The equation of the tangent line to the graph of g(x) = x · f(x) at x = 2, where f(x) = 3x^2 + 5x + 8, is y = 64x - 68. The slope of the tangent line is 64, and the y-intercept is -68.
To find the equation of the tangent line to the graph of the function g(x) = x · f(x), where f(x) = 3x^2 + 5x + 8, at x = 2, we first need to find the derivative of g(x), g'(x), which will give us the slope of the tangent line at any point x. The derivative of g(x) can be found by using the product rule:
g'(x) = f(x) + x · f'(x), where f'(x) is the derivative of f(x).
The derivative of f(x) is f'(x) = 6x + 5.
Using these derivatives, we find g'(2) = f(2) + 2 · f'(2):
f(2) = 3(2)^2 + 5(2) + 8 = 12 + 10 + 8 = 30
f'(2) = 6(2) + 5 = 12 + 5 = 17
g'(2) = 30 + 2(17) = 30 + 34 = 64
The slope of the tangent line is therefore 64. The point on the graph of g(x) at x=2 is (2, g(2)) = (2, 2 · f(2)) = (2, 60). The equation of the tangent line is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
Plugging in the values we found:
y - 60 = 64(x - 2)
y = 64x - 128 + 60
y = 64x - 68
The equation of the tangent line at x = 2 is y = 64x - 68. The slope is 64, and the y-intercept is -68.
The probable question may be:
Consider the technology function g(x)=x⋅f(x), where f(x) is a tech-related function. Find the equation of the tangent line to the graph of g at x=2. Assume f(x) represents the number of tech gadgets sold per day, and x represents the number of days since the launch of a new product. If the initial sales function f(x) is given by f(x)=3x2+5x+8, determine the equation of the tangent line at x=2. Provide the slope and y-intercept of the tangent line.
Additional Information:
The function f(x) models the daily gadget sales.
x represents the number of days since the launch of the product.
Assume the product was launched 5 days ago, so x=0 corresponds to the day of the launch.
Calculate the values needed to find the equation of the tangent line.