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

a. Graph f with a graphing utility.

f(x) = (x−1) sin^−1 x on [−1,1]

{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by

f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>

Let f (x) = √x.

a. Find the exact value of f' (4).

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

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

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)

Derivatives and tangent lines

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

f(x) = 1/ √x; a= 1/4

Use definition (1) (p. ​133​) to find the slope of the line tangent to the graph of f at P.

f(x) = 2/√x; P(4,1)

Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. <IMAGE>

a. At which points is the slope of the curve negative?

Use definition (1) (p. ​133​) to find the slope of the line tangent to the graph of f at P.

f(x) = -3x² - 5x + 1; P(1,-7)

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

³√x+³√y⁴ = 2;(1,1)

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

a. Verify that the given point lies on the curve.

(x²+y²)²=25/4 xy²; (1, 2)​​

{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>

a. What is the rate of change of the angle of elevation dθ/dx when the plane is x=500 m past the observer?

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by $y=10e^{-\frac{t}{2}}\cos \left(\frac{\pi t}{8}\right)$.

a. Graph the displacement function. 

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

f(x) = 2x + 1; P(0,1)

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

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

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.

Determine her velocity at t = 1 and t = 3. 

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

a. Find equations of all lines tangent to the curve at the given value of x.

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

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

cos y = x; (0, π/2)

{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(t) = 1/√t; a=9, 1/4

{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.

a. Graph and interpret the mileage function.

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

f(x) = 8 - 2x²; P(0, 8)

Comparing velocities Two stones are thrown vertically upward, each with an initial velocity of 48 ft/s at time t=0. One stone is thrown from the edge of a bridge that is 32 feet above the ground, and the other stone is thrown from ground level. The height above the ground of the stone thrown from the bridge after t seconds is f(t) = − 16t²+48t+32. and the height of the stone thrown from the ground after t seconds is g(t) = −16t²+48t.

a. Show that the stones reach their high points at the same time.

58–59. Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

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

21–30. Derivatives

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

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

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

f(x) = x² - 4; P(2, 0)

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)

Derivatives and tangent lines

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

f(x) = √2x+1; a= 4

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>

a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)

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.

a. Find db/da for a torus with a volume of 64π².

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

a. Verify that the given point lies on the curve.

x³+y³=2xy; (1, 1)​​