πŸƒ Flashcards Β· Topic 1.2

Rates of
Change

8 cards across 1 print sheets. Print double-sided, cut along dashed lines.

πŸ“‹ Notes πŸƒ Flashcards 🧠 Quiz
Print double-sided (flip on long edge) β†’ cut along dashed lines.
8 cards Β· 1 print sheet
πŸ“„ Page 1 β€” Questions FRONT Β· Sheet 1/1
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Topic 1.2 Β· ROC at a Point
What does it mean when the rate of change at a point is POSITIVE?
Think roller coaster
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Topic 1.2 Β· ROC at a Point
What does it mean when the rate of change at a point is ZERO?
Where does this happen on a graph?
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Topic 1.2 Β· ROC at a Point
What does it mean when the rate of change at a point is NEGATIVE?
Think roller coaster
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Topic 1.2 Β· Comparing ROC
Points A and B are on a graph. 'Least' rate of change means which value β€” the more positive or more negative one?
Think about the number line
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Topic 1.2 Β· Average ROC
What is the formula for Average Rate of Change (AROC) over [a, b]?
It's the slope of the secant line
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Topic 1.2 Β· AROC Positive/Negative
AROC over [a, b] is positive when… and negative when… and zero when…?
Compare f(a) and f(b)
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Topic 1.2 Β· AP Precalc Limit
In AP Precalculus, what CAN we do vs. what CAN'T we do with the rate of change at a single point?
Calculus is needed for exact values
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Topic 1.2 Β· Example 6
Life expectancy: 1900β†’53.63, 1950β†’70.65, 2000β†’81.83 (50-year intervals). Which interval has greater AROC: 1900–1950 or 1950–2000?
Compute (Ξ”y)/50 for each
πŸ“„ Page 2 β€” Answers BACK Β· columns swapped
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βœ“ ROC = Zero at Peaks/Valleys
At a LOCAL MAX or LOCAL MIN
ROC = 0 at peaks (local max) and valleys (local min). The function transitions from going up to going down (or vice versa) β€” at that instant, slope = 0.
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βœ“ Positive ROC
Function is INCREASING (going uphill)
Positive ROC = going uphill on the graph. The output values are increasing as inputs increase. Think: roller coaster going up.
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βœ“ Comparing ROC β€” 'Least'
'Least' = most NEGATIVE
On the number line, βˆ’7 < βˆ’3. The steepest downhill slope is the most negative = smallest (least) value. Don't confuse 'least' with 'smallest absolute value'.
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βœ“ Negative ROC
Function is DECREASING (going downhill)
Negative ROC = going downhill. Output values are decreasing. Steeper downhill = more negative ROC.
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βœ“ AROC Positive/Negative/Zero
Positive: f(b)>f(a) Β· Negative: f(b)
Only the endpoints determine the sign. The function can wiggle in between β€” only start and end values matter for AROC sign.
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βœ“ AROC Formula
AROC = (f(b) βˆ’ f(a)) / (b βˆ’ a)
Change in output Γ· change in input. It is the slope of the secant line connecting (a, f(a)) to (b, f(b)).
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βœ“ Example 6 β€” AROC Comparison
1900–1950: 0.3404 > 1950–2000: 0.2236
1900–1950: (70.65βˆ’53.63)/50 = 0.3404
1950–2000: (81.83βˆ’70.65)/50 = 0.2236
1900–1950 has greater AROC.
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βœ“ AP Precalc vs Calculus
CAN: pos/neg/zero + compare Β· CAN'T: exact value
In AP Precalculus, we determine if ROC at a point is positive, negative, or zero and compare two points. The exact value requires calculus (called the derivative or instantaneous ROC).