Determine a simplified expression, for the slope of the secant #PQ# with #P(1, 1/3)# and #Q(1+h, f(1+h))# where #f(x) = 1/sqrt(2x+7)# where #x gt -7/2#. State restrictions on #h#?
1 Answer
# "slope "PQ = (-2)/((3sqrt(2h+9))(3+sqrt(2h+9))) \ \ \ (h gt 0)#
Explanation:
We have;
# f(x) = 1/sqrt(2x+7) #
First we check that
# x=1=> f(1)= 1/sqrt(9) = 1/3 #
Confirming
Next we find the gradient of the chord
# m_(PQ) = (Delta y)/(Delta x) #
# \ \ \ \ \ \ \ = (f(1+h) - 1/3)/(1+h-1) #
# \ \ \ \ \ \ \ = (1/sqrt(2(1+h)+7) - 1/3)/(h) #
# \ \ \ \ \ \ \ = (3 - sqrt(2h+9))/(3hsqrt(2h+9)) * (3 + sqrt(2h+9))/(3 + sqrt(2h+9))#
# \ \ \ \ \ \ \ = (3^2 - (2h+9))/((3hsqrt(2h+9))(3+sqrt(2h+9))) #
# \ \ \ \ \ \ \ = (-2h)/((3hsqrt(2h+9))(3+sqrt(2h+9))) #
# \ \ \ \ \ \ \ = (-2)/((3sqrt(2h+9))(3+sqrt(2h+9))) #
Conclusion:
If we take the limit, as
# f'(1) = lim_(h rarr 0) m_(PQ) #
# \ \ \ \ \ \ \ \ = lim_(h rarr 0) (-2)/((3sqrt(2h+9))(3+sqrt(2h+9))) #
# \ \ \ \ \ \ \ \ = (-2)/((3sqrt(9))(3+sqrt(9))) #
# \ \ \ \ \ \ \ \ = (-2)/((9)(6)) #
# \ \ \ \ \ \ \ \ = -1/27 #