TinyProf's Step-by-Step Explanation 🎓

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1. What We're Solving

We need to unpack the phrase "decreases at an increasing rate" to figure out how consecutive differences compare — and connect this language to derivative concepts. We also need to identify what the prepositional phrase "at an increasing rate" grammatically modifies.

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2. The Approach

This problem is about translating everyday language into calculus language. Before we work with numbers, we must understand what the sentence is actually telling us about the function's behavior.

Think of it like decoding a message: every word is a clue about the shape of the graph! 🔍

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3. Step-by-Step Solution

Step 1: Break Down the Sentence Grammatically

The sentence says:

> "A quantity decreases at an increasing rate."

The prepositional phrase "at an increasing rate" modifies the verb "decreases" — it describes how the decreasing is happening.

✅ It tells us about the manner of the decrease, not the quantity itself.

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Step 2: Translate "Decreases" into Calculus Language

If a quantity is decreasing, this tells us about its derivative:

> The first derivative is negative: f′(t) < 0

The function is going down as time moves forward.

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Step 3: Translate "At an Increasing Rate" into Calculus Language

The rate of decrease is increasing. The rate of change = the first derivative, so the first derivative itself is becoming more negative over time, meaning:

> The second derivative tells us whether the rate is increasing or decreasing.

Since the rate of decrease is increasing (getting steeper downward):

• If f′(t) is going from, say, −2 to −5, it's becoming more negative
• That means f′(t) is decreasing as a number
• So f′′(t) < 0

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Step 4: Visualize This on a Graph

Imagine the function plotted over time:

``` Value | | * | * | * | * +-------------------------> Time 1 2 3 ```

Notice how the steps down get bigger each second? The curve is bending downward more steeply — that's a concave down shape!

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Step 5: Compare the Differences

Let's use this insight to compare measured differences.

Let's call:

• Drop A = decrease from second 1 to second 2 = f(1) − f(2)
• Drop B = decrease from second 2 to second 3 = f(2) − f(3)

Since the rate of decrease is getting larger over time, the function is falling faster and faster.

Therefore:

> Drop B > Drop A > > The decrease from second 2 to second 3 is greater than the decrease from second 1 to second 2.

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Step 6: Connect to Derivatives Formally

| Language | Calculus Meaning | |---|---| | "Decreases" | f′(t) < 0 | | "At an increasing rate" | f′(t) is becoming more negative | | Combined effect | f′′(t) < 0 (concave down) | | Consequence | Consecutive drops get larger |

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4. The Answer

• Grammatically: "At an increasing rate" modifies the verb "decreases"
• Calculus translation: f′(t) < 0 AND f′′(t) < 0
• Comparison: The decrease from second 2 to 3 is GREATER than from second 1 to 2

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5. Memory Tip 💡

"Decreasing at an increasing rate" = Falling faster and faster = Concave DOWN

Think of a ski jump ramp — you're going down (decreasing) AND picking up speed (increasing rate). Each moment, you drop more distance than the last! ⛷️

> 🌟 The trickiest part of calculus is often the language — once you can decode phrases like this, the math follows naturally!