(a) In the diagram, a hockey player is D feet from the net...

(a) In the diagram, 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 one side 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 (i) $w$ or (ii) $D$.

Key Concepts

Parameter Sensitivity Analysis

Examining the simplified expression ? ? (3 - w/2)/D allows us to understand how changes in the parameters affect the shooting angle. An increase in w directly decreases the numerator, thus reducing ?, while an increase in D increases the denominator, thereby also reducing ?. This analysis is important for understanding how geometric and physical constraints influence the angle.

Inverse Trigonometric Function Approximation

The approximation tan?¹(x) ? x for small x is a key concept that enables the simplification of expressions involving arctan. By applying this approximation to the formula for ? under the condition d = 0, we obtain a linear relationship between ?, w, and D, showing that ? ? (3 - w/2)/D.

Linear Approximation

Linear approximation involves approximating a function near a specific point using the value of the function and its derivative at that point. In this case, the function tan?¹(x) is approximated by x when x is close to 0, which simplifies the algebra and shows how ? can be approximated when d = 0.

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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