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Chapter 6: Problem 57

Find the equation of the tangent line to the parabola at the given point. $$x^{2}=2 y,(4,8)$$

Short Answer

Expert verified
The equation of the tangent line to the parabola \(x^{2} = 2y\) at the point (4,8) is \(y = 4x - 8\).

Step by step solution

01

Rewrite the function in y=f(x) form

The function is given as \(x^{2}=2 y\). Reordering terms, this can be rewritten as \(y=\frac{x^2}{2}\).
02

Compute the derivative of y

The derivative of \(y=\frac{x^2}{2}\) is \(\frac{dy}{dx} = x\). This derivative gives the rate of change of \(y\) with respect to \(x\), which corresponds to the slope of the tangent line to the graph of the function at any point \((x, y)\).
03

Evaluate the derivative at the point (4,8)

Evaluating the derivative \(\frac{dy}{dx}=x\) at \(x=4\) gives \(\frac{dy}{dx} = 4\). This is the slope of the tangent line to the graph of the function at the point \((4,8)\).
04

Write the equation of the tangent line

Using the slope of the tangent line and the point-slope form equation (y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point at which the tangent touches), the tangent line equation can be written as \(y - 8 = 4(x - 4)\). Simplifying this equation gives \(y = 4x - 8\).

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Most popular questions from this chapter

Consider a line with slope \(m\) and \(y\) -intercept \((0,4)\) (a) Write the distance \(d\) between the origin and the line as a function of \(m\) (b) Graph the function in part (a). (c) Find the slope that yields the maximum distance between the origin and the line. (d) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

Find the distance between the point and the line. Point \((1,4)\) Line \(y=4 x+2\)

Think About It \(\quad\) Explain what each of the following equations represents, and how equations (a) and (b) are equivalent. A. \(y=a(x-h)^{2}+k, \quad a \neq 0\) B. \((x-h)^{2}=4 p(y-k), \quad p \neq 0\) C. \((y-k)^{2}=4 p(x-h), \quad p \neq 0\)

A projectile is launched at a height of \(h\) feet above the ground at an angle of \(\theta\) with the horizontal. The initial velocity is \(v_{0}\) feet per second, and the path of the projectile is modeled by the parametric equations $$x=\left(v_{0} \cos \theta\right) t$$ and $$y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}.$$ Use a graphing utility to graph the paths of a projectile launched from ground level at each value of \(\boldsymbol{\theta}\) and \(v_{0} .\) For each case, use the graph to approximate the maximum height and the range of the projectile. (a) \(\theta=15^{\circ}, \quad v_{0}=50\) feet per second (b) \(\theta=15^{\circ}, \quad v_{0}=120\) feet per second (c) \(\theta=10^{\circ}, \quad v_{0}=50\) feet per second (d) \(\theta=10^{\circ}, \quad v_{0}=120\) feet per second

Determine whether the statement is true or false. Justify your answer. If \(D \neq 0\) and \(E \neq 0,\) then the graph of \(x^{2}-y^{2}+D x+E y=0\) is a hyperbola.
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