미분법 전체 문제

For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

[T] $xy+\sin (x)=1,\left(\frac{\pi }{2},0\right)$

[T] The graph of a folium of Descartes with equation $2{x}^{3}+2{y}^{3}-9xy=0$ is given in the following graph.

a. Find an equation of the tangent line at the point $(2,1).$ Graph the tangent line along with the folium. b. Find an equation of the normal line to the tangent line in a. at the point $(2,1).$

For the equation ${x}^{2}+2xy-3{y}^{2}=0,$

a. Find an equation of the normal to the tangent line at the point $(1,1).$ b. At what other point does the normal line in a. intersect the graph of the equation?

Find all points on the graph of ${y}^{3}-27y={x}^{2}-90$ at which the tangent line is vertical.

Find an equation of the tangent line to the graph of the equation ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{6}$ at the point $\left(0,\frac{1}{2}\right).$

Find an equation of the tangent line to the graph of the equation ${\tan }^{-1}(x+y)={x}^{2}+\frac{\pi }{4}$ at the point $(0,1).$

For the following exercises, find an equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.

Find ${y}^{'}$ and ${y}^{''}$ for ${x}^{2}+6xy-2{y}^{2}=3.$

[T] Find an equation of the tangent line to the graph of $f(x)=4x{e}^{({x}^{2}-1)}$ at the point where

$x=-1.$ Graph both the function and the tangent line.

[T] Find an equation of the tangent line to the graph of ${x}^{3}-x\ln y+{y}^{3}=2x+5$ at the point (2, 1). (Hint: Use implicit differentiation to find $\frac{dy}{dx}.)$ Graph both the curve and the tangent line.

Consider the function $y={x}^{1/x}$ for $x>0.$

a. Determine the points on the graph where the tangent line is horizontal. b. Determine the points on the graph where ${y}^{'}>0$ and those where ${y}^{'}<0.$

The formula $I(t)=\frac{\sin t}{{e}^{t}}$ is the formula for a decaying alternating current.

a. Complete the following table with the appropriate values.

[T] Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year.

[T] Using the exponential best fit for the data, write a table containing the second derivatives evaluated at each year.

A 10-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?

Two airplanes are flying in the air at the same height: airplane A is flying east at 250 mi/h and airplane B is flying north at $300\,\text{mi/h}.$ If they are both heading to the same airport, located 30 miles east of airplane A and 40 miles north of airplane B, at what rate is the distance between the airplanes changing?

A 6-ft-tall person walks away from a 10-ft lamppost at a constant rate of $3\,\text{ft/sec}.$ What is the rate that the tip of the shadow moves away from the pole when the person is $10\,\text{ft}$ away from the pole?

A trough has ends shaped like isosceles triangles, with width 3 m and height 4 m, and the trough is 10 m long. Water is being pumped into the trough at a rate of $5\,{\text{m}}^{3}\text{/min}.$ At what rate does the height of the water change when the water is 1 m deep?

A tank is shaped like an upside-down square pyramid, with base of 4 m by 4 m and a height of 12 m (see the following figure). How fast does the height increase when the water is 2 m deep if water is being pumped in at a rate of $\frac{2}{3}$ m³/sec?

A lighthouse, L, is on an island 4 mi away from the closest point, P, on the beach (see the following image). If the lighthouse light rotates clockwise at a constant rate of 10 revolutions/min, how fast does the beam of light move across the beach 2 mi away from the closest point on the beach?

In Exercises 37–40, find the equations of the tangent and normal lines to the graph of $g$ at the indicated point.

$g(s) = e^s(s^2+2)$ at $(0,2)$.