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Chapter 11 Graphs and Functions

Chapter 11 — Graphs and Functions

Concepts, worked examples and 18 CSIR NET-JRF Part A questions (December 2016 – December 2025) with full solutions. Click Show solution under any question.

Concepts

11.1 Reading a graph: slope and area

A graph of a quantity \(y\) against a quantity \(x\) carries two kinds of information beyond the individual values it shows. The first is the slope. Between two points \((x_1,y_1)\) and \((x_2,y_2)\) the average rate of change is \((y_2-y_1)/(x_2-x_1)\), and as the points draw together this becomes the slope of the tangent, \(dy/dx\). A rising curve has a positive slope, a falling curve a negative one, and a horizontal stretch means that \(y\) is momentarily not changing at all. The way the slope itself changes is visible as curvature: a curve that bends upwards (concave up) has an increasing slope, \(d^2y/dx^2>0\), and one that bends downwards has a decreasing slope.

The second is the area under the curve. If \(y\) is a rate, the area \(\int y\,dx\) is the total accumulated. Dividing that area by the width of the interval gives the average value of the function,

\[\boxed{\langle y\rangle = \frac{1}{b-a}\int_a^b y\,dx},\]

which is the height of the rectangle having the same base and the same area. For a straight segment the average is simply the mean of its two end values, so a graph made of straight pieces of equal width can be averaged by inspection.

A third quantity, the ratio \(y/x\) at a point, is easily confused with the slope. Geometrically \(y/x\) is the slope of the chord joining the origin to the point \((x,y)\). To make this ratio as large as possible, we rotate a line about the origin until it last touches the curve; at that point the line is tangent to the curve. Formally,

\[\frac{d}{dx}\left(\frac{y}{x}\right) = \frac{x\,y' - y}{x^2} = 0\quad\Longrightarrow\quad \boxed{y' = \frac{y}{x}},\]

so the ratio is extremal where the tangent passes through the origin. This is the condition for the best efficiency, such as the greatest return per unit of effort, and it is in general neither the steepest point nor the highest point of the curve.

When only two values of a quantity are known and the relation between them is taken to be linear, intermediate values follow by linear interpolation. The change in \(y\) per unit change in \(x\) is \((y_2-y_1)/(x_2-x_1)\), and a change \(\Delta x\) produces the change \(\Delta y = \Delta x\,(y_2-y_1)/(x_2-x_1)\).

11.2 Graphs of motion

The ideas of Section 11.1 take a concrete form for a body moving along a line. On a distance–time graph the slope is the speed. A straight line means constant speed, a curve bending upwards means the body is speeding up, and a curve that flattens to the horizontal means the body is coming to rest. Distance travelled can never decrease, so a distance–time graph never falls. If the acceleration \(a\) is constant,

\[s = s_0 + ut + \tfrac12 a t^2,\qquad \frac{d^2 s}{dt^2} = a,\]

so the distance–time graph is a parabola whose curvature measures the acceleration: of two such curves on the same axes, the one that bends more sharply belongs to the body with the larger acceleration, whatever the initial distance \(s_0\).

On a velocity–time graph the slope is the acceleration and the area under the curve is the distance travelled. A straight line through the origin, \(v = at\), describes uniform acceleration from rest. The velocity then increases by the same amount, not by the same factor, in each equal interval, and the distance covered in successive intervals grows, because the area under each successive strip is larger.

When a speed graph and a distance graph are drawn together, they must be consistent at every instant. Where the speed is zero the distance curve is horizontal; where the speed is greatest the distance curve is steepest, which is a point of inflection; and where the speed is rising the distance curve bends upwards. A constant speed requires a straight distance line, and a speed rising linearly in time requires a parabolic distance curve.

The same reasoning applies to any quantity accumulated at a given rate. A vessel filled at a constant volume rate \(Q\) contains \(V = Qt\), whatever its shape. The shape of the vessel affects only how the height of the liquid grows: for an inverted cone of half-angle \(\alpha\), \(V = \tfrac13\pi h^3\tan^2\alpha\)

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