Including the term $x^3$, hence use the expansion to determine the value of $\left(\frac{41}{40}\right)^{10}$.
Evaluate: $\tanh 4x$.
Answer (1)
Use binomial expansion to approximate ( 40 41 ) 10 as ( 1 + 40 1 ) 10 .
Expand ( 1 + x ) 10 up to the x 3 term: 1 + 10 x + 45 x 2 + 120 x 3 .
Substitute x = 40 1 into the expansion: 1 + 10 ( 40 1 ) + 45 ( 40 1 ) 2 + 120 ( 40 1 ) 3 .
Calculate the approximate value: 1.28 .
Explanation
Problem Analysis We are given a two-part problem. The first part requires us to use binomial expansion to approximate the value of ( 40 41 ) 10 , including the term x 3 . The second part asks us to evaluate tanh 4 x .
Binomial Expansion Setup For the first part, we recognize that 40 41 = 1 + 40 1 . Thus, we can express ( 40 41 ) 10 as ( 1 + 40 1 ) 10 . We will use the binomial expansion of ( 1 + x ) n with n = 10 and x = 40 1 . The binomial expansion is given by: ( 1 + x ) n = 1 + n x + 2 ! n ( n − 1 ) x 2 + 3 ! n ( n − 1 ) ( n − 2 ) x 3 + ...
Substituting x = 1/40 Substituting n = 10 , we get: ( 1 + x ) 10 = 1 + 10 x + 2 10 ( 9 ) x 2 + 6 10 ( 9 ) ( 8 ) x 3 + ... = 1 + 10 x + 45 x 2 + 120 x 3 + ... Now, we substitute x = 40 1 into the expansion: ( 40 41 ) 10 ≈ 1 + 10 ( 40 1 ) + 45 ( 40 1 ) 2 + 120 ( 40 1 ) 3
Calculating the Approximation Calculating the terms: 1 + 40 10 + 1600 45 + 64000 120 = 1 + 4 1 + 320 9 + 1600 3 = 1 + 0.25 + 0.028125 + 0.001875 = 1.27999 ≈ 1.28 Therefore, ( 40 41 ) 10 ≈ 1.28
Evaluating tanh(4x) For the second part, we need to evaluate tanh 4 x . Recall that the hyperbolic tangent function is defined as: tanh u = cosh u sinh u = e u + e − u e u − e − u Substituting u = 4 x , we get: tanh 4 x = e 4 x + e − 4 x e 4 x − e − 4 x This is the expression for tanh 4 x . Without a specific value for x , we cannot evaluate it further.
Final Answer Thus, the approximate value of ( 40 41 ) 10 using the binomial expansion up to the x 3 term is approximately 1.28 , and tanh 4 x = e 4 x + e − 4 x e 4 x − e − 4 x .
Examples
Binomial expansion is used in various fields such as physics, engineering, and finance. For instance, in finance, it can be used to approximate the future value of an investment with a small rate of return. In physics, it can be used to simplify complex equations involving small perturbations. Understanding binomial expansion allows for quick and accurate approximations in these scenarios, saving time and resources.
