1) Given that f(x)=x^9h(x) h(-1)=2 h'(-1)=5 Calculate f'(-1).

Question

HELP PLEASE

Answer 1

f(x) = x⁹h(x)

 

h(-1) = 2

 

h'(-1) = 5

 

 

We need to find h(x) first.  Once we have h(x), we can find the derivative of f(x).  Then evaluate the derivate when x = -1.

 

 

We know that derivative is the slope of the tangent line.   In this case, the slope of the line tangent to h(x) at x=-1 is 5.

 

h(-1) = 2      has a coordinate point of  (-1, 2).

 

 

2 = 5(-1) + b

 

2 = -5 + b

 

7 = b

 

 

h(x) = 5x + 7     -----> equation of the tangent line

 

 

 

f(x) = x⁹(5x + 7)

 

f(x) = 5x¹⁰ + 7x⁹

 

 

Take the derivative of f(x).

 

f'(x) = 50x⁹ + 63x⁸

 

f'(-1) = 50(-1)⁹ + 63(-1)⁸

f'(-1) = -50 + 63

f'(-1) = 13

Answer 2

Lauren,

           Using the chain rule on the above gives f'=x⁹*h'+9x⁸*h. If we now evaluate this at x=-1

f'(-1)=-1*5+9*2=18-5=13.

 

Hope this helps

 

Jim