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Linear Approximation Calculator
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Linear approximation calculator is an online tool to find out the approximation of a curve to straight line with best fit. Operating with the given formula we get the straight line equation which can be plotted in a graph. This is also termed as the tangent line approximation. The standard equation for the linear approximation is given by,
f(x)=f(a)+f'(a)(x-a) at x=a
where,
f(x) is the given function,
f'(x) is the first derivative of the given function.

You can see a calculator with default function with its expansion point given below. The calculator will find the first derivative of the function and substitute the expansion point in the equation. The substituted equation will give you the linear approximation, when u click on "Find the Linear Approximation".
 

Steps for Linear Approximation Calculator

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Step 1 :  

Calculate the value of f(x=a)



Step 2 :  

Determine the first derivative f'(x) of the given function



Step 3 :  

Find out the value of f'(x=a)



Step 4 :  

Substitute the above values in the given equation and simplify it and arrive at the straight line equation.

f(x)=f(a)+f'(a)(x-a)     at x=a



Problems on Linear Approximation Calculator

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  1. Find the linear approximation to the function f(t)=t3+5t2-6t+7 at t=2?


    Step 1 :  

    The function is given by,


    f(t)=t3+5t2-6t+7


    f(t=2)=23+5×22-6×2+7=23



    Step 2 :  

    The first derivative is given by,


    f'(t)=3t2+10t-6



    Step 3 :  

    f'(t=2)=3×22+10×2-6=26



    Step 4 :  

    Linear approximation is given by,


    f(x)=f(a)+f'(a)(x-a)     at x=a


    So,


    f(t)=23+26(t-2)


    f(t)=23+26t-52=26t-29



    Answer  :  

    Linear approximation is given by,

     

    f(t)=26t-29



  2. Find the linear approximation to the function f(x)=√x at x=4?


    Step 1 :  

    The function is given by,


    f(x)=√x


    f(x=4)=√4=2



    Step 2 :  

    The first derivative is given by,


    f'(x)=$\frac{1}{2}x^{-\frac{1}{2}}$


    f'(x)=$\frac{1}{2\sqrt{x}}$



    Step 3 :  

    f'(x=4)=$\frac{1}{2\sqrt{4}}$=$\frac{1}{4}$



    Step 4 :  

    f(x)=f(a)+f'(a)(x-a)     at x=a


    So,


    f(x)=2+$\frac{1}{4}$(x-4)


    f(x)=2+$\frac{1}{4}$x-1


    f(x)=$\frac{1}{4}$x-1


    f(x)=x+4



    Answer  :  

    Linear approximation is given by,

     

    f(x)=x+4



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