. Sketching of Curves When we are sketching the graph of functions either by hand or through any graphing software we cannot show the entire graph. Only a part of the graph can be sketched. Hence a crucial question is which part of the curve we need to show and how to decide that part.
To decide on this we use the derivatives of functions. We enlist few guidelines for determining a good viewing rectangle for the graph of a function. They are : (i) The domain and the range of the function. (ii) The intercepts of the cure (if any).
(iii) Critical points of the function. (iv) Local extrema of the function. (v) Intervals of concavity. (vi) Points of inflexions (if any).
(vii) Asymptotes of the curve (if exists) Example . Sketch the curve y . Factorising the given function, we have )( . ( ) The domain of the given function f x ( ) is the entire real line.
( ) Putting y = we get x = − , . Therefore the x -intercepts are ( , ) − and ( , ) putting x = we get y = − . Therefore the y -intercept is ( , ( ) ′ and hence the critical point of the curve occurs at x = . ( ) ′′ > ∀ .
Therefore at x = the curve has a local minimum which is f = − ( ) The range of the function is y ≥− ( ) Since ′′ > ∀ the function is concave upward in the entire real line. ( ) Since f x ≠ ∀ the curve has no points of inflection. ( ) The curve has no asymptotes. The rough sketch of the curve is shown on the right side.
Fig. . − − − − − − − − − − _ - - Applications of Differential Calculus Example . Sketch the curve y .
Factorising the given function, we have )( ( ) The domain and the range of the given function f x ( ) are the entire real line. ( ) Putting y = , we get the x = . The other two roots are imaginary. Therefore, the x -intercept is ( , ) .
Putting x = , we get y = − . Therefore, the y -intercept is ( , ( ) ′ and hence the critical points of the curve occur at x = ± . ( ) ′′ . Therefore at x = the curve has a local minimum because