Concept explainers
Finding and Analyzing Derivatives Using Technology In Exercises 49-54, (a) use a computer algebra system to differentiate the function, (b) sketch the graphs of f and
on the same set of coordinate axes over the given interval, (c) find the critical numbers of f in the open interval, and (d) find the interval(s) on which
is positive and the interval(s) on which
is negative. Compare the behavior of f and the sign of
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Chapter 3 Solutions
Calculus (MindTap Course List)
- Sketching the Graph of a Sine orCosine Function In Exercises 31–52,sketch the graph of the function. (Includetwo full periods.)31. y = 5 sin x 32. y = 14 sin x33. y = 13 cos x 34. y = 4 cos x35. y = cosx2 36. y = sin 4x37. y = cos 2πx 38. y = sin πx439. y = −sin 2πx3 40. y = 10 cos πx641. y = cos(x − π2) 42. y = sin(x − 2π)43. y = 3 sin(x + π) 44. y = −4 cos(x +π4)45. y = 2 − sin 2πx3 46. y = −3 + 5 cos πt1247. y = 2 + 5 cos 6πx 48. y = 2 sin 3x + 549. y = 3 sin(x + π) − 3 50. y = −3 sin(6x + π)51. y = 23cos(x2 − π4) 52. y = 4 cos(πx +π2) − 1arrow_forwardDetermining Concavity In Exercises 3–14,determine the open intervals on which the graphof the function is concave upward or concavedownward.' 3. f (x) = x2 − 4x + 8arrow_forwardTesting for Functions RepresentedAlgebraically In Exercises 11–18,determine whether the equation represents yas a function of x.11. x2 + y2 = 4 12. x2 − y = 913. y = √16 − x2 14. y = √x + 515. y = 4 − ∣x∣ 16. ∣y∣ = 4 − x17. y = −75 18. x − 1 = 0arrow_forward
- Complex Analysis - Derivatives Show that f'(z) does not exists any point z when f(z)= Im zarrow_forwardDetermining Differentiability In Exercises77–80, describe the x-values at which f isdifferentiable.arrow_forwardProof of the Quotient Rule Let F = ƒ/g be the quotient of twofunctions that are differentiable at x.arrow_forward
- Finding and Analyzing InverseFunctions In Exercises 45–54, (a) find theinverse function of f, (b) graph both f andf −1 on the same set of coordinate axes,(c) describe the relationship between thegraphs of f and f −1, and (d) state the domainsand ranges of f and f −1.45. f(x) = x5 − 2 46. f(x) = x3 + 847. f(x) = √4 − x2, 0 ≤ x ≤ 248. f(x) = x2 − 2, x ≤ 049. f(x) = 4x50. f(x) = −2x51. f(x) = x + 1x − 2 52. f(x) = x − 23x + 553. f(x) = √3 x − 1 54. f(x) = x35arrow_forwardDescribe and explain a. continuity of a function at a point. b. continuity of a function on an interval c. Intermediate value theoremarrow_forwardTest the functions in Exercises 65–66 for local maxima and minima and saddle points. Find each function’s value at these points. 65. ƒ(x, y) = x2 - xy + y2 + 2x + 2y - 4 66. ƒ(x, y) = 5x2 + 4xy - 2y2 + 4x - 4yarrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage