Motion
Related Rates
Linearization
L'Hospitals
Random
100
No Calc
A particle moves along the x-axis. The function x(t) gives the particle's position in cm at any time t≥0 in seconds.
x(t)=t4-2t2-4
What is the particle's velocity at t=1? Include units with your answer.
0
100
No Calc
Ohm’s law states that if the resistance of the path of current between two points is constant, then the voltage difference V between the points and the current I flowing between the points, measured in amperes, satisfy the relationship V=cI, where c is a constant. Which of the following best describes the relationship between the rate of change with respect to time t of the voltage and the rate of change with respect to time t of the current?
A: dv/dt=c(dI/dt)
B: dv/dt = dI/dt
C: c(dv/dt) = dI/dt
A: dv/dt=c(dI/dt)
100
No Calc
The line tangent to the graph of the twice-differentiable function f at the point x=3 is used to approximate the value of f(3.25). Which of the following statements guarantees that the tangent line approximation at x=3.25 is an underestimate of f(3.25) ?
A: The function f is decreasing on the interval 3≤x≤3.25
B: The function f is increasing on the interval 3≤x≤3.25
C: The graph of the function f is concave down on the interval 3≤x≤3.25
D: The graph of the function ff is concave up on the interval 3≤x≤3.25
D
100
No Calc
limx→0 x2/(1−cosx) is
2
100
Given velocity, in cm/seconds, v(x)=x2 - 6x-7, determine if the particle is speeding up or slowing down at x = 10 seconds. Justify your answer.
Particle is speeding up at x = 10 since velocity and acceleration have the same signs at x = 10.
v(10) = 33
a(x) = 2x - 6 so a(10) = 14
200
No Calc
A particle moves along the x-axis. The graph of the particle’s velocity v(t) at time t is shown above for 0 < t < 4.5. How many times does the particle change direction over the time interval 0 < t < 4.5 ? Justify your answer.
Two times
When the particle changes direction, v(t) then changes its sign. It doesn't change signs at 2!
200
No Calc
A triangle has base b centimeters and height h centimeters, where the height is three times the base. Both b and h are functions of time t, measured in seconds. If A represents the area of the triangle, which of the following gives the rate of change of A with respect to t ?
dA/dt=3b(db/dt)cm2/sec
200
No Calc
The locally linear approximation of the differentiable function f at x=3 is used to approximate the value of f(3.2). The approximation at x=3.2 is an overestimate of the corresponding function value at x=3.2. Which of the following could be the graph of f?
A:
B:
C:
D:
D
200
Which of the following limits does not yield an indeterminate form?
A: limx→0 4x3/(cos(x)−1)
B: limx→3 ln(x/3)/(x2−7x+12)
C: limx→π (π−x)/(sin(2x)−1)
D: limx→∞ x10/(e2x+x)
C
200
No Calc
A particle moves along the x-axis. The graph of the particle’s velocity v(t) at time t is shown above for 0 < t < 4.5. How many times is the acceleration = 0? Justify your answer.
5 times
every time where v(t) has the slope of the tangent line = 0, which is a relative min or max of v(t).
300
A rock thrown vertically upward from the surface of the moon at a velocity of 32 meters per second reaches a height of s(t) = 32t - 0.8t2 meters in t seconds.
How long did it take the rock to reach its highest point? Include units.
t=20 seconds
at rest means v(t)=0
32-1.6t=0
300
No Calc
A particle moves on the circle x2+y2=100 in the xy-plane for time t≥0. At the time when the particle is at the point (8,6), the value of dx/dt is 5. What is the value of dy/dt at this time?
dy/dt=-20/3
300
No Calc
The function f is twice differentiable with f(2) = 1 f′(2) = 4 , and f″(2) = 3 . What is the value of the approximation of f(1.9) using the line tangent to the graph of f at x = 2 ?
1.4
300
Given: x = 2, f(x) = 4, f'(x)=3, g(x) = 2, g'(x) = 1
Selected values of the twice-differentiable functions f and g and their derivatives are given above. The value of limx→2 (x2f(x)−16)/(g(x)−2) is
28
300
Find the derivative of the function. (hint: rewrite it)
𝑦 = csc 𝑥 cos x - e2x
-csc2x - 2e2x
400
No Calc
A particle moves along the y-axis so that at time t ≥ 0 its position is given by y(t) = (2/3)t3 -5t2 + 8t. Over the time interval 0 < t < 5, for what values of t is the speed of the particle increasing?
0<t<1 and 4<t<5
SPEED INCREASING Acceleration and velocity have to have the same signs
SPEED DECREASING Acceleration and velocity have to have the opposite signs
400
No Calc
A 10-foot ladder is leaning straight up against a wall when a person begins pulling the base of the ladder away from the wall at the rate of 1 foot per second. Which of the following is true about the distance between the top of the ladder and the ground when the base of the ladder is 9 feet from the wall?
The distance is decreasing at a rate of 9/(√19) feet per second.
400
No Calc
Let f be the function given by f(x) = 2 cos x + 1. What is the approximation for f(1.5) found by using the line tangent to the graph of f at x = π/2 ?
π-2
400
No Calc
The figure above shows the graph of the twice-differentiable function f and the line tangent to the graph of f at the point (0,2). The value of
limx→0 (f(x)e-x−2)/(x2−2x) is
2
first time you'll get 0/0 so it is Hospital-able...take the derivative and retry
400
No Calc
Determine f'(x) in simplified form if f(x) = 2cos(x)*ln(x2)
4cos(x)/x - 2sin(x)*ln(x2)
500
A particle moves along the x-axis so that at time t its position is given by:
x(t)=t3 - 6t2 + 9t + 11
where t is measured in seconds and x is measured in feet. What is the displacement over the first six weeks? Include units.
54 feet
x(6)-x(0)
65-11
500
An ice cube is melting at a rate of 5 cubic cm per hour. At what rate is the edge of the cube changing when the edge of the cube is 3 cm.
-5/27 cm/hour
500
Let 𝑓 be a function with 𝑓(2)=-2 such that for all points (x,y) on the graph of 𝑓 the slope is given by x2-1/2y. Write an equation for the line tangent to the graph of 𝑓 at x=2 and use it to approximate 𝑓(2.2).
y+2=0.75(x - 2)
y = -2.15
500
limx→∞ (ln(e3x+x))/x=
3
500
No Calc
Determine the velocity, in cm/sec, at t = 2 seconds if position along the x-axis is given by x(t) = (2t2 - 6t + 7)3.
v(2) = 54 cm/sec