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.

$\underset{x\to 0}{\lim }\frac{\sqrt{4+\sin \left(x\right)}-2}{x}$

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)

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.

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

b. Compute and graph f'.

f(x)=e^−x tan^−1 x on [0,∞)

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

b. Graph the tangent lines on the given graph.

4x³ =y²(4−x); x=2 (cissoid of Diocles)​​

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = − 0.01x²+40x+100, 0≤x≤1500, a=1000

{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.  

{Use of Tech} Computing limits with ​​angles in degrees Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.

b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

b. Graph the tangent lines on the given graph.

x+y³−y=1; x=1

Use a graphing utility to plot the curve and the tangent line.

y = cos x / 1−cos x; x = π/3

13-26 Implicit differentiation Carry out the following steps.

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

tan xy = x+y; (0,0)

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

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

(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

{Use of Tech} Fuel economy Sup​​pose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with g gallons of gas remaining in the tank on a particular stretch of highway is given by m(g) = 50g−25.8g²+12.5g³−1.6g⁴, for 0≤g≤4.

b. Graph and interpret the gas mileage m(g)/g. 

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?

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>

{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.

b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

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?

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)

Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².

b. Evaluate this derivative when r=30 and h=40.

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.

x⁴-x²y+y⁴=1; (−1, 1)

{Use of Tech} Bungee jumpe​​r A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e^−t cos t), for t ≥ 0.

b. Use a graphing utility to determine when she is moving downward and when she is moving upward during the first 10 s.  

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.

$\underset{x\to 2}{\lim }\frac{{(x^2-3)}^5-1}{x-2}$

58–59. Carry out the following steps.

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

xy^5/2+x^3/2y=12; (4, 1)

21–30. Derivatives

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

f(x) = 4x²+1; a= 2,4

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

13-26 Implicit differentiation Carry out the following steps.

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

(x+y)^2/3=y; (4, 4)

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. Use the table to compute the following derivatives.

<IMAGE>

b. $h^{\prime }\left(2\right)$

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

b. d/dx(tan^−1 x) =sec² x

Deriving​​ trigonometric identities

b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

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>

c. (f^-1)'(1)

Highway travel A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 A.M. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 A.M. (see figure). Assume s is positive when the car is north of the patrol station. <IMAGE>

c. Find the average velocity of the car over the interval [1.75, 2.25]. Estimate the velocity of the car at 11:00 A.M. and determine the direction in which the patrol car is moving.