Unit 2

Curve sketching is another practical application of differential calculus. 

Curve Sketching

Chapter 4,6,7,8

Functions of the general form y=ax+q are called hyperbolic functions.

y=h(x)=1x

You just need to substitute the values into the equation.

h(14)h(12)h(1)h(2)h(3)=====114112111213=====4211213

From the table we get the following points: (−3;−13), (−2;−12), (−1;−1), (−12;−2), (−14;−4), (14;4), (12;2), (1;1), (2;12), (3;13).

Functions of the form y=mx+c

The effect of m

As m increases, the gradient of the graph increases.

If m>0 then the graph increases from left to right (slopes upwards).

If m<0 then the graph increases from right to left (slopes downwards). 

For this reason, m is referred to as the gradient of a straight-line graph.

The effect of c

We also notice that the value of c affects where the graph cuts the y-axis. For this reason, c is known as the y-intercept.

If c>0 the graph shifts vertically upwards.

If c<0, the graph shifts vertically downwards.

Gradient and y-intercept method

We can draw a straight line graph of the form y=mx+c using the gradient m and the y-intercept c.The gradient of a line is the measure of steepness. Steepness is determined by the ratio of vertical change to horizontal change:

m=changeinychangeinx=verticalchangehorizontalchange

For example, y=32x−1, therefore m>0 and the graph slopes upwards.

m=changeinychangeinx=3↑2→=−3↓−2←

Increasing and decreasing functions

The derivative of a function can tell us where the function is increasing and where it is decreasing. If

a) f'(x) > 0 on an interval I, the function is increasing on I.

b) f'(x) < 0 on an interval I, the function is decreasing on I.

The intervals of increase and decrease will occur between points where f'(x) 

= 0 or f'(x) is undefined. 

From A to B, the slope of the tangent lines are all negative, so the derivative, f'(x) is negative from A to B. The theorem above states that the function is decreasing from A to B. Similarly, the function is also decreasing between C and D. From B to C however, the slopes of the tangent lines are positive.

If:

a) f' changes from positive to negative at c, there is a local maximum at c.

b) f' changes from negative to positive at c, there is a local minimum at c.

c) f' does not change sign at c, there is no maximum or minimum at c.