In the diagram (on the following page), a hockey player is D...

In the diagram (on the following page), a hockey player is $D$ feet from the net on the central axis of the rink. The goalie blocks off a segment of width $w$ and stands $d$ feet from the net. The shooting angle to the left of the goalie is given by $\phi=\tan ^{-1}\left[\frac{3(1-d / D)-w / 2}{D-d}\right] .$ Use a linear approximation of $\tan ^{-1} x$ at $x=0$ to show that if $d=0,$ then $\phi \approx \frac{3-w / 2}{D}$ Based on this, describe how $\phi$ changes if there is an increase in (a) $w$ or (b) $D$ GRAPH CANT COPY

In the diagram (on the following page), a hockey player is $D$ feet from the net on the central axis of the rink. The goalie blocks off a segment of width $w$ and stands $d$ feet from the net. The shooting angle to the left of the goalie is given by $\phi=\tan ^{-1}\left[\frac{3(1-d / D)-w / 2}{D-d}\right] .$ Use a linear approximation
of $\tan ^{-1} x$ at $x=0$ to show that if $d=0,$ then $\phi \approx \frac{3-w / 2}{D}$ Based on this, describe how $\phi$ changes if there is an increase in (a) $w$ or (b) $D$
GRAPH CANT COPY

Key Concepts

Linear Approximation

Linear approximation is a method for estimating the value of a function near a specific point using the tangent line at that point. In general, if a function f(x) is differentiable at x = a, then f(x) can be approximated by f(a) + f'(a)(x - a) for values of x near a. This simplifies analysis by reducing nonlinear behavior to linear behavior near the point of interest.

Small-Angle Approximation for Inverse Trigonometric Functions

For small values of x, many trigonometric functions can be approximated by their linear terms. Specifically, the inverse tangent function, tan?¹x, is approximated by x when x is near zero. This small-angle approximation is particularly useful for simplifying expressions in physics and engineering when the angle in question is small.

Sensitivity Analysis of Parameters

Sensitivity analysis involves examining how the output of a function or model changes when its input parameters are varied. In approximation formulas, understanding parameter sensitivity helps in predicting the effect of marginal increases or decreases in variables. For example, in a linearized model, the impact of changes in a variable is directly reflected by its coefficient, clarifying how adjustments such as an increase in a blocking width or a change in distance affect the overall result.

Transcript

00:01 In problem 67 for a hockey game where we have the goal with equals w and we have a goalie here that stands at a distance d from the net d small and from the net we have the shooter or the hockey player at distance d capital and we have five which is the left and of the galley equals this expression we want to use linear approximation of ten inverse of x at x equal 0 when this term is about to be 0 we want to get the linear approximation at this point linear approximation is about to get the graph of a function add the specified value we get the tangent equation of the function at this point.

01:24 This means we get we want to get the tangent equation of 10 inverse of x when x equals you.

01:32 The linear approximation function equals or the linear approximation equation equals f of x node plus f dash of x node multiplied by x minus x node.

01:53 What is the function f of x? f of x is the equation we need to get its tangent if x is 10 inverse of x and what is f dash of x sorry what is f of x is the function we want to get its tangent which is then inverse of x and what is f dash of x f dash of x is the differential of this function f dash of x is a differentiation of 10 inverse x which is 1 divided by 1 plus x squared now we can evaluate f of x node and f dash of x node to get the equation of the tangent f of x node or f of 0 equals 10 inverse of 0 equals 0 and f dash of x node or f dash of 0 equals 1 plus 1 plus 0 1 divided by 1 plus 0 equals 1 by 1 by substituting here we get the equation of the tangent l of x equals f of x node which is 0 plus f dash of x node which is 1 multiply by x minus x node which is x minus 0 this sums up to be x this means the linearization of 10m of x at x equal 0 equals x and let's related to our problem what is x x is this inner function then we have phi approximately equals 3 multiplied by 1 minus d divided by d minus w divided by 2 divided by 2 divided by d and when d equals 0 where the guli is here d equals 0 we have 5 approximately equals 3 multiplied by 1 minus 0 minus w divided by 2 divided by d minus 0 which equals 3 minus w divided by 2 divided by 2 divided by d to be proven let's see what happens when w changes for example if w changes for example increases we can see what happens to the nominator the nominator decreases this means the the overall fraction will decrease phi will decrease the angle decrease and it means if if w decreases the angle increases an inverse relationship.

05:48 Let's see what happens if d changes...

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