Question

Estimate the slope of the tangent line to the curve f(x) = x2 at the point (1, f(1)) by examining the trend in the secant slopes calculated.

Answer 1

f'(1) = 2

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

• Function Notation

Calculus

The definition of a derivative is the slope of the tangent line.

Derivative Notation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Explanation:

Step 1: Define

f(x) = x²

Point (1, f(1))

Step 2: Differentiate

1. Basic Power Rule: f'(x) = 2 · x²⁻¹
2. Simplify: f'(x) = 2x

Step 3: Find Slope

Use the point (1, f(1)) to find the instantaneous slope

1. Substitute in x: f'(1) = 2(1)
2. Multiply: f'(1) = 2

This tells us that at point (1, f(1)), the slope of the tangent line is 2. We can write an equation using point slope form as well: y - f(1) = 2(x - 1)