Errors in Products, Quotients and Powers

Transcript

0:00Introduction

Hello everyone! Okay, one small question before we start. You measure the side of a cube, and your measurement has only a one percent error. Very good measurement, right? Now you calculate the volume of the cube. What is the error in the volume? Most students say one percent, same as the side. But no! It is not one percent. And when you divide one quantity by another, do the errors subtract? Also no! Today we will find out exactly how errors grow when you multiply, divide, or raise to a power. And look, in JEE, every question on the error in a formula is built on exactly this rule. So, let's begin.

0:45Multiply or divide: relative errors add

So, first, a quick recall from the last lesson. When Z is A plus B, or A minus B, the absolute errors add: delta Z equals delta A plus delta B, okay? This little triangle, delta, means the error in a quantity. Clear? Now, Z equals A times B, or Z equals A divided by B, the fraction on screen. Do the absolute errors add again? Think about it. For a product, which kind of error adds? Yes, the relative errors! Delta Z by Z equals delta A by A, plus delta B by B, okay? Here, delta A by A means the error in A, divided by the value of A itself: the relative error. Multiply by a hundred and you get percentage errors, so the same rule works in percent too. Write this down. Alright, now the division. Z equals A by B, okay? Division is the opposite of multiplication, so for a quotient the errors should subtract, right? No, no, careful! This is the JEE trap. Errors always add, even for a quotient. Why? Because we want the maximum possible error. We never know whether a reading was a little high or a little low, so we take the worst case, where both errors push the answer the same way. And in the worst case, they add. Got it?

2:10Powers: the power multiplies the error

Now, powers. Suppose Z equals A squared. What is its relative error? Look, A squared is just A times A, right? So, use the product rule. Delta A by A, plus delta A by A, okay? That is two times delta A by A, isn't it? The power two simply comes in front, as a multiplier. Really simple! Now the general case. Z equals A to the power p, times B to the power q, divided by C to the power r, okay? Each power comes in front of its own relative error. Delta Z by Z equals p times delta A by A, plus q times delta B by B, plus r times delta C by C, okay? And notice, it is plus r, not minus r, even though C sits in the denominator. Errors always add. One more thing. What about pi, or an exact number like two or six, in a formula? They carry no error at all, so they simply drop out. Only the measured quantities bring errors. With me?

3:18Topper's trick: take the log, then differentiate

Now let me share a trick that toppers use, so you never forget this rule. Take the natural log of both sides; that is the l n you see on screen. Log Z equals p log A, plus q log B, minus r log C, okay? Now differentiate both sides. d Z by Z equals p d A by A, plus q d B by B, minus r d C by C, okay? Here d means a small change, and a small error is just a small change, right? But look, there is a minus sign in front of r. What do we do with it? Remember our rule for errors? Yes, exactly, you got it! Make every term positive. Errors always add, remember? And there you have the power rule, in just two lines. As simple as that. And look at the multipliers. The quantity with the highest power gets the biggest multiplier, so it contributes most to the error. So in the lab, measure that quantity most carefully. Got it?

4:20Example 1: error in volume and surface area of a cube

Now, remember our cube from the start? Let's finish it. The side of a cube is measured with a one percent error. Find the percentage errors in its volume and in its surface area. Pause the video, and try it first. Done? Okay, let's check. The volume V equals a cubed, where small a is the side. The power is three, so three times one percent: three percent. See? Not one percent, three percent! That is the answer to our puzzle. Now the surface area, S equals six a squared. The power two gives two percent, and then there is a six in front. So six times two, twelve percent, right? No, no, careful! The six is an exact number. It carries no error, so it drops out. Only the power counts: two times one percent, two percent. So, three percent for the volume, and two percent for the surface area.

5:20Example 2: percentage error with powers and a square root

Now a slightly harder one, JEE level. P equals a cubed, b squared, divided by, root c times d. The percentage errors in a, b, c and d are one, three, four and two percent. Find the percentage error in P, okay? Pause and try. Hint: write the square root as a power. Done? Okay, let's check. First, the powers. The square root of c is c to the power half. And d, in the denominator, has power one. So the powers are three, two, half and one. Now, c and d sit in the denominator. So we subtract their errors, right? No, no! What did we say? Errors always add, every single time! So, three times one percent, plus two times three percent, plus half of four percent, plus one times two percent. Each power multiplies its own error. That is three, plus six, plus two, plus two. Thirteen percent. And notice, b had only a three percent error, but its power two turned it into six percent. That is why powers matter so much.

6:36Quick revision: product, quotient and power rules

So, quick revision. Multiply or divide: the relative errors add, never subtract. Powers: each power multiplies its own relative error. And constants like pi, and exact numbers, carry no error. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!