Homework help!!

Question

The point 𝑃 (5, 1/ 3 ) on the graph of 𝑦 = 𝑓(𝑥) = 1 /√𝑥+4 . If Q is the point with coordinates (𝑥, 1 /√𝑥+4 ),

(a) Find a formula for the slope of the secant line PQ in terms of x.

(b) Find the slope of the secant line 𝑃𝑄⃗⃗⃗⃗⃗ (rounded to 3 decimal places) for the given values of 𝑥. (i) 𝑥 = 4.5 (ii) 𝑥 = 4.9 (iii) 𝑥 = 4.99 (iv) 𝑥 = 4.999 (v) 𝑥 = 5.5 (vi) 𝑥 = 5.1 (vii) 𝑥 = 5.01 (viii)𝑥 = 5.001

(c) What number would you use as an estimate for the slope of the tangent line at 𝑃 (5, 1 /3 )?

Answer 1

As stated in the problem, the secant line is the line connecting P and Q. So the slope of the secant line can be given by the rise over the run, or the change in f(x) over the change in x. In this case, the slope of the secant line is given by

( f(x) - f(5) )/ (x - 5).

Now the slope of the tangent line is given by

lim_x→5 ( f(x) - f(5) )/ (x-5).

This is the definition of the derivative! So as x gets really close to 5 (but not equal to 5),

lim_x→5 ( f(x) - f(5) )/ (x-5) = f'(5).

So for which of the values given in the problem is the slope of the secant line almost like the slope of the tangent line (hint: it's going to be very close to 5. Can you see why)? You can find the exact value of f'(5) if you know the rules of differentiation (no need to use the limit definition). So which values of x is the slope of the secant line closest to the value of f'(5)?