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Calculus/Tangent lines in triangle

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Expert: Alon Mandes - 10/20/2009


Question
Find the area of the triangle formed by the x-axis, the tangent to the
graph of f(x) = 6x-x^2 at the point (5; 5), and the normal through this
point (the line through this point that is perpendicular to the tangent).

Answer

drawing
I have attached a drawing to illustrate the problem.

# The Blue Line , is the Tangent line , has equation y=mx+n . Let's find m %26 n :

Finding m :
---------
m=f'(5) . f'(x)=6-2x ; f'(5)=6-10=-4 . Therefore m=-4 .

Finding n :
---------
The line passes through the point (5,5), therefore :
5=-4*5+n --> n=25 .
So, y=-4x+25 .


# The Red Line , is the perpendicular line , has equation y=ax+b . Let's find a %26 b :

Finding a :
---------
a=-1/m --> a=1/4 .

Finding b :
---------
The line passes through the point (5,5), therefore :
5=(1/4)*5+b --> 5=(5/4)+b --> 15/4 .
So, y=(x/4)+(15/4) .

# To calculate the area of the triangle , we need to find : the base B , %26 the height H .

Finding H:
---------
H=5 . The height is the value of y .

Finding B :
---------
B = Blue line intersect - Red line intersect
Blue line intersect : To find intersect point of y=-4x+25 with the x-axis , we substitute y=0 :
0=-4x+25 --> x=25/4 .
Red line intersect : To find intersect point of y=(1/4)x+(15/4) with the x-axis , we substitute y=0 : 0=(x/4)+(15/4) --> x=-15 .
Thus, B=25/4-(-15)=105/4 .

So, Area=B*H/2=(5*105/2)/2=525/4 .

Alon.  

Alon Mandes

Expertise

Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience

1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

Organizations
Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

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