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.
Each of Exercises 81–84 shows the graphs of the first and second
derivatives of a function y = f(x). Copy the picture and add to it a
sketch of the approximate graph of f, given that the graph passes
through the point P.
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.
11. ƒ(x) = x2 + 1, (2, 5) 12. ƒ(x) = x - 2x2, (1, -1)
13. g(x) = x/(x - 2) , (3, 3) 14. 8/ x2 , (2, 2)
15. h(t) = t3, (2, 8) 16. h(t) = t3 + 3t, (1, 4)
17. ƒ(x) = sqrt(x), (4, 2) 18. ƒ(x) = sqrt(x + 1), (8,3).
Precalculus Enhanced with Graphing Utilities (7th Edition)
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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