Curve Cracker Worksheet The Great Approach
Navigating Average & Instantaneous Rates
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Part 1: The 100-Mile Question
"If you drive 100 miles in exactly 2 hours, did you go 50 mph the entire time?"
Explain your reasoning. What is the difference between your "trip speed" and what you see on the speedometer at any given second?
Part 2: Micah's Hike
Based on the video, describe the relationship between the curved function (Micah's actual path) and the straight secant line (the average speed calculation).
Key Definitions
Secant Line: A line connecting two points on a curve. Its slope represents Average Rate of Change.
Tangent Line: A line that "touches" the curve at exactly one point. Its slope represents Instantaneous Rate of Change.
Part 3: The Shrinking Interval Challenge
Consider the function \( f(x) = x^2 \), which represents the distance (meters) a falling object travels over time (seconds). We want to find the speed exactly at \( t = 2 \). To do this, we'll calculate the Average Rate of Change (ARC) over smaller and smaller intervals starting at \( x = 2 \).
\[ \text{ARC} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
Interval \([2, x_2]\) \( \Delta x \) (Width) Calculation: \( \frac{x_2^2 - 2^2}{x_2 - 2} \) Avg. Rate (m/s) \([2, 4]\) \( 2 \) \( \frac{16 - 4}{4 - 2} = \frac{12}{2} \) \( 6 \) \([2, 3]\) \( 1 \) \([2, 2.5]\) \( 0.5 \) \([2, 2.1]\) \( 0.1 \) \([2, 2.01]\) \( 0.01 \)
Reflect:
As the width of the interval (\( \Delta x \)) gets closer to zero, what single number does the average rate seem to be approaching?
Based on your table, what would you estimate is the instantaneous speed at exactly \( t = 2 \)? Why can't we just plug in \( x = 2 \) into our ARC formula directly?
The Great Approach Slides The Great Approach
Unlocking the Secret of Instantaneous Speed
Algebra II • Introduction to Calculus
Warm-Up Discussion
"If you drive 100 miles in 2 hours, did you go 50 mph the entire time?"
Average Speed
The "Big Picture" view of the whole trip.
Instantaneous Speed
The "Snap Shot" view at one specific moment.
Video: Micah's Hike
Embedded media
Watch from 0:00 to 2:07
Look For:
The Curve vs. the Secant Line .
The "Rise" and "Run" on the graph.
"If the rate keeps changing, use one big rate that covers everything."
Pause & Discuss
The Hiking Graph Observation:
Where was the graph the steepest ? What did that mean for Micah's speed at that moment?
Thinking Ahead:
If we zoomed in on a tiny part of that curve, would it start to look like a straight line?
The Shrinking Interval
We can't find instantaneous speed with just one point because:
\(\text{Slope} = \frac{y-y}{x-x} = \frac{0}{0}\)
Mathematical Error!
The Secret Strategy:
Use two points that are insanely close together.
"Average" speed over a 0.0001 second interval is basically "Instantaneous" speed.
Visualizing the Finish Line
S
Secant Line
Cuts through two points on the curve.
Represents Average Rate
T
Tangent Line
Kisses the curve at one single point .
Represents Instantaneous Rate
Rate Navigator Reference Sheet Rate Navigator
Scientific Calculator Quick-Reference
1. The Master Formula
Average Rate of Change (ARC)
\( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \)
2. Calculator Workflow
A
Use the Fraction Key \( \frac{n}{d} \)
Always set up your "Big Fraction" first. It prevents order-of-operation errors!
B
Parentheses are Your Friends
When inputting \( f(x) = x^2 \), type it as (2.01)² to avoid sign errors, especially with negatives.
The "Shrink" Trick
To save time when calculating for intervals like \([2, 2.1], [2, 2.01], [2, 2.001]\), use the Arrow Key (up) to highlight your previous calculation and press ENTER to copy it.
Just change 2.1 to 2.01 and hit enter!
Interpretation Key
Positive Result
Object is speeding up or moving forward.
Negative Result
Object is slowing down or returning.
Blueprint for Algebra II • Designed for Accuracy
