ACT Science interpolation and extrapolation flashcards

What is interpolation?

Interpolation is the method of estimating values within the range of a discrete set of known data points.

Example of interpolation in real life?

Determining a temperature at noon based on temperatures at 11 AM and 1 PM.

True or False: Interpolation can predict values outside known data.

False. Interpolation estimates values within the known data range only.

Fill in the blank: Interpolation helps find values between __________.

known data points.

When is linear interpolation used?

Linear interpolation is used when data points form a straight line.

What are common methods of interpolation?

- Linear - Polynomial - Spline

Calculate the interpolated value: (2, 3) and (4, 7), find value at x=3.

Using linear interpolation: $\displaystyle y = 3 + \\frac{(7-3)}{(4-2)}(3-2) = 5$

Comparison: Interpolation vs. Extrapolation.

Interpolation estimates within known data; extrapolation predicts outside known data.

What is the formula for linear interpolation?

$\displaystyle y = y_1 + \\frac{(y_2 - y_1)}{(x_2 - x_1)}(x - x_1)$

True or False: More data points improve interpolation accuracy.

True. More data points typically lead to better accuracy in estimates.

What is a practical application of interpolation?

Predicting a stock price based on previous prices at set intervals.

Fill in the blank: Interpolation assumes that data changes __________.

smoothly between points.

What does spline interpolation involve?

Using piecewise polynomials to create a smooth curve through the data points.

How do you visualize interpolation?

Graph data points and draw a line or curve to estimate intermediate values.

What is a common software used for interpolation?

Excel, Python (with libraries like SciPy), and MATLAB.

True or False: Interpolation requires a minimum of three data points.

False. Two points are sufficient for linear interpolation.

What is extrapolation?

Extrapolation is predicting values beyond a known data range based on established trends.

True or False: Extrapolated values are always accurate.

False. Extrapolated values can be uncertain and depend on the validity of the trends.

Fill in the blank: Extrapolation is often used in __________ to forecast future trends.

economics, science, and engineering.

What is a common application of extrapolation?

Predicting future sales based on past data trends.

Cause → Effect: Why is extrapolation risky?

Because it relies on assumptions that may not hold true outside the known data range.

How does extrapolation differ from interpolation?

Extrapolation predicts beyond known data; interpolation estimates within the data range.

Example of extrapolation: If 100 miles equals 2 hours, how long for 150 miles?

Set up a ratio: $\displaystyle 150/100 = x/2$. Solve: $\displaystyle x = 3$ hours.

What is linear extrapolation?

A method using straight lines to estimate values outside a data set, assuming a constant rate of change.

True or False: Extrapolation can be used for any data set.

False. It’s best for linear or predictable trends.

What are potential pitfalls of extrapolation?

- Overconfidence in predictions - Ignoring external factors - Linear assumption errors

If a car travels 60 miles in 1 hour, how far in 3 hours?

Extrapolate: $\displaystyle 60 imes 3 = 180$ miles.

What is polynomial extrapolation?

Using polynomial functions to predict values, which can fit more complex trends.

Fill in the blank: Extrapolation assumes that the __________ continues as before.

existing trend or pattern.

How can you check the validity of an extrapolated value?

- Compare with new data - Analyze trend consistency - Assess surrounding conditions

What is time series extrapolation?

Using historical time data to make future predictions, considering cycles and trends.

True or False: Extrapolation is only useful in scientific contexts.

False. It is used in finance, marketing, and more.

How do you estimate temperature tomorrow?

Use interpolation with today's temperature and the forecasted trend.

True or False: Extrapolation can predict future gas prices.

True. It uses past data to estimate future values.

If today is 70°F, what's tomorrow at 75°F?

Interpolation gives a reasonable guess of 72.5°F.

Fill in the blank: Extrapolation helps predict future values by extending _______.

existing trends.

What's an example of distance extrapolation?

If a car travels 60 miles in 1 hour, it can be projected to travel 120 miles in 2 hours.

Compare interpolation vs extrapolation.

Interpolation estimates within known data; extrapolation estimates outside known data.

How to estimate travel time at 50 mph for 150 miles?

Extrapolate from 1 hour for 50 miles: 3 hours for 150 miles.

What is the formula for linear interpolation?

$\displaystyle y = y_1 + \\frac{(y_2 - y_1)}{(x_2 - x_1)}(x - x_1)$.

If gas prices rose from $\displaystyle 2 to$3 in a month, what's next month?

Using extrapolation, if the trend continues, it might be $4.

True or False: Interpolation is less reliable than extrapolation.

False. Interpolation is generally more reliable since it is based on known data.

Estimate temperature if 60°F today, 65°F next day.

Interpolation suggests tomorrow could be around 62.5°F.

Cause → Effect: Why use interpolation?

To make informed estimates using known data points.