Derivative at a Point question

Question

Let w(x)=2x+ 4/x

1 Fill in the blanks:

w'(5) is the limit at x approaches ______ of the expression  _____. The value of this limit is _____.

2.Fill in the right-hand side of the equation below so it is the equation of a line tangent to the graph of y=w(x) at x=5

y=____.

Answer 1

1. w(5) is the limit as x approaches 0 of the expression [w(x + 5) - w(5)] / x. The value of this limit is 46/25.

That is one form of the limit definition of the derivative of a function (here, w) at a point (here, 5). The expression:

2(x +5 ) + 4/(x+5) - ( 2(5) + 4/5 )

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x

which eventually simplifies to

lim (10x + 46) / (5x + 25) = 46/25

x→0

2. The slope of the tangent to w(x) at x=5 is the first derivative there, i.e. w'(5)=46/25, from above. So, in slope-intercept form, the equation of this tangent is y = mx + b, i.e. y = (46/25)x + b . To find b, first find y when x=5: y = w(5) = 2(5) + 4/5 = 54/5 . Therefore, b = y - mx = 54/5 - 5(46/25) = 8/5, so the equation of the tangent is y = (46/25)x + 8/5.

Answer 2

1. w'(5) = lim x→5 (w(x) - w(5))/(x - 5) = lim x→5 (2x + 4/x - 10 - 4/5)/(x - 5) ) = lim x→5 (10x² - 54x + 20)/(x(x - 5)) = lim x→5 2(x - 5)(5x - 2)/(x(x - 5)) = lim x →5 2(5x - 2)/x = 46/5
2. w(5) = 10 + 4/5 = 54/5; y = 54/5 + 46/5(x - 5); y = 46x/5 - 176/5