Units Of Measure
A sphere is expanding such that its volume is increasing at a rate of 15 cubic inches per second. What are the units of the rate of change of the radius?
The units of the rate of change of the radius are inches per second (in/s). Since the volume of a sphere is proportional to the cube of the radius (V = 4/3πr³), the rate of change of the radius will have units of length per time.
Find the limit of X as x approaches 0
sin(x)/x
Answer: 1
A particle moves along a line with position x(t)= t^3-6t^2+9t.
a) Find the velocity at t=2
b) Find acceleration at time t=2
a) -3
b) 0
A circular puddle is growing in such a way that its radius increases at a rate of 2 cm/min. How fast is the area of the puddle increasing when the radius is 5 cm?
20π cm^2/min
The height of a ball thrown in the air is modeled by the function h(t)=-5t^2+20t+2, where h is the height in meters and t is time in seconds.
Find the average rate of change of the ball's heights between t=1 seconds and t=3 seconds.
What is 0 meters per second?
A ladder is leaning against a wall. The top of the ladder is sliding down the wall at a rate of 2 feet per second. What are the units of the rate at which the bottom of the ladder is moving away from the wall?
The units of the rate at which the bottom of the ladder is moving away from the wall are feet per second (ft/s), as this describes the speed of the bottom of the ladder in relation to time.
Limit of x as x approaches infinity
2x2+3/5x2-7
v(t)=10sin(.4t^2)/t^2-t+3 [0,3.5]
a) Speed at t=2?
b) What's the displacement of the particle?
c ) Total distance traveled?
a) 1.999
b) 2.843
c) 3.737
A 10-foot ladder is leaning against a vertical wall. The bottom of the ladder is sliding away from the wall at 1 ft/sec. How fast is the top of the ladder sliding down the wall when the bottom is 6 feet from the wall?
:-¾ ft/sec
The temperature in a city over a day can be modeled by T(x)=-0.5x^2+4x+20, where x is the number of hours after midnight.
Find the average rate of change of the temperature from 2 a.m. to 6 a.m.
What is 0 degrees per hour?
The area of a circle is increasing at a rate of 10 square meters per second. What are the units of the rate of change of the radius of the circle?
The units of the rate of change of the radius are meters per second (m/s). Since the area of a circle is proportional to the square of the radius (A = πr²), the rate of change of the radius is in units of length per time.
When the limit of x as x approaches 0 or infinity equals 0/0 or infinity/infinity, what should you do?
Use L'hopitals rule
v(t)= -2+(t^2+3t)^6/5-t^3 position at t=0 is 10.
a) What is the position of the particle after 4 seconds?
b) What is the total distance from 0 to 4 seconds?
a) 16.531
b) 14.739
A balloon is rising vertically at a rate of 3 m/s. A person is standing 20 meters away from the point directly beneath the balloon. How fast is the distance between the person and the balloon increasing when the balloon is 15 meters high?
1.8 m/s
The position of a car along a straight road at time t is given by s(t)=2t^3-9t^2+12t (in meters).
Estimate the instantaneous rate of change of the car's position at t=2 seconds using a difference quotient with change in time =0.1.
What is approximately 0.32 meters per second?
A balloon is being inflated, and its volume is increasing at a rate of 30 cubic feet per minute. What are the units of the rate of change of the radius of the balloon?
The units of the rate of change of the radius are feet per minute (ft/min). The volume of the balloon is related to the cube of the radius, so the radius changes in terms of feet per minute.
Find the limit of x as x approaches 0+
ln(x) / 1/x
Answer: 0
For t≥0 is given by a(t)= t+3/√(t^3+1)
a) Velocity at t=0 is 5, what is the velocity of the particle at t=3?
a) 11.70
Water is being poured into a conical tank (point down) at a rate of 5 m³/min. The tank has a height of 6 m and a base radius of 3 m. How fast is the water level rising when the water is 4 m deep?
5/4π m/min
Given the function f(x)=x^3-6x^2+5x,
find the exact instantaneous rate of change at x=2 by finding the derivative and evaluating it.
What is -7?
Water is being poured into a cone-shaped tank at a rate of 5 cubic feet per minute. If the height of the water in the tank is increasing at a rate of 0.3 feet per minute, what are the units of the rate of change of the radius of the water surface?
The units of the rate of change of the radius of the water surface are feet per minute (ft/min). The radius is increasing with respect to time, and since the tank is shaped like a cone, the rate of change of the radius will be in units of length per time.
find the limit of x as x approaches 0
tan(x) / x
answer: 1
v(t)=5+e^t/3, time t≥0
What is the acceleration at t=4?
1.265
Car A is traveling east at 60 km/h, and Car B is traveling north at 80 km/h. They are both heading toward the same intersection. At what rate are the two cars approaching each other when Car A is 3 km and Car B is 4 km from the intersection?
100 km/h
A population of bacteria grows according to P(t)=100e^0.03t, where t is measured in hours.
(a) Find the average rate of change of the population between t=0 and t=5.
(b) Find the instantaneous rate of change at t=5 by finding P′(5).
A): ~3.24 bacteria/hour
B): ~3.49 bacteria/hour