62​–​65. {Use of Tech} Graphing f and f'

b. Compute and graph f'.

f(x) = (x−1) sin^−1 x on [−1,1]

A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.

b. Show that y = B cos t satisfies the equation for any constant B.

For what values of x does g(x) = x−sin x have a slope of 1?

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

x⁴+y⁴ = 2;(1,−1)

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.​

Find the velocity of the oscillator, v(t) =y′(t).

Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = (x + 5) / (x - 1); a = 3

79–82. {Use of Tech} Visualizing tangent and normal lines

b. Graph the tangent and normal lines on the given graph.

x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

x = e^y; (2, ln 2)

Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.

b. Evaluate this derivative when a=6 and b=10.

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

b. (f^-1)'(6)

45–50. Tangent lines Carry out the following steps. <IMAGE>

b. Determine an equation of the line tangent to the curve at the given point.

x³+y³=2xy; (1, 1)​​

Velocity of a car The graph shows the position s=f(t) of a car t hours after 5:00 P.M. relative to its starting point s=0,where s is measured in miles. <IMAGE>

b. At approximately what time is the car traveling the fastest? The slowest?

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).

Find an equation of the line tangent to y = h(x) at x = 3.

Vertical​ ​tangent lines

b. Does the curve have any horizontal tangent lines? Explain.

{Use of Tech} Hours of daylight The number o​​f hours of daylight at any point on Earth fluctuates throughout the year. In the Northern Hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40° north latitude, the length of a day is approximated by D(t) = 12−3 cos (2π(t+10) / 365), where D is measured in hours and 0≤t≤365 is measured in days, with t=0 corresponding to January 1.

b. Find the rate at which the daylight function changes.

Product Rule for three functi​​ons Assume f, g, and h are differentiable at x.

b. Use the formula in (a) to find d/dx(e^x(x−1)(x+3))

{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure). <IMAGE>

b. Graph dθ/dx as a function of x and determine the point at which θ changes most rapidly.

109-112 {Use of Tech} Calculating limits​ The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\underset{h\to 0}{\lim }\frac{\frac{1}{3{({(1+h)}^5+7)}^{10}}-\frac{1}{3{\left(8\right)}^{10}}}{h}$

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>

b. What is the rate of change of θ with respect to the bird’s height when it is 60 ft above the ground?

{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.

b. Graph the volume function and verify that the tank is empty when t=1000 min. 

Use the graph of g in the figure to do the following. <IMAGE>

b. Find the values of x in (-2,2) at which g is not differentiable.

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.

A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and ϕ(t), respectively, where 0≤t≤4 and t is measured in minutes (see figure). These angles a​​re measured in radians, where θ=ϕ=0 represent the starting position and θ=ϕ=2π represent the finish position. The angular velocities of the runners are θ′(t) and ϕ′(t). <IMAGE>

b. Which runner has the greater average angular velocity?

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

sin y = 5x⁴−5; (1, π)

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>

b. Estimate the value of L'(a) for a ≥ 4 . What does this tell you about the talon lengths on these birds? (Source: ZooKeys, 132, 2011)

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

b. When does the stone reach its highest point?

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(x) = 1/x+1; a = -1/2;5

{Use of Tech} Spring oscillations A spr​​ing hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10sin t - 10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>

b. Find dx/dt and interpret the meaning of this derivative.  

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

b. ln(x + 1) + ln(x − 1) = ln(x² − 1), for all x.

Explain why or why not Determine whether the following statements are true and give an explanation or counter example.

b. d²/dx² (sin x) = sin x.