To find:The slope of secant line connecting two points of the curve of given quadratic function, and the slope tangent line as a function of x in both differential and prime notation.
Answer to Problem 14E
The slope of secant to the curve
Explanation of Solution
Given information:
The given quadratic function is
Method/Formulaused:
The slope of a line is tangent of the angle, the line makes with x -axis. Let
Calculations:
Let us consider two neighboring points
The line PQ is a secant line connecting points
In figure, the coordinates of point N are
Hence, the slope of the secant PQ is
For the given function,
Thus, the slope of secant to the curve
Further, we may write
The secant PQbecomes a tangent at pointP when point Q coincides with point P that is when
The slope of tangent is the limit of the slope of secant as
Now, taking limit as
Thus, the slope of the tangent at point
(Differential form), or
(Prime notation)
Conclusion:
The slope of secant to the curve
Want to see more full solutions like this?
Chapter 2 Solutions
Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
- Determine whether each of the following statements is true or false, and explain why. The derivative of a product is the product of the derivatives.arrow_forwardDetermine whether each of the following statements is true or false, and explain why. The chain rule is used to take the derivative of a product of functions.arrow_forwardFind the derivative of each function defined as follows. y=12x3-8x2+7x+5arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage