Calculus Unit 2 Progress Check

Derivative and Tangent Line

• The derivative function is expressed as f′(x) = 0.1x + e^(0.25x).
• For x > 0, a tangent line to the graph of f has a slope of 2.
• Upon solving the equation f′(x) = 2, the value of x is determined to be approximately 2.287.

Function Approximation

• The function f is defined as f(x) = 2x³.
• To approximate f′(0.5) using table values, a difference quotient is applied.
• The calculated derivative at x = 0.5 is f′(0.5) = 0.567.
• An estimate derived from the table yields a derivative approximation of 1, resulting in an error of 0.433 between the actual and estimated values.

Function Analysis

• The function f is represented by f(x) = (1/7)x^7 + 12x^6 − x^5 − (15/4)x^4 + (4/3)x^3 + 6x^2.
• An important value in this context is f′(0.4), indicating the evaluation of the derivative at x = 0.4.