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Problem 3.11.45
Watching an elevator An observer is 20 m above the ground floor of a large hotel atrium looking at a glass-enclosed elevator shaft that is 20 m horizontally from the observer (see figure). The angle of elevation of the elevator is the angle that the observer’s line of sight makes with the horizontal (it may be positive or negative). Assuming the elevator rises at a rate of 5 m/s, what is the rate of change of the angle of elevation when the elevator is 10 m above the ground? When the elevator is 40 m above the ground? <IMAGE>
Problem 3.11.47
The bottom of a large theater screen is 3 ft above your eye level and the top of the screen is 10 ft above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of 3 ft/s while looking at the screen. What is the rate of change of the viewing angle θ when you are 30 ft from the wall on which the screen hangs, assuming the floor is horizontal (see figure)? <IMAGE>
Problem 3.11.60
A ship leaves port traveling southwest at a rate of 12 mi/hr. At noon, the ship reaches its closest approach to a radar station, which is on the shore 1.5 mi from the port. If the ship maintains its speed and course, what is the rate of change of the tracking angle θ between the radar station and the ship at 1:30 P.M. (see figure)? (Hint: Use the Law of Sines.) <IMAGE>
Problem 3.11.30
Parabolic motion An arrow is shot into the air and moves along the parabolic path y=x(50−x) (see figure). The horizontal component of velocity is always 30 ft/s. What is the vertical component of velocity when (a) x=10 and (b) x=40? <IMAGE>
Problem 3.11.33
Piston compression A piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 3 cm/s (see figure). What is the rate of change of the volume of the cylinder when the piston is 2 cm from the base of the chamber? <IMAGE>
Problem 3.14
9–61. Evaluate and simplify y'.
y = (2x−3)x^3/2
Problem 3.15
If f(t)=t¹⁰, find f′(t), f′′(t), and f′′′(t).
Problem 3.17
The line tangent to the graph of f at x=5 is y = 1/10x-2. Find d/dx (4f(x)) |x+5
Problem 3.19
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
y = x⁵
Problem 3.2.33
Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2
Problem 3.2.42
Consider the line f(x)=mx+b, where m and b are constants. Show that f′(x)=m for all x. Interpret this result.
Problem 3.2.43a
Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.
Problem 3.2.64
A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.
y= √x; P(4, 2)
Problem 3.2.2
If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at x=1, 2, and 3.
Problem 3.2.7
If f is differentiable at a, must f be continuous at a?
Problem 3.2.8
If f is continuous at a, must f be differentiable at a?
Problem 3.2.10
Use the graph of f(x)=|x| to find f′(x).
Problem 3.2.11
Use limits to find f' (x) if f(x) = 7x.
Problem 3.2.23a
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 4x²+1; a= 2,4
Problem 3.2.23b
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(x) = 4x²+1; a= 2,4
Problem 3.2.25a
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
Problem 3.2.25b
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(x) = 1/x+1; a = -1/2;5
Problem 3.2.27a
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
Problem 3.2.27b
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(t) = 1/√t; a=9, 1/4
Problem 3.2.29a
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(s) = 4s³+3s; a= -3, -1
Problem 3.2.29b
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(s) = 4s³+3s; a= -3, -1
Problem 3.2.30a
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(t) = 3t⁴; a= -2, 2
Problem 3.2.30b
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(t) = 3t⁴; a= -2, 2
Problem 3.2.31a
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
Problem 3.2.31b
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).
b. Determine the instantaneous velocity of the projectile at t=1 and t = 2 seconds.
s(t)= −16t²+100t