Find the linearization l(x) of y=e8xln(x) at a=1

 

Explanation:

The linearization L(x) of y = f(x) at x = a is given by

     (1)

Where f'(a) is the derivative of y = f(x) evaluated at x = a. So, we need to find the derivative of y = f(x) and evaluate the derivative at x = a. 

The derivative of y = f(x) is computed using the product rule for the derivatives because f(x) is the product of two functions: logarithmic and exponential. 

So, the derivative of y = f(x) is given by



Since a = 1, the derivative of y = f(x) evaluated at x = a = 1 is given by



Moreover, note that



Using equation (1), the linearization L(x) of y = f(x) at x = a = 1 is given by

Answer

1 number

explanation

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

 

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.