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>

a. Estimate L' (1.5) and state the physical meaning of this quantity.

Airline travel The following figure shows the ​​position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>

a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 ≤ t ≤ 1.5).

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = √3x; a= 12

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

a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)

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

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

a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)

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

{Use of Tech} Tree growth ​Let​ ​b ​​represent ​the base diameter of a conifer tree and ​let ​​h ​represent the height of the tree, where ​b ​is measured in centimeters ​and ​​h ​is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².

a. Graph the height function.

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(x) = 4x²+1; a= 2,4

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 1/x; P (1,1)

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.

a. Find dr/dh for a cone with a lateral surface area of A=1500π.

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

a. Graph f with a graphing utility.

f(x)=e^−x tan^−1 x on [0,∞)

Deriving​​ trigonometric identities

a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

Find an equation of the line tangent to the given curve at a.

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

Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.

31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).

a. For the following functions s(t), find the instantaneous velocity function v(t). (Recall that the velocity function v is the derivative of the position function s.)

s(t)= −16t²+100t

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>

a. (f^-1)'(4)

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 8x; a = −3

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

a. Use implicit differentiation to find dy/dx.

The volume V of a sphere of radius r changes over time t.

a. Find an equation relating dV/dt to dr/dt.

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 1/ √x; a= 1/4

Shrinking isosceles triangle The hypotenuse of an isosceles right triangle decreases in length at a rate of 4 m/s.

a. At what rate is the area of the triangle changing when the legs are 5 m long?

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = -7x; P(-1,7)

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

<IMAGE>

a. $h^{\prime }\left(3\right)$

13-26 Implicit differentiation Carry out the following steps.

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

³√x+³√y⁴ = 2;(1,1)

Derivatives and tangent lines

b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.

f(x) = √3x; a= 12

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)​​

The energy (in joules) released by an earthquake of magnitude M is given by the equation E=25,000 ⋅ 10^1.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)

Compute dE/dM and evaluate it for M=3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)

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

b. Compute and graph f'.

f(x)=e^−x tan^−1 x on [0,∞)

Vertical​ ​tangent lines

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

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 bug is moving along the right side of the parabola y=x² at a rate such that its distance from the origin is increasing at 1 cm/min.

b. Use the equation y=x² to find an equation relating dy/dt to dx/dt.