Finding the secant line of a gradient, thus finding the tangent line

Question

Given the function f(x) = x³ + 2x² - 2x - 1, find the gradient of the secant joining the points P (1, f(1)) and Q (1+h, f(1+h))

 

 

 

Then find the gradient of the function at P.

 

 

 

I understand what a secant line is, i am just struggling putting it together to find the secant, then the tangent and so on and so forth.

THANK YOU FOR YOUR HELP!

Answer 1

First, we need to calculate the y-coordinate of P:

f(1)=1³+2(1)²-2(1)-1=0

 

-> P(1,0)

 

the gradient is defined as Δy/Δx=(y_Q-y_P)/(x_Q-x_P)

Let's plug in our values: ((1+h)³+2(1+h)²-2(1+h)-1-0)/(1+h-1)

Expand it...   (h³+3h²+3h+1+2(1+2h+h²)-2-2h-1-0)/(1+h-1)

...and do some more simplifying

 

For the gradient at P, either calculate the limit of your result for h->0, or calculate the first derivative of f and plug in x=1

 

Hope this helps!