient of x4 in the expansion of (3−4x+x2)/e2x . Solution
(10) Find the value of
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