Derivatives
What is the derivative of f(x)=5x3
f'(x)=15x^2
What is the integral of ∫1 dx
∫1dx=x+C
If the height of a ball is given by h(t) = -5t^2 + 20t, when will the ball hit the ground?
When h(t) = 0 → Solve -5t^2 + 20t = 0 → t = 0 or t = 4 seconds
Who is Albert Einstein
Developed the theory of relativity, which revolutionized our understanding of space, time, and gravity.
How is calculus used when driving a car?
To calculate speed and acceleration from distance-time data.
Find the derivative of f(x)=cos(x)
f'(x)=-sin(x)
Find ∫sin(x) dx
∫sin(x)dx=−cos(x)+C
The population of a town is modeled by P(t) = 1000e^{0.03t}. What does the 0.0
The growth rate (3% per year)
Who is Marie Curie
Discovered the radioactive elements polonium and radium, and was the first woman to win a Nobel Prize.
How does NASA use calculus to land a spacecraft on the Moon
They use integrals and derivatives to calculate changing positions, speeds, and fuel use during descent.
Apply the Product Rule to find the derivative of f(x)=x2sin(x)
f′(x)=2xsin(x)+x2cos(x)
Compute ∫1/x dx
∫1/x dx=ln ∣x∣ +C
A rectangle has one side of fixed length 10 cm. What width maximizes the area if the perimeter is 40 cm?
Width = 10 cm (because 2(10 + w) = 40 → w = 10)
Beth A. Brown
(February 4, 1969 – October 5, 2008) was a NASA astrophysicist with a research focus on X-ray observations of elliptical galaxies and black holes. She earned a Ph.D. in Astronomy from the University of Michigan in 1998, becoming the first African-American woman to do so
How does an engineer use calculus when designing a roller coaster?
To calculate the curves and slopes that make the ride safe and thrilling.
Find the second derivative of f(x)=x3−5x2+4x−2
f′′(x)=6x−10
Compute ∫e2x dx
∫e2x dx=e2x/2 + C
Water is leaking from a tank at a rate of 3 liters per minute. What is the total amount leaked in 30 minutes?
90 liters
Isabella Aiona Abbott
from Hawaii, was an ethnobotanist who studied Pacific marine algae. Abbott is especially known for her research on edible seaweed (limu), of which there are more than 70 types, and has been called the "First Lady of Limu." (Fun fact! Seaweed is a form of algae, so limu is edible algae.)
How is calculus used in weather forecasting
To model and predict changes in temperature, wind, and pressure over time.
Differentiate f(x)=e3xsin(2x) using the Product Rule and Chain Rule.
f′(x)=e3x(3sin(2x)+2cos(2x))
Compute ∫1/x2 dx
∫1/x2 dx=-1/x+C
A balloon is rising at a constant rate of 4 meters per second. After 5 seconds, how high is the balloon? What is its velocity?
• Height = 4 \times 5 = 20 meters
• Velocity = 4 m/s (constant)
Dilhan Eryurt
Turkish astrophysicist Dilhan Eryurt researched how the Sun affects the Moon’s environment, aiding NASA in developing technology for lunar missions, including Apollo 11. She discovered the Sun was once brighter and hotter, challenging prior beliefs and influencing understanding of Earth and Moon chemistry. Her work earned her the Apollo Achievement Award in 1969.
How does Spotify use calculus to improve your music recommendations?
It helps optimize algorithms that track your listening habits and adjust suggestions in real-time using rate-of-change models.