Kinematics
These are the first and second derivatives of position with respect to time, respectively.
What are velocity and acceleration?
This is the integral of -cos(x).
What is -sin(x) + C?
These are the assumptions we utilized to help us derive the exponential growth/decay function y = y0ekt.
What are (1) the population growth/decay is proportional to the size of the population and (2) there is no lack of resources.
This is the constant solution for dy/dx = (y-1)x.
What is y = 1?
You invest $5,000 into an investment account with an interest rate of 0.035%, compounded continuously. This is the amount you will have in the account after 5 years.
What is $5,956.23?
A car moving at 40 m/s applies the brakes, decelerating at -5 m/s2. This is the distance the car covers before it stops.
What is 160 meters?
This is the integral of (1/x)dx.
What is ln|x| + C?
A town has a population of 67,824. The population is decreasing at an annual rate of about 2.5% per year. This is what the approximate population of the town will be after 20 years.
What is about 41,137 people?
This is the general solution for dy/dx = 4x2.
What is 4x3/3 + C?
You invest $10,000 into a savings account with a 4.44% interest rate, which is compounded monthly. This is the approximate amount of money that will be in the account after 5 years.
What is $12,455.75?
A car is speeding down the road at a constant 50m/s. A police officer starts from rest and begins to chase the car after 4 seconds, accelerating uniformly at 5m/s2. How far down the road do they meet?
What is approximately 1370.82 meters down?
This is the integral of x4 - 2x2 + 44dx.
What is x5/5 - 2x3/3 + 44x + C?
A village has an initial population of 5,100 people that is growing at about 2.75% annually. This is the approximate number of years, to the nearest year, it will take for the population to triple.
What is 40 years?
This is the general solution for dy/dx = yx2.
What is y = ±Ce(x^3)/3?
You invest $1000 into a savings account with a 5.15% interest rate, which gets compounded quarterly. This is the approximate amount of money you will have in the savings account after 10 years.
What is $1,668.15?
A soccer ball is kicked at an angle of 25 degrees with respect to the ground with an initial velocity of 35m/s. This is about how long the ball is in the air before hitting the ground.
What is 3 seconds?
This is the derivative of 2x3 · sin(x)
What is 2x2(3sin(x) + xcos(x))?
A fossil is found to have only 7.4% of its original Carbon-14 remaining. Given the half-life of Carbon-14 is 5,730 years, this is the age of the fossil.
What is approximately 21,523.78 years?
These are the solutions to dy/dx = (y + 5)x.
What are y = -5 and y = ±Ce(x^2)/2?
You invest $2000 into a bank account with an 0.48% interest rate, compounded continuously. This is approximately how much money you will have in the account after 20 years.
What is $2,423.34?
An object is dropped (without any initial speed) from the top of a building that is 100 meters tall. This is the velocity of the object at the point of impact with the ground.
What is -44.27 m/s?
This is the derivative of (cos(3x))2.
What is −6cos(3x)sin(3x)?
Substance A has a half-life of 4 hours, and Substance B has a half-life of 7 hours. Given both substances begin with an initial weight of 250 grams, this is how much more of B than A remains after 3 hours.
What is approximately 37.1 grams?
This is the solution to the initial value problem dy/dx = x3/y with y(0) = 6.
What is y = sqr(x4/2 + 36)?
You can invest $5,000 into one of two savings accounts. Account A has an interest rate of 2.51%, compounded monthly. Account B has an interest rate of 3.45%, compounded quarterly. This is the account that will net you more money after 5 years, and this is how much money it will have.
What is Account B, with $5,936.97?