Question

Please ASAP

With explanation PLEASE

( a ) For the curve C with equation y = f(x); dy dx =x^ 3 +2x-7. Given that the point P(2, 4) lies on C find an equation for the normal to C at P in the form ax + by + c = 0 , where a , b and c are integers

(b) y = x ^ p + 2x ^ q where is the derived function . dy dx =11x^ 10 +10x^ 4 ,; (dy)/(dx) Find the value of p and the value of q .

Answer 1

(a) x +5y = 22

(b) p = 11, q = 5

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