Thread: functions

Using the graph determine the following
1)intervals on which the function is increasing
2) intervals on which the function is decreasing
3) intervals on which the function is constant



any help in explaining the steps necessary in solving this question would be appreciated

Originally Posted by zeratul

Using the graph determine the following
1)intervals on which the function is increasing
2) intervals on which the function is decreasing
3) intervals on which the function is constant



any help in explaining the steps necessary in solving this question would be appreciated

A function is said to be increasing over an interval if its first derivative is positive over that interval. i.e. the gradient of the tangents to the curve are positive, i.e sloping upwards.

A function is said to be decreasing over an interval if its first derivative is negative over that interval. i.e. the gradient of the tangents to the curve are negative, i.e. sloping downwards.

A function is said to be constant over an interval if its first derivative is 0 over that interval. i.e. the gradient of the tangents to the curve are 0, i.e. parallel to the horizontal.

Do these definitions help you?

Equation of the shown curve is

y+4=(x+1)^2

differentiating both sides wrt. x

dy/dx=2(x+1)

for increasing fn. dy/dx>0 i.e. slope of tangent is positive

2(x+1)>0

i.e. x>-1

for decreasing fn. dy/dx<0 i.e. slope of tangent is negative

2(x+1)<0

i.e. x<-1

Originally Posted by sumit2009

Equation of the shown curve is

y+4=(x+1)^2

differentiating both sides wrt. x

dy/dx=2(x+1)

for increasing fn. dy/dx>0 i.e. slope of tangent is positive

2(x+1)>0

i.e. x>-1

for decreasing fn. dy/dx<0 i.e. slope of tangent is negative

2(x+1)<0

i.e. x<-1

This assumes that the curve goes off to infinity in each direction. The part of the graph we can see stops at x values of -4 and 2. We can hence, only conclude that the interval for decease is [-4,-1), increase (-1,2], and constant [-1,-1]. These are verifiable by the eye alone without the aid of equations. To assume that the curve is continuous after these intervals is far too presumptuous for mathematical rigor.

Originally Posted by Mush

This assumes that the curve goes off to infinity in each direction. The part of the graph we can see stops at x values of -4 and 2. [snip]

It DOESN'T stops at x=-4 and x=2
we can see from the graphs that there are ARROW HEADS on both ends which means that it extends from -∞ to +∞

Originally Posted by sumit2009

This assumes that the curve goes off to infinity in each direction. The part of the graph we can see stops at x values of -4 and 2.

It DOESN'T stops at x=-4 and x=2
we can see from the graphs that there are ARROW HEADS on both ends which means that it extends from -∞ to +∞

Ahh. My eyesight wasn't sharp enough to catch that. Jolly good.

In any instance, equations aren't necessary. The values can be deduced by graph inspection.

Originally Posted by Mush

[snip]In any instance, equations aren't necessary. The values can be deduced by graph inspection.

yes sir I agree with u

thanks alot for the replies, but differentiation was not thought as yet so can't use the method given my sumit2009.

As for the definitions given by Mush thanks alot for them but i'm still not sure how to deduce the answers for the graph, can anyone simplify it a little more for me

thanks