Newton's theory of gravitation states that the weight of a...

Newton's theory of gravitation states that the weight of a person at elevation $x$ feet above sea level is $W(x)=P R^{2} /(R+x)^{2}$ where $P$ is the person's weight at sea level and $R$ is the radius of the earth (approximately 20,900,000 feet). Find the linear approximation of $W(x)$ at $x=0 .$ Use the linear approximation to estimate the elevation required to reduce the weight of a 120 -pound person by $1 \%$

Key Concepts

Percentage Change Estimation via Linearization

This concept involves using the linear approximation to relate small changes in the input variable to proportional changes in the function's output. By equating a given percentage change in the function’s value to the approximated change derived from the derivative, one can estimate the corresponding variation in the input variable required to achieve that percentage change.

Tangent Line Approximation

The tangent line approximation uses the line that just touches the curve at a point (the tangent line) to approximate the function near that point. It is a practical tool for estimating small variations in the function's value, especially when changes are minimal relative to the reference point.

Linear Approximation

Linear approximation is a method for estimating the value of a function near a given point using the tangent line at that point. It involves evaluating the function and its derivative at a specific point, then constructing a linear function that approximates the behavior of the original function for small changes around that point.

Derivative

The derivative of a function measures the rate at which the function's value changes with respect to changes in its independent variable. This concept is crucial for determining the slope of the tangent line used in linear approximations, and it underpins many techniques in differential calculus.

Transcript

00:01 In provening 63 we have w of x functions that gives us the weight of a person at different elevations above sea level where b is the person's weight at sea level b equals w at x equal 0 at sea level and r is the radius of the earth.

00:29 We want to use linear approximation to get an equation near x equals 0 near c level linear approximation is about to get the tangent equation of this function when we have x equal 0 or at the specified point the linear approximation equation equals l of x equals the function at the subsify point x node plus the value of the differential of the function and the subsified point w dash of x node multiplied by x minus x node we have said that we want elevations near c level then we will make the approximation at x node equal 0 this means we need to define w of x node and w dash of x node to get l of x first we get w of x node we substitute here by x equals x node or x equal 0 to get w of x node we have here w of 0 and we've said that w of 0 equals b the person's weight at c level 2 second step is to get this value but first we need to get the differential of the function w dash of x the differentiation of p r squared divided by r plus x squared we have b r squared constant we keep it as it is and we have here a function with the power of minus 2 then we put we multiply by the power minus 2 and we subtract 1 from the power we have r plus x to the bar of minus 2 minus 1 equals 3 equals minus 2 b r squared divided by r plus x cubed now we can evaluate w dash of x node which equals w dash of 0 equals minus 2 p r squared divided by r cubed equals minus 2b divided by r the third step is to use these values and substitute here to get the linearization l of x equals w of x node w of x node equals b plus w dash of x node equals minus 2b divided by r minus 2b divided by r multiplied by x minus x node x minus 0 it equals b multiplied by 1 minus 2 x divided by r this is the linearization or the linear approximation at x equals you for the second part of the question we want to get the elevation we want to get x which reduces the weight of a body by 1 % if b equals 120 pounds we want to use this equation when the weight of the body reduces to b by 1 % this means we have w equals p minus 1 % of b % of b we have b equals p minus 1 % of b we have b equals 120 pounds minus 120 divided by 100 equals 100 and 18 .8 bounce.

06:01 We have the linearization l of x and the linearization is an approximation for the weight.

06:11 This means this number represents l of x.

06:15 We have l of x equals 118 bounce and now we can use the equation in stem number 3 to get x we have 100 the 18 .8 equals p which is 120 multiplied by 1 minus 2x divided by r and we have r given in the problem equals 20 million and 900 ,000s...

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