Find an equation of the line that is tangent to the graph of the function and parallel to the line .

Question

Find an equation of the line that is tangent to the graph of the function  and parallel to the line  . ​​

Quizplus · Accepted Answer

Since we need the line to be tangent to the graph of the function, its slope must be equal to the derivative of the function at the point of tangency. 
The derivative of the given function is: f'(x) = 4xe^x + 2e^x. 
We want the line to be parallel to the given line with slope 4, so our line must also have slope 4. 
So, we need to find the x-value at which f'(x) = 4. 
4xe^x + 2e^x = 4 
e^x(4x+2) = 4 
e^x = 2 / (2x+1) 
x = ln(2/(2x+1)) 
Using a calculator, we find that x is approximately 0.359. This is the x-value at which the line will be tangent to the graph. 
To find the corresponding y-value, we evaluate the function at x=0.359: 
f(0.359) = (0.359^3 + 2.359^2 + 1.359 + 3)e^(0.359) 
≈ 6.07 
So, the point of tangency is (0.359, 6.07). 
Now we have the slope (4) and a point (0.359, 6.07) and we can write the equation of the line using point-slope form: 
y - 6.07 = 4(x - 0.359) 
Simplifying and rearranging, we get: 
y = 4x + 4.394 
This matches choice D.

Find an Equation of the Line That Is Tangent to the Graph