Concept explainers
In Example 6 in this section, we were given
(a) the instantaneous rate of change of f(x) at x = 6.
(b) the slope of the tangent to the graph of y = f(x) at x = 6.
(c) the point on the graph of y = f(x) at x = 6.
EXAMPLE 6 Tangent Line
Given
(a) the derivative of f(x) at any point (x, f(x)).
(b) the slope of the tangent to the curve at (1, 16).
(c) the equation of the line tangent to
Solution
(a) The derivative of f(x) at any value x is denoted by f'(x) and is
(b) The derivative is
(c) The equation of the tangent line uses the given point (1, 16) and the slope m = 8. Using
The discussion in this section indicates that the derivative of a function has several interpretations.
Want to see the full answer?
Check out a sample textbook solutionChapter 9 Solutions
Mathematical Applications for the Management, Life, and Social Sciences
- Find the differential equation of the Family of lines where x-intercept is equal to y-intercept.arrow_forwardA ladder 5 meters long is leaning against a vertical wall. Due to the smooth floor, the foot of the ladder is sliding away from the wall. Let x be the horizontal distance of the foot of the ladder from the wall and y be the height of the top of the ladder from the floor. Determine when will the rate of change of y with respect to x be numerically equal to the negative of the current value of x?arrow_forwardThe tank shown in the figure was filled with 100 liters of substance A in a concentration of 10g / L. Material A is fed into the tank at a flow rate of 5L / min in 1g / L variation. At the exit of the tank, there is an outlet flow at a rate of 5L / minute and the tank is continuously mixed. In the tank, substance A is converted into substance B. Derive the equation that ties the concentration of substance B in the tank to time. ( Reaction constant = 0.1)arrow_forward
- Estimate the instantaneous rate of change of P(x)=5/x+2 at the point x=2arrow_forwardAt a certain instant, the radius of a right circular cylinder is 6 inches and is increasing at the rate 0.2 in/sec, while the altitude is 8 inches and is decreasing at the rate 0.4 in/sec. Find the time rate of change of volume.arrow_forwardIt is known that a bacterial culture increases at a rate that is proportional to its current amount. 1000 bacterial fibers were observed in the culture after 1 hour and 3000 fibers after 4 hours. Accordingly, initially find the number of fibers in the culture.arrow_forward
- Discrete Mathematics and Its Applications ( 8th I...MathISBN:9781259676512Author:Kenneth H RosenPublisher:McGraw-Hill EducationMathematics for Elementary Teachers with Activiti...MathISBN:9780134392790Author:Beckmann, SybillaPublisher:PEARSON
- Thinking Mathematically (7th Edition)MathISBN:9780134683713Author:Robert F. BlitzerPublisher:PEARSONDiscrete Mathematics With ApplicationsMathISBN:9781337694193Author:EPP, Susanna S.Publisher:Cengage Learning,Pathways To Math Literacy (looseleaf)MathISBN:9781259985607Author:David Sobecki Professor, Brian A. MercerPublisher:McGraw-Hill Education