Consider the following.
y =
x2 − 9x
x2 + 5x
at (3, −
3
4
)
(a) At the indicated point, find the slope of the tangent line.
(b) At the indicated point, find the instantaneous rate of change of the function.
To find the slope of the tangent line at a specific point on a curve, you need to find the derivative of the function and evaluate it at that point. The derivative represents the rate of change of the function at any given point.
In this case, you have a rational function y = (x^2 - 9x)/(x^2 + 5x). To find the derivative of this function, you can use the quotient rule of differentiation:
1. Differentiate the numerator: (d/dx) (x^2 - 9x) = 2x - 9
2. Differentiate the denominator: (d/dx) (x^2 + 5x) = 2x + 5
3. Apply the quotient rule: (f/g)' = (f'g - g'f)/(g^2)
Using the quotient rule, the derivative of y with respect to x is:
dy/dx = [(2x - 9)(x^2 + 5x) - (2x + 5)(x^2 - 9x)] / (x^2 + 5x)^2
Next, substitute the x-coordinate of the indicated point (3) into the derivative expression to find the slope of the tangent line:
slope = dy/dx evaluated at x = 3
slope = [(2(3) - 9)(3^2 + 5(3)) - (2(3) + 5)(3^2 - 9(3))] / (3^2 + 5(3))^2
Simplifying this expression will give you the slope of the tangent line at the point (3, -3/4).
To find the instantaneous rate of change of the function at the indicated point, you can also use the derivative. The derivative of a function represents the instantaneous rate of change at any point on the curve.
To find the instantaneous rate of change, substitute the x-coordinate of the indicated point (3) into the derivative expression:
rate of change = dy/dx evaluated at x = 3
rate of change = [(2(3) - 9)(3^2 + 5(3)) - (2(3) + 5)(3^2 - 9(3))] / (3^2 + 5(3))^2
Simplifying this expression will give you the instantaneous rate of change of the function at the point (3, -3/4).