SOLUTION: find the equation of the tangent line: 1) f(x)=√(25-x^2) at the point where x=4 2) f(x)=x^2+√x at the point where x=1

Algebra ->  Functions -> SOLUTION: find the equation of the tangent line: 1) f(x)=√(25-x^2) at the point where x=4 2) f(x)=x^2+√x at the point where x=1      Log On


   

Question 1107335: find the equation of the tangent line:
1)
f(x)=√(25-x^2) at the point where x=4
2)
f(x)=x^2+√x at the point where x=1
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
1) f%28x%29=sqrt%2825-x%5E2%29 is define only for -5%3C=x%3C=5
f%284%29=sqrt%2825-4%5E2%29=sqrt%2825-16%29=sqrt%289%29=3

As a calculus problem:
The derivative of f%28x%29=sqrt%2825-x%5E2%29=%2825-x%5E2%29%5E%221%2F2%22
df%2Fdx=%281%2F2%29%2825-x%5E2%29%5E%22-1%2F2%22%28-2x%29=-x%2Fsqrt%2825-x%5E2%29
For x=4 , df%2Fdx%284%29=-4%2Fsqrt%2825-16%29=-4%2Fsqrt%289%29=-4%2F3
is the slope of the tangent.

Otherwise,
y=sqrt%2825-x%5E2%29 is the y%3E=0 half of the circle with equation
y%5E2=25-x%5E2 <--> x%5E2%2By%5E2=5%5E2
The tangent at the point with system%28x=4%2Cy=3%29
is perpendicular to the radius at that point.
The equation of the line containing that radius is y=%283%2F4%29x .
The slope of a line perpendicular to that radius is -1%2F%223%2F4%22=-4%2F3

Either way, the equation of the tangent
at the point with system%28x=4%2Cy=3%29 , with slope -4%2F3 is
y-3=%28-4%2F3%29%28x-4%29
y-3=%28-4%2F3%29x%2B16%2F3
y=%28-4%2F3%29x%2B16%2F3%2B3
highlight%28y=%28-4%2F3%29x%2B25%2F3%29


2) f%28x%29=x%5E2%2Bsqrt%28x%29=x%5E2%2Bx%5E%221%2F2%22 is defined for x%3E=0 .
f%281%29-1%5E2%2Bsqrt%281%29=1%2B1=2
df%2Fdx=2x%2B%281%2F2%29x%5E%22-1%2F2%22=2x%2B1%2F2sqrt%28x%29
The slope of the tangent at x=1 is
df%2Fdx%281%29=2%2A1%2B1%2F2sqrt%281%29=2%2B1%2F2=5%2F2
The equation of the tangent line
passing through the point with system%28x=1%2Cy=2%29 ,
with slope 5%2F2 is
y-2=%285%2F2%29%28x-1%29
y-2=%285%2F2%29x-5%2F2
y=%285%2F2%29x-5%2F2%2B2
highlight%28y=%285%2F2%29x-1%2F2%29
graph%28300%2C300%2C-1%2C4%2C-1%2C9%2C-2%2C%285%2F2%29x-1%2F2%2Cx%5E2%2Bsqrt%28x%29%29