Concept explainers
In Exercises 31–34 find k so that the function is continuous on any interval.
Want to see the full answer?
Check out a sample textbook solutionChapter 1 Solutions
Calculus: Single And Multivariable
Additional Math Textbook Solutions
University Calculus: Early Transcendentals (3rd Edition)
Precalculus Enhanced with Graphing Utilities (7th Edition)
University Calculus: Early Transcendentals (4th Edition)
Calculus: Early Transcendentals (2nd Edition)
Calculus Early Transcendentals, Binder Ready Version
Precalculus
- In Exercises 51–54, graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0. If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function’s value(s) should be?arrow_forwardIn Exercises 15–22, calculate the approximation for the given function and interval.arrow_forwardIn Exercises 83–85, you will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Per-form the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where ƒ′ = 0. (In some exercises, you may have to use the numerical equation solver to ap-proximate a solution.) You may want to plot ƒ′ as well. c. Find the interior points where ƒ′ does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function’s absolute extreme values on the interval and identify where they occur. 83. ƒ(x) = x4 - 8x2 + 4x + 2, [-20/25, 64/25] 84. ƒ(x) = -x4 + 4x3 - 4x + 1, [-3/4, 3] 85. ƒ(x) = x^(2/3)(3 - x), [-2, 2]arrow_forward
- In Exercises 25–30, give a formula for the extended function that iscontinuous at the indicated point.arrow_forwardIn Exercises 37–40, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?arrow_forwardUse graphs to determine if each function f in Exercises 45–48 is continuous at the given point x = c. [2 – x, if x rational x², if x irrational, 45. f(x) c = 2 x² – 3, if x rational 46. f(x) = { 3x +1, if x irrational, c = 0 [2 – x, if x rational 47. f(x) = { x², if x irrational, c = 1 x² – 3, if x rational 3x +1, if x irrational, 48. f(x) : c = 4arrow_forward
- In Exercises 6–10, let f(x) = cos x, g(x) = Vx+ 2, and h(x) = 3x?. Write the given function as a composite of two or more of f, g, and h. For example, cos 3x? is f(h(x)). 6. V cos x + 2 1. V3 cos?x + 2 8. 3 cos x + 6 ). cos 27x* 10. cos V2 + 3x²,arrow_forwardIn Exercises 83–86, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false. If the graph of a function has three x-intercepts, then it musthave at least two points at which its tangent line is horizontalarrow_forwardEach of Exercises 15–30 gives a function f(x) and numbers L, c, and ɛ > 0. In each case, find an open interval about c on which the inequal- ity |f(x) – L| 0 such that for all x satisfying 0 0, L= 2m, c = 2, — тх, ɛ = 0.03 28. f(x) = mx, ɛ = c > 0 L = 3m, c = 3, m > 0, L = (m/2) + b, 29. f(x) c = 1/2, m> 0, ɛ = c > 0 = mx + b, 30. f(x) 3D тх + b, m> 0, L%3Dm+ b, с %3D 1, &%3D 0.05arrow_forward
- In Exercises 27–32, use a graphingutility to graph the function on the closed interval [a, b].Determine whether Rolle’s Theorem can be applied to f on theinterval and, if so, find all values of c in the open interval (a, b)such that f '(c= ' 0.) f(x)=|x|-1,[-1,1]arrow_forwardIn Exercises 181–184, determine whetherthe statement is true or false. If it is false, explain why or givean example that shows it is false. The slope of the function f (x) = cos bx at the origin is −b.arrow_forwardIn Exercises 49–51, sketch a graph of ƒ and identify the points c such that f'(c) does not exist. In which cases is there a corner at c?arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage