Math Problem Statement
Solution
The problem asks to find the slope predictor function for and then use it to estimate the slope of the tangent line at .
Step 1: Find the slope predictor function (derivative)
To find the slope predictor function, we need to differentiate with respect to :
Differentiate term by term:
-
The derivative of is .
-
The derivative of requires using the power rule. Recall that .
So, the slope predictor function (derivative) is:
Step 2: Estimate the slope of the tangent line at
Now, we need to plug into the slope predictor function to find the slope of the tangent line at that point:
First, calculate :
- is (approx).
- Then is .
So,
Final Answer
The slope of the tangent line at is approximately .
Would you like to see more details, or do you have any further questions?
Here are five related questions to further explore the topic:
- How do you interpret the slope of a tangent line in the context of a function?
- What does it mean if the slope of the tangent line is positive or negative?
- How would the slope predictor function change if the original function were ?
- How can you use the second derivative to determine if the tangent line slope is increasing or decreasing at a point?
- How does the slope of a tangent line relate to the concept of instantaneous rate of change?
Tip: Always check your derivative by simplifying and breaking down complex expressions to avoid calculation errors.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Slope of Tangent Line
Formulas
Derivative formula: f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]
Power rule: d/dx (x^n) = n * x^(n-1)
Theorems
Power Rule for Differentiation
Suitable Grade Level
Grades 11-12
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