Questions and Answers
At what value of x for x > 0 does the line tangent to the graph of f at x have slope 2?
2.287
What is the difference between the approximation and the actual value of f′(0.5)?
0.433
What is the result of f′(0.4) for the function f(x) = (1/7)x^7 + 12x^6 − x^5 − (15/4)x^4 + (4/3)x^3 + 6x^2?
f′(0.4)
Study Notes
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of derivative concepts and tangent lines with this multiple-choice quiz. This quiz focuses on determining slopes and evaluating functions in the context of calculus. Perfect for students studying Unit 2 of their calculus curriculum.