Question

Approximate the value for x in the equation :

e^5x + x = 2


using the local linearization of the left-hand side of the inequality near x = 0

Answer 1

1/6

Explanation:

For f(x) = e^(5x) +x, the derivative is ...

f'(x) = 5e^(5x) +1

Then the linearization at x=0 is ...

f(x) ≈ f'(x)(x -0) + f(0)

f(x) ≈ 6x +1

Then for f(x) = 2, we have

6x +1 = 2

x = (2 -1)/6 = 1/6

So, an approximate value of x is x = 1/6.

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As exponential functions do, the curve grows steeper very quickly, so this value of x is an overestimate.