Vertical tangent lines If a function f is continuous at a and lim x → a | f ′ ( x ) | = ∞ , then the curse y = f ( x ) has a vertical tangent line at a and the equation of the tangent line is x = a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 31–32) is used. Use this information to answer the following questions. 36. Graph the following curves and determine the location of any vertical tangent lines. a. x 2 + y 2 = 9 b. x 2 + y 2 + 2 x = 0
Vertical tangent lines If a function f is continuous at a and lim x → a | f ′ ( x ) | = ∞ , then the curse y = f ( x ) has a vertical tangent line at a and the equation of the tangent line is x = a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 31–32) is used. Use this information to answer the following questions. 36. Graph the following curves and determine the location of any vertical tangent lines. a. x 2 + y 2 = 9 b. x 2 + y 2 + 2 x = 0
Solution Summary: The author illustrates how the function x2+y 2=9 has a vertical tangent at x=3, and the derivative is infinite at these points.
Vertical tangent linesIf a function f is continuous at a and
lim
x
→
a
|
f
′
(
x
)
|
=
∞
, then the curse y = f(x) has a vertical tangent line at a and the equation of the tangent line is x = a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 31–32) is used. Use this information to answer the following questions.
36. Graph the following curves and determine the location of any vertical tangent lines.
In Exercises 73–78, the graph of f is shownin the figure. Sketch a graph of the derivative of f. To print anenlarged copy of the graph, go to MathGraphs.com.image5
PART B:
a) One of the submissions for the roller coaster proposal contains the following information:
.
"
.
.
The function representing the roller coaster is y = -x + 2x³ + 13x² - 14x - 24.
Two of the x-intercepts are at x = -3 and x = 4. Determine the other x-intercept(s)
Determine the first derivative of the function
There are three extrema at x = -2.193 x = 0.5, and x = 3.193. Determine the y-values at the extrema.
(round the answer to the nearest hundredth)
Determine the first derivative of the function
There are two possible point of inflection at x =-1.055 and x = 2.055. Determine the y-values at the points of
inflection. (round the answer to the nearest hundredth)
Use a chart, to complete the first and second derivative tests in the chart below to get a better idea of what the roller coaster
would look like.
The sign of the FIRST derivative
The behaviour of the function
The sign of the SECOND derivative
The behaviour of the function
The SHAPE of the graph
b) Draw a sketch of the…
1. Write a function of x, f(x), such that the following graph represents y=f(x).
X
Precalculus Enhanced with Graphing Utilities (7th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY