Question

For the function P(x) = x3 − 9x, at the point (2, −10), find the following. (a) the slope of the tangent to the curve (b) the instantaneous rate of change of the function

Answer 1

Answer:

3, in both a), b)

Step-by-step explanation:

a) The slope of the line tangent to the curve that passes through the point (2,-10) is equal to the derivative of p at x=2.

Using differentiation rules (power rule and sum rule), the derivative of p(x) for any x is $p'(x)=3x^2-9$. In particular, the value we are looking for is $p'(2)=3(2^2)-9=12-9=3$.

If you would like to compute the equation of the tangent line, we can use the point-slope equation to get $y=3(x-2)-10=3x-16$

b) The instantaneus rate of change is also equal to the derivative of P at the point x=2, that is, P'(2). This is equal to $p'(2)=3$.