Use linear approximation to approximate [tex]$\sqrt{81.3}$[/tex] as follows.
Let [tex]$f(x)=\sqrt{x}$[/tex]. The equation of the tangent line to [tex]$f(x)$[/tex] at [tex]$x=81$[/tex] can be written in the form [tex]$y=m x+b$[/tex]. Compute [tex]$m$[/tex] and [tex]$b$[/tex].
[tex]$\begin{array}{l}
m= \\
b=
\end{array}$[/tex]
Using this find the approximation for [tex]$\sqrt{81.3}$[/tex].
Answer:
Answer (2)
Find the derivative of f ( x ) = x , which is f ′ ( x ) = 2 x 1 .
Evaluate the derivative at x = 81 to find the slope: m = f ′ ( 81 ) = 18 1 .
Find the y-coordinate at x = 81 : f ( 81 ) = 9 .
Use the tangent line y = m x + b to approximate 81.3 , where m = 18 1 and b = 2 9 , so the approximation is 9.016666666666666 .
Explanation
Problem Analysis We are asked to use linear approximation to estimate 81.3 . We will use the tangent line to the function f ( x ) = x at x = 81 to approximate the value of f ( 81.3 ) .
Finding the Derivative First, we need to find the derivative of f ( x ) = x . Using the power rule, we have f ′ ( x ) = 2 1 x − 2 1 = 2 x 1
Calculating the Slope Next, we evaluate the derivative at x = 81 to find the slope m of the tangent line at that point: m = f ′ ( 81 ) = 2 81 1 = 2 ⋅ 9 1 = 18 1 ≈ 0.0556
Finding the y-coordinate Now, we find the y -coordinate of the point on the curve at x = 81 :
f ( 81 ) = 81 = 9
Point-Slope Form We use the point-slope form of a line to find the equation of the tangent line: y − f ( 81 ) = m ( x − 81 ) y − 9 = 18 1 ( x − 81 )
Finding the Tangent Line Equation We rewrite the tangent line equation in the form y = m x + b :
y = 18 1 x − 18 81 + 9 y = 18 1 x − 2 9 + 9 y = 18 1 x + 2 9 So, m = 18 1 and b = 2 9 = 4.5 .
Approximation Finally, we use the tangent line equation to approximate 81.3 by plugging in x = 81.3 into the tangent line equation: y = 18 1 ( 81.3 ) + 2 9 y = 18 81.3 + 4.5 y = 4.516666... + 4.5 y = 9.016666...
Final Approximation Therefore, the approximation for 81.3 is approximately 9.016666...
Examples
Linear approximation is used in various fields like physics and engineering to simplify complex calculations. For instance, when analyzing the motion of a pendulum, for small angles, we approximate sin ( θ ) ≈ θ . This simplifies the equations of motion, allowing for easier solutions and a good approximation of the pendulum's behavior. This technique is also used in economics to estimate changes in demand or supply based on small changes in price or income.
To find the tangent line for x at x = 81 , the slope m is 18 1 and the y-intercept b is 4.5 . Using this, we approximate 81.3 to be about 9.01667 .
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