Tangent Line to a Function

Calculus I Project

Objective:

To understand the connection between the derivative of a function at a point,   f  ' ( a ) ,  and the tangent line to the function at the point  ( a, f ( a ) ).

Maple will help with graphing the function and the tangent line.

Solved Example:

 Consider  f ( x )   = sqrt(x)/(1+x)    .

A ) Find an equation  of  the tangent line to f ( x )  at the point where   x  = 4.

B ) Use Maple to plot the graphs of the function and the tangent line in one window.

    

Solution:

  A ) The point the tangent line passes through  is  ( 4 , f (4) ). Since   f ( 4 ) = 0   .4 , the coordinates of the point are   (4, 0.4 ).  We can see this using Maple.

We define the function.

>    f:=x->sqrt(x)/(x+1);

f := proc (x) options operator, arrow; sqrt(x)/(x+1) end proc

>    f(4);

2/5

>    evalf(%);

.4000000000

The slope of the tangent line at the point  ( 4, f (4) ) is equal to   f ' (4 ).

Then     f ' ( x )  = (1-x)/(2*sqrt(x)(1+x)^2)   , so that     f ' (4)  = -0.03.    

Let us do this using Maple.  We find the derivative at x = 4 using Maple.

>    fprimex:= diff(f(x),x);

fprimex := 1/(2*x^(1/2)*(x+1))-x^(1/2)/(x+1)^2

>    fprime4:= subs(x=4,fprimex);

fprime4 := -3/200*4^(1/2)

>    evalf(%);

-.3000000000e-1

So the equation of the tangent line in question is    y - 0.4  = -0.03( x - 4) ,   or       y = -0.03 x + 0.52 .

B ) Now, draw the tangent line

>    y:=x->-0.03*x+0.52;

y := proc (x) options operator, arrow; -.3e-1*x+.52 end proc

y := proc (x) options operator, arrow; -.3e-1*x+.52 end proc

>    plot({f(x),y(x)},x=-1..10);

[Maple Plot]

Remark:  Notice that a tangent to a curve may intersect the curve at some other point of the curve.  

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ASSIGNMENT

Apply questions  ( A )  and  ( B ) of  the solved example to the curves at the indicated points. When plotting, select the interva l of  x  appropriately.  Label, by hand, the   curves, the tangent lines, and the points of tangency.  

Problem 1:

  y   =   x/(1+x^2)          at    x  = 3

      

Problem 2:

      y  = x^2*(sqrt(2)-sqrt(x))       at      (i)   x  = 2         (ii)        x = 4  

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MSIP Grant #P120A80089-98:  "Three Urban Calculus Reform Programs: Adopting the Best," 1998-2001; MSEIP Grant #P120AA010031:  "Four Colleges: Calculus + Enhancements", 2001-2004