Find a function f such that f^'(x)=x^2 and y=4 x+7 is a tangent line to the graph of f.

Name: Problem 61, Chapter 5: Calculus, Early Transcendentals
Uploaded: 2021-08-26T18:08:27-08:00
Duration: 5 min 48 s
Channel: Stark Ledbetter
Description: Find a function f such that f^'(x)=x^2 and y=4 x+7 is a tangent line to the graph of f.

step 1 Right. We're given that f prime of x is x squared, and that y equals 4x plus 7 is tangent to the

Find a function f such that f^'(x)=x^2 and y=4 x+7 is a...

Key Concepts

Constant of Integration Determination

Upon integration of the derivative to find the original function, the resulting expression includes an arbitrary constant. When additional conditions, like the existence of a specific tangent line, are imposed, this constant can be determined by substituting the coordinates of the tangency point into the original function and equating it to the value given by the tangent line equation. This ensures that the function and its tangent line coincide at the particular point.

Antiderivatives and Integration

This concept involves finding the original function from its derivative by integrating. In general, if you are given a derivative function f'(x), the corresponding original function f(x) can be found by calculating the antiderivative (indefinite integral) of f'(x). The process always introduces an arbitrary constant, known as the constant of integration, because the derivative of any constant is zero.

Tangent Lines and Tangency Conditions

A tangent line to a curve at a given point has two key characteristics: its slope is equal to the derivative of the function at that point, and it touches the curve without crossing it. When given a potential tangent line, you must match the line’s slope with the value of the derivative and ensure that at the point of tangency the function’s value equals the function value given by the line. This creates a system of equations that can be used to determine unknown parameters.

Transcript

Please give Ace some feedback