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

  1/6

Step-by-step 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.

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step-by-step explanation: