Particle Motion
Extrema and Concavity
Related Rates and Optimization
f' analysis
Misc. Topics
100
A particle moves along a straight line. The graph of the particle's position x(t) from 0<t<6 is graphed above. The graph has horizontal tangents at x=1 and x=5, and a point of inflection at x=2. On what interval(s) is the particle's velocity increasing?
(0,2)
100
The function f(x)=3x5-4x3-3x has a relative maximum at what value of x?
x=-1
100
The volume of a cube with edges of length x centimeters is increasing at a constant rate of 40 cubic centimeters per minute. At the instant when x=2, what is the rate of change of x, with respect to time?
10/3 cm per minute
100
The graph of f', the derivative of f, is shown above. At what value of x does f attain a relative maximum?
x=-2
100
Find the value of c that satisfies the Mean Value Theorem on the interval [0,3] for f(x)=3x2-5x.
c=3/2
200
Two particles have position x1(t)=-3t2+5t+20 and x2(t)=t2+4t. At time t=3, are the particles moving towards each other or away from each other?
away from each other
200
List the open interval(s) the function f(x)= -3x4+6x3+10x is concave down.
(-∞,0)u(1,∞)
200
A particle moves along the hyperbola xy=15 for time t>0 seconds. At a certain instant, x=3 and dx/dt=6. How is y changing at that instant?
decreasing by 10 units per second
200
The graph of f', the derivative of f, is shown in the figure above for the interval (a,b). Which of the following are true? Select all that apply.
I. f is continuous on the open interval (a,b)
II. f is decreasing on the open interval (a,b)
III. f is concave down on the open interval (a,b)
I. and III.
200
A differentiable function has the property that f(5)=3 and f'(5)=4. What is the estimate for f(4.8) using the tangent line approximation to f at x=5?
2.2
300
A car is traveling on a road with its acceleration shown in the graph above. Is the velocity of the car increasing, decreasing, or constant at time t=5?
Increasing
300
A function g is given by g(x)=4x3+3x2-6x+1. What is the absolute minimum value of g on the interval [-2,1]?
-7
300
A rectangular area is to be enclosed by a wall on one side and fencing on the other three sides. If 18 meters of fencing are used, what is the maximum area that can be enclosed?
81/2 m2
300
The table gives values of a twice-differentiable function f(x). Describe both the sign (+/-) and the behavior (increasing/decreasing) of f'(x) on the open interval (1,5).
f' is negative and increasing
300
The limit as x approaches zero of (x2)/(1-cosx) =
2
400
A particle is moving along the x-axis with position given by x(t)=cos(t)+sin(2t). What is the acceleration of the particle at time t=π?
1
400
Let the function f be defined by f(x)= -3+6x2-2x3. On what open interval is the graph of f both increasing and concave up?
(0,1)
400
The area of an isosceles right triangle with legs length s is increasing at a rate of 12 square centimeters per second at the instant when s=√32. At what rate is the length of the hypotenuse of the triangle increasing, in centimeters per second, at that instant?
3 cm/sec
400
The graph of f', the derivative of f, is shown above on the interval [1,4]. f'(x) has zeros at x=0, 2, 3, and 4, and horizontal tangents at x=1, 2.5 and 3.5. Determine the interval(s) that f is both increasing and concave down.
(1,2)u(3.5,4)
400
Let f be defined by f(x)=(lnx)/x. What is the absolute maximum value of f?
1/e
500
A particle is moving along the x-axis. For 0<t<8, the position is given by v(t)=4t3-10t2+27. Is the particle speeding up or slowing down at time t=1?
slowing down
500
If g is a differentiable function such that g(x)<0 for all real numbers x, and if f'(x)=(x2-4)g(x), which of the following is true?
A. f has relative maxima at x= -2 and x=2
B. f has relative minima at x= -2 and x=2
C. f has a relative maximum at x= -2 and a relative minimum at x=2
D. f has a relative minimum at x= -2 and a relative maximum at x=2
D.
500
Consider all right circular cylinders for which the sum of the height and circumference is 30 centimeters. What is the radius of the one with maximum volume?
10/π cm
500
For x>0, f is a function such that f'(x)=(lnx)/x and f''(x)=(1-lnx)/x2. Which of the following is true?
A. f is decreasing for x>1, and the graph of f is concave down for x>e
B. f is decresaing for x>1, and the graph of f is concave up for x>e
C. f is incresaing for x>1, and the graph of f is concave down for x>e
D. f is increasing for x>1, and the graph of f is concave up for x>e
C.
500
A car travels on a straight track. During the interval 0<t<60, the car's velocity and acceleration are continuous functions. The table above shows selected values. Must there be a time, t, on the interval 0<t<60, such that a(t)=0? Justify your answer.
Yes.
The Mean Value Theorem (or Rolle's Theorem) guarantees an t value on the interval (0,25) such that a(t)=0, since v(0)=v(25)=0. Since (0,25) is within the larger interval (0,60) there will be a time t on the interval (0,60) such that a(t)=0.