Finding and Analyzing Derivatives Using Technology In Exercises 63-70, (a) use a computer algebra system to differentiate the function, (b) sketch the graphs of f and f' 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 f is positive and the interval(s) on which f' is negative. Compare the behavior of f and the sign of f'.
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Chapter 4 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
- Determining Differentiability In Exercises85–88, find the derivatives from the left and fromthe right at x = 1 (if they exist). Is the functiondifferentiable at x =1? \text { 8. } f(x)=(1-x)^{2 / 3}arrow_forwardSketching 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_forwardUse a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and points of inflection. (c) Graph f, f′, and f ″ on the same set of coordinate axes and state the relationship between the behavior of f and the signs of f′ and f ″.arrow_forward
- Finding an Equation of a Tangent Line In Exercises49–52, use implicit differentiation to find an equation of thetangent line to the graph at the given point. 50. x2 + xy + y2 = 4, (2, 0)arrow_forwardFinding a Derivative In Exercises 13–32, findthe derivative of the function.g(x) = [2 + (x2 + 1)4]3arrow_forwardUsing the Alternative Form of theDerivative In Exercises 69–76, use thealternative form of the derivative to find thederivative at x c, if it exists. g(x)=x^{2}-x, c=1arrow_forward
- Finding a Derivative In Exercises 13–32, findthe derivative of the function. g(x)=3(4-9 x)^{5 / 6}arrow_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_forwardFinding a Derivative In Exercises 7–26, usethe rules of differentiation to find the derivative ofthe function.\text { 23. } y=x^{2}-\frac{1}{2} \cos xarrow_forward
- Determining Differentiability In Exercises77–80, describe the x-values at which f isdifferentiable.arrow_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_forwardComplex Analysis - Derivatives Show that f'(z) does not exists any point z when f(z)= Im zarrow_forward
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