Suppose that 𝑓 of two equals seven and 𝑓 prime of 𝑥 equals five 𝑥 squared minus six over 𝑥 minus nine. Using tangent line approximation, what is the best approximation for the value of 𝑓 of 1.9? a) 6.8, b) 7.2, c) 7.1, or d) 6.9.
Let’s first consider what tangent line approximation means. If we have some function 𝑓 of 𝑥 and a point at 𝑎, 𝑓 of 𝑎, the tangent line to 𝑎 𝑓 of 𝑎 would look like this. The formula for this tangent line is 𝑦 equals 𝑓 of 𝑎 plus 𝑓 prime of 𝑎 times 𝑥 minus 𝑎. And since the curve of 𝑓 of 𝑥 and the tangent line are close to each other for all the points where 𝑥 is near 𝑎, we can use this tangent line formula to approximate the value of 𝑓 of 𝑥 for any value where 𝑥 is near 𝑎.
The tangent line approximation tells us that 𝑓 of 𝑥 is approximately 𝑓of 𝑎 plus 𝑓 prime of 𝑎 times 𝑥 minus 𝑎. In our case, we’re given 𝑓 of a for 𝑓 of two. Our 𝑎-value is two. And that means our 𝑥 that is close to two is 1.9. We want to know what 𝑓 of 1.9 is. So we’ll take 𝑓 of two. We’ll add 𝑓 prime of two times 1.9 minus two. We already know that 𝑓 of two is seven.
To find 𝑓 prime of two, we’ll plug in two for 𝑓 prime of 𝑥, five times two squared minus six over two minus nine. And 1.9 minus two is negative 0.1. Five times two squared is 20. 20 minus six equals 14. two minus nine equals negative seven. 14 over negative seven equals negative two. 𝑓 prime of two equals negative two. 𝑓 of 1.9 is approximately seven plus negative two times negative 0.1. Negative two times negative 0.1 equals positive 0.2. Seven plus 0.2 is 7.2. 𝑓 of 1.9 is approximately 7.2, which is answer choice b.