Overview

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle first. The graph of the absolute value of x has one sharp corner, at zero. The graph of x squared has no corner at all. Now, at every x, take the larger of the two. How many corners does the new graph have? One plus zero, so one, right? Hmm, not so fast. By the end, you will count them in a few lines. Mixed questions like this come up often in JEE Main. So, let's begin!

0:31How this chapter is built

So, where are the real exam questions? The real JEE Main questions, from twenty twenty one to twenty twenty six, are at the end of Chapters one to seven. The short dash means to. Okay? This chapter has original problems in the same style, in three mixed sets: eight solved plus three practice problems each. Mixed means any chapter can come next, so you spot the type yourself. Got it?

0:56What gets mixed: limits

Alright, these questions mix several kinds of limits. Think back over Chapters one and two: which limit tools do you expect to need? Let's check your list. First, standard limits. The arrow means tends to. As x tends to zero, sine x over x tends to one, and e to the power x, minus one, over x, also tends to one. Conditions: x in radians, and the argument tends to zero. Okay? Second, expansions: e to the power x equals one plus x, plus x squared over two, plus x cubed over six, and the dots mean and so on. Expand each term up to the order of the denominator, that is, up to its power of x. Clear? Third, the one to the power infinity form; the sideways eight is infinity. The short form lim means limit. The limit of f to the power g equals e to the power, the limit of g times, f minus one, only when f tends to one and g tends to infinity. Never replace the base by one too early. With me? Fourth, greatest integer limits. The square brackets around x mean the greatest integer less than or equal to x. So, what is the limit of greatest integer of x, as x tends to one? It depends on the side! From the left, the small minus, x is just below one, so greatest integer of x is zero. From the right, the small plus, it is one. The two sides differ, so the limit at one does not exist.

2:32What gets mixed: continuity, differentiability

Right? Next, continuity and differentiability, usually of piecewise and modulus functions. Continuous at a: the left-hand limit, LHL, equals the right-hand limit, RHL, equals the value f of a. Okay? Differentiable at a: the left-hand derivative, L f prime of a, equals the right-hand derivative, R f prime of a, both finite. The little dash is called prime. Clear? Now, if f is continuous at a, must it be differentiable there? No, no! The arrow means implies: differentiable implies continuous, not the reverse. Left graph: y equals the absolute value of x, the vertical bars. No break, but a corner at zero: slope minus one on the left, slope one on the right. Right? Right graph: y equals the cube root of x, the small three on the root sign. No break, but at zero the curve stands straight up: a vertical tangent, an infinite slope. Not differentiable. Got it? For modulus functions, with g differentiable: the absolute value of g fails to be differentiable at a simple root of g, where g changes sign and g prime is not zero. With me? Max means the larger of two functions, min the smaller. Check the points where the two cross, because the slopes there usually differ.

3:57What gets mixed: derivatives and applications

Okay? Then derivatives: the basic rules, and implicit, parametric and logarithmic differentiation, from Chapter five. Clear? And short applications. Monotonicity: if f prime of x is greater than zero on an interval, f is strictly increasing there. Right? Tangents: at the point x one, y one, small ones below, the tangent is y minus y one equals m times, x minus x one, with slope m equal to f prime of x one. Okay? Extreme values: where f prime of c is zero, f double prime of c less than zero gives a local maximum, greater than zero a local minimum. Two dashes mean double prime. Got it? Last, the greatest or least value of a continuous function on the closed interval from a to b. Here the square brackets mean the endpoints are included; they are not the greatest integer. Compare the values at the critical points with the values at the endpoints too. Forgetting the endpoints is a classic trap!

4:56Example 1: k for continuity of (e^kx - 1)/sin 2x at x = 0

Now, Example one, from Set one, marked easy. f of x equals e to the power k x, minus one, over sine two x, when x is not zero, the slashed equals sign; and f of zero equals three. Find k so that f is continuous at zero. Pause the video, and try it. Done? Okay, let's check. Limits meet continuity here: the limit of f of x, as x tends to zero, must equal f of zero, which is three. Clear? Many students say: the top is like k x, the bottom like x, so the limit is k, and k is three, right? Oh no, careful! The bottom is sine of two x, not sine x. And k cannot be zero, or f would be zero near zero. Write the fraction as: e to the power k x, minus one, over k x, times two x over sine two x, times k over two. The dot means times. With me? As x tends to zero, k x and two x tend to zero, so the standard limits apply: the first two factors tend to one. The limit is k over two. Okay? Set k over two equal to three. So, k equals six. Perfect!

6:13Example 2: non-differentiable points of max(|x|, x squared)

Now, remember our puzzle? Let's finish it. From Set three, marked medium: find the number of points where f of x, equal to max of the absolute value of x and x squared, is not differentiable. The curly brackets hold the two functions. Pause the video, and try it. Done? Okay, let's check. Our tip: first find where they cross. The absolute value of x equals x squared at minus one, zero and one. Right? Between minus one and one, the absolute value of x is larger: at one half, a half beats a quarter. Outside, x squared is larger. So f is the absolute value of x when the absolute value of x is at most one, the sign like a less than with a line below, and x squared when it is more than one. Clear? The absolute value of x has one corner, and x squared has none. So the answer is one point, right? Let's check each point. At zero, the corner of the absolute value of x: slopes minus one and one. Not differentiable, as expected. With me? But only one point? No, no, wait! Look at the crossing points. At one, the left piece x has slope one, and the right piece x squared has slope two x, that is two. At minus one: minus two on the left, from x squared, and minus one on the right, from minus x. The slopes do not match: two more corners! Got it? So, not differentiable at minus one, zero and one: three points. Everywhere else, each piece is smooth. One corner plus no corners gave three corners. Puzzle solved!

7:58Quick revision: limit tools, continuity, corners of max/min

So, quick revision. Limits: standard limits, expansions, the one to the power infinity form, and greatest integer limits from each side. Continuity: left limit, right limit and value all equal; differentiable implies continuous, not the reverse. For modulus, max and min, check the roots and the crossing points for corners. Now open Set one and start solving. All the best!