Find an equation for the tangent line to the following curve,
[tex]$h(x)=18 x-2 \ln x$[/tex]
at the point where [tex]$x=e$[/tex].
Answer (1)
Evaluate the function at x = e to find the y-coordinate of the point of tangency: h ( e ) = 18 e − 2 .
Find the derivative of the function: h ′ ( x ) = 18 − x 2 .
Evaluate the derivative at x = e to find the slope of the tangent line: h ′ ( e ) = 18 − e 2 .
Use the point-slope form to find the equation of the tangent line: y = ( 18 − e 2 ) x .
y = ( 18 − e 2 ) x
Explanation
Problem Analysis We are given the function h ( x ) = 18 x − 2 ln x and we want to find the equation of the tangent line at x = e . To do this, we need to find the point ( e , h ( e )) and the slope of the tangent line at that point, h ′ ( e ) .
Finding the Point of Tangency First, let's find the y -coordinate of the point of tangency by evaluating h ( e ) .
h ( e ) = 18 e − 2 ln e = 18 e − 2 ( 1 ) = 18 e − 2 Using a calculator, we find that 18 e − 2 ≈ 18 ( 2.71828 ) − 2 ≈ 49 − 2 = 46.929 . So the point of tangency is approximately ( e , 46.929 ) .
Finding the Derivative Next, we need to find the derivative of h ( x ) .
h ′ ( x ) = d x d ( 18 x − 2 ln x ) = 18 − x 2
Finding the Slope Now, we evaluate the derivative at x = e to find the slope of the tangent line at that point. h ′ ( e ) = 18 − e 2 ≈ 18 − 2.71828 2 ≈ 18 − 0.73576 = 17.264
Finding the Tangent Line Equation Now we use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the point of tangency ( e , 18 e − 2 ) and m is the slope 18 − e 2 .
y − ( 18 e − 2 ) = ( 18 − e 2 ) ( x − e ) y − 18 e + 2 = ( 18 − e 2 ) x − ( 18 − e 2 ) e y − 18 e + 2 = 18 x − e 2 x − 18 e + 2 y = ( 18 − e 2 ) x So the equation of the tangent line is y = ( 18 − e 2 ) x .
Final Answer Therefore, the equation of the tangent line to the curve h ( x ) = 18 x − 2 ln x at x = e is y = ( 18 − e 2 ) x .
Examples
Consider a scenario where you're optimizing the production cost of a factory. The function h ( x ) = 18 x − 2 ln x might represent the cost of producing x units, where 18 x is the cost of raw materials and − 2 ln x represents the savings from economies of scale. Finding the tangent line at a specific production level x = e helps you estimate the marginal cost at that level. This marginal cost approximation is crucial for making informed decisions about increasing or decreasing production to maximize profit.
