# Thread: Linear Approximation and the Derivitive

1. ## Linear Approximation and the Derivitive

2 Questions for you

1. Find a formula for the error E(x) in the tangent line approximation to the function near x=a. Using a table of values for E(x)/(x-a) near x=a, find a value for k such that E(x)/(x-a) is approximately equal to k(x-a). Check that, approximately, k=f''(a)/2 and that E(x) is approximately equal to (f''(a)/2)(x-a)^2.
The problem is : f(x)= square root of x, a=1

2. Writing g for the acceleration due to gravity, the period t, of a pendulum of length l is given by T=2pi times the root of l/g.
a) show that if the length of the pendulum changes by delta l, the change of the period, delta T, is given by delta T= T/2l x delta l
b) If the length of the pendulum increases by 2%, by what percent does the period change?

THANKSSSSSSSSSS

2. Originally Posted by Paige05
2 Questions for you

1. Find a formula for the error E(x) in the tangent line approximation to the function near x=a. Using a table of values for E(x)/(x-a) near x=a, find a value for k such that E(x)/(x-a) is approximately equal to k(x-a). Check that, approximately, k=f''(a)/2 and that E(x) is approximately equal to (f''(a)/2)(x-a)^2.
The problem is : f(x)= square root of x, a=1

2. Writing g for the acceleration due to gravity, the period t, of a pendulum of length l is given by T=2pi times the root of l/g.
a) show that if the length of the pendulum changes by delta l, the change of the period, delta T, is given by delta T= T/2l x delta l
b) If the length of the pendulum increases by 2%, by what percent does the period change?

THANKSSSSSSSSSS
In general your linear approximation function is
$\displaystyle f_{approx}(x) \approx f(a) + f^{\prime}(a) \cdot (x - a)$

So your error function will be:
$\displaystyle E(x) = f(x) - f_{approx}(x)$

-Dan