Calculus help on intro to the derivative. For the function f'(x)=3^x, estimate f'(1). From the graph of f'(x), would you expect your estimate to be greater than or less than the true value of...

Calculus help on intro to the derivative.

For the function f'(x)=3^x, estimate f'(1). From the graph of f'(x), would you expect your estimate to be greater than or less than the true value of f'(1)?

Asked on by hellome234

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

For f'(x) = `3^x`, the value of f'(1) can be estimated by substituting x by 1. That gives f'(1) = `3^1` = 3

The value of f'(1) obtained from the graph of f'(x), would be exactly equal to the estimated value of f'(1). This is verified from the graph of f'(x) = `3^x` that has been given below :

The graph passes through the point (1, 3) or f'(1) = 3.

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

The derivative of a function at a point is:

`f'(a) = lim_(h-gt0) (f(a+h)- f(a))/h`

`` For `a = 1 =gt f'(1) = lim_(h-gt0) (f(1+h)- f(1))/h`

`` You need to find f(x) integrating f'(x).

`int 3^x dx = (3^x)/ln3 + c`

`` `f'(1) = lim_(h-gt0) (3^(1+h) - 3^1)/(h*ln3)`

f'(1) = `3^1`  = 3

The derivative of the function f(x) at the point x=1 is f'(1)=3.

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