Answer:
(a) x +5y = 22
(b) p = 11, q = 5
Step-by-step explanation:
(a)
The derivative of a function tells you the slope of its curve at every point. Then the slope of C at x=2 is ...
dy/dx = 2³ +2(2) -7 = 5
The normal to the curve at the point of interest will have a slope that is the opposite reciprocal of this: -1/5. Then the point-slope equation of the normal line can be written as ...
y -k = m(x -h) . . . . line with slope m through point (h, k)
y -4 = -1/5(x -2) . . . . line with slope -1/5 through point (2, 4)
5y -20 = -x +2 . . . . multiply by 5
x +5y = 22 . . . . . . add x+20 to put in standard form
The graph shows curve C and the desired normal line.
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(b)
The power rule for derivatives tells you ...
(d/dx)(a·x^n) = a·n·x^(n-1)
This relation gives you two ways to find the values of p and q.
a) using the exponent of the term
b) using the coefficient of the term
Either way, the values are ...
p = 10 +1 = 11
q = 4 +1 = 10/2 = 5
