Question

Consider the function f(x)=
`4√(x) +6`

(a) Simplify the following difference quotient as much as possible

(f(9+h)−f(9))/ h

(b)Use your result from (a) and the limit definition of the derivative to calculate

f′(9)= (f(9+h)−f(9))/h
lim h→0

(c)Use your answer from part (b) to find the equation of the tangent line to the curve at the point (9,f(9))
.
.

Answer 1

(a) First, we need to find f(9+h) and f(9):

f(9+h) = (9+h)^2 + 3 = 81 + 18h + h^2 + 3 = h^2 + 18h + 84

f(9) = 9^2 + 3 = 84

Now, we can substitute these values into the difference quotient and simplify:

(f(9+h) - f(9))/h = ((h^2 + 18h + 84) - 84)/h = (h^2 + 18h)/h = h + 18

(b) Using the limit definition of the derivative:

f′(9) = lim(h→0) (f(9+h) - f(9))/h = lim(h→0) (h + 18) = 18

(c) The equation of the tangent line to the curve at the point (9,f(9)) is given by:

y - f(9) = f′(9)(x - 9)

Substituting f(9) and f′(9) into this equation, we get:

y - 84 = 18(x - 9)

y = 18x - 54 + 84

y = 18x + 30