Find the slope of the tangent line to the graph of the function
f(x)=x*e^x
at x=3 correct to two decimal places.
To find the slope of the tangent line to the graph of the function f(x) = x * e^x at x = 3, we can use the concept of differentiation.
Step 1: Differentiate the function with respect to x to find the derivative.
The derivative of f(x) with respect to x is given by the product rule, which states that the derivative of the product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.
f'(x) = (x * e^x)' = x * (e^x)' + e^x * (x)'
The first derivative of e^x is simply e^x, and the derivative of x is 1.
f'(x) = e^x * x + 1 * e^x = (x + 1) * e^x
Step 2: Evaluate the derivative f'(x) at x = 3.
By substituting x = 3 into the derivative, we can find the slope of the tangent line at that point.
f'(3) = (3 + 1) * e^3 = 4 * e^3
Step 3: Calculate the value of the slope as a decimal to two decimal places.
To obtain the numerical value, we need to multiply 4 by the numerical value of e^3.
Using a calculator, we find that e^3 ≈ 20.0855.
Therefore, the slope of the tangent line at x = 3 is approximately:
4 * 20.0855 ≈ 80.34 (rounded to two decimal places).
Hence, the slope of the tangent line to the graph of f(x) = x * e^x at x = 3, correct to two decimal places, is approximately 80.34.