Question

find (a) the slope of the curve at the given point p, and (b) an equation of the tangent line at p. ( y = \frac{4}{x} ); ( p(-2, -2) )

a. the slope of the curve at p is (square).
(simplify your answer.)

Explanation:

Step1: Rewrite the function

The function is \( y = \frac{4}{x} \), which can be rewritten as \( y = 4x^{-1} \).

Step2: Find the derivative

Using the power rule for differentiation, if \( y = ax^n \), then \( y' = nax^{n - 1} \). For \( y = 4x^{-1} \), the derivative \( y' \) is \( y' = - 4x^{-2}=-\frac{4}{x^{2}} \).

Step3: Evaluate the derivative at \( x=-2 \)

Substitute \( x = - 2 \) into the derivative \( y'=-\frac{4}{x^{2}} \). We get \( y'|_{x = - 2}=-\frac{4}{(-2)^{2}}=-\frac{4}{4}=- 1 \). The slope of the curve at the point \( P(-2,-2) \) is the value of the derivative at \( x=-2 \).

Answer:

-1