Find the linear approximation of the function 𝑓 of 𝑥 equals 𝑥 cubed minus 𝑥 squared plus three at the point 𝑥 equals negative two.
To find a linear approximation at 𝑥 equal to 𝑎, you solve for your original function at 𝑥 equal to 𝑎, 𝑓 of 𝑎. You add that value to the solution of 𝑥 equal to 𝑎 for the derivative of your original function. And we multiply that value times 𝑥 minus 𝑎. We’re gonna take this linear approximation and break it up into three parts, solving for each of the parts separately.
First, we’ll solve for our original function at our point 𝑎. Our 𝑎 value, the point at which we want to find the linear approximation, is negative two. We’re gonna solve our original function at negative two: negative two cubed minus negative two squared plus three. Negative two cubed is negative eight, minus negative two squared, four, plus three. 𝑓 of negative two equals negative eight minus four plus three. The value of our function at 𝑥 equals negative two is negative nine. We found our first piece, negative nine.
Now we want to find the derivative of our original function. The derivative with respect to 𝑥 of 𝑥 cubed minus 𝑥 squared plus three is equal to three times 𝑥 squared minus two 𝑥 to the first power plus zero. Our 𝑓 prime of 𝑥 is equal to three 𝑥 squared minus two 𝑥. And we’re solving for this at point negative two, which is equal to three times negative two squared minus two times negative two.
Three times negative two squared is three times four, which equals 12. Negative two times negative two is positive four. 12 plus four equals 16. The derivative function at negative two is equal to 16. We can go ahead and add that piece to our linear approximation.
Our third and final piece is 𝑥 minus 𝑎. 𝑥 minus 𝑎 for us is 𝑥 minus negative two, which we simplify to 𝑥 plus two. 16 times 𝑥 plus two is the missing piece. Our linear approximation is then negative nine plus 16 times 𝑥 plus two.
We want to simplify, so we distribute that 16 to 𝑥 and two: 16𝑥 plus 32. Our new expression: negative nine plus 16𝑥 plus 32. We can combine like terms. Negative nine plus 32 equals 23 plus 16𝑥. We want to lead with our 𝑥 term, so we move it around and say 16𝑥 plus 23. The linear approximation of the function 𝑓 of 𝑥 is equal to 𝑥 cubed minus 𝑥 squared plus three at 𝑥 equals negative two is 16𝑥 plus 23.