Log Function Derivative and Tangent/Normal Lines
Consider the function \(y = \ln(x)\). (a) Find the derivative of \(y = \ln(x)\). (b) Determine the equation of the normal line to the curve at the point where \(x = e\).
Minimizing Distance Using the Normal Line
Given the curve \(y=x^2\) and an external point \(A(0,3)\), find the coordinates of the point \(Q\) on the curve such that the normal line at \(Q\) passes through \(A\). Hence, determine the minimum distance from \(A\) to the curve.
Part b
Determine the point of intersection (other than \((1, 3)\)) between the normal line at \(x=1\) and the curve \(f(x)= x^2 + 2\).
Analysis of Tangent and Normal Lines in Real Context
A roadway curve is modeled by the function \(f(x) = x^2 + 2\). In the context of designing safe road curves, both the tangent and normal lines to the curve are used to assess vehicle turning behavior. Determine the following: (a) Find the derivative \(f'(x)\) at \(x = 1\) and hence determine the equation of […]
Normal Line X-Intercept for f(x)=x^2 at x=2
Consider the function \(f(x)=x^2\). Find the x-coordinate of the x-intercept of the normal line to the curve at the point where \(x=2\).
Finding the Normal Line from Parametric Equations
Consider the parametric functions \(x(t) = t^2 + 1\) and \(y(t) = 2t – 3\). At \(t = 2\), find the slope of the normal line to the curve and provide its equation.
Tangent and Normal Slopes for a Quadratic Function
Given the function \(f(x) = x^2\), find the slope of the tangent line to the graph of \(f\) at \(x = 2\). Then, determine the slope of the normal line at the same point.
Normal Line and Temperature Gradient Analysis
A heated metal rod has a temperature distribution modeled by the function \(T(x) = k e^{-0.5x} + c\), where \(k\) and \(c\) are constants and \(x\) is the position along the rod in meters. This function represents how the temperature in degrees Celsius changes along the rod. (a) Model derivative: Determine the derivative \(T'(x)\) to […]
Part b
Find the equation of the normal line to \(P(x) = 800 – 3x\) at \(x = 20\), and explain how this normal line can be used to estimate the change in price if production increases by 5 (hundred units).
Business Revenue Tangent Analysis
A company’s demand function is given by \(P(x) = 800 – 3x\), where \(x\) represents the number of hundreds of units sold and \(P(x)\) is the price per unit in dollars. The company wants to analyze its revenue model around a production level of \(x = 20\). (a) Determine the slope of the tangent line […]