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Area under a curve Suppose the function y = h(x) is nonnegative and continuous on [α, β], which implies that the area bounded by the graph of h and x-axis on [α, β] equals
103. Find the area under one arch of the cycloid x = 3(t − sin t), y = 3(1 − cos t) (see Example 5).
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Chapter 12 Solutions
Calculus: Early Transcendentals (3rd Edition)
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- Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle rolls along the x -axis given that P is at a maximum when x=0.arrow_forwardA function f of two variables has a function equation of the form f(x, y) = ln(ax²y + bxy + c) where a, b and c are real numbers. It is given that the tangent plane to the graph off at the point (−1, 3, ƒ(−1, 3)) has equation z = -6x-y-3. a) Explain why the information given tells you that f(-1, 3) = 0. b) Consider the contour line of the function f through the point (-1, 3) in the (x, y)-plane. Find the equation of the tangent line to this contour line at the point (-1, 3). You do not need to find the values for a, b and c to answer this question! c) Find the values for the numbers a, b and c.arrow_forwardDo part b in detailarrow_forward
- Find parametric equations for the line tangent to the curve of intersection of the surfaces tan-1(x? + y? + z²) + e(2 Inx+3In y) – xvz and 15 sec-1(2v2x + 2v2y + v2z) at the point (1, 1, 1).arrow_forwardhi My question is about Complex Derivative. I showed in the upload photo. z is a complex number and z* is complex conjugate of z. f(z)=u(x,y)+i.v(x,y); In section a, we want to obtain the derivative of the function f with respect to z and z * in the Cartesian coordinate system. In section b, I want to obtain the derivative of the function f with respect to z and z * in the polar coordinate system. . In one of the photos, I put a pattern to solve the question. Thank you very much.arrow_forward4. Let f (x) = x³ -x² + 5. a) Find the y-intercept of f. y-intercept: b) Find f' and f", and determine where each are 0 and/or do not exist (DNE). If none, write "none". f' = 0: f' DNE: f" = 0: f" DNE: c) E Do a sign analysis on f' and f". d) Find the intervals on which f is increasing and decreasing. Increasing: Decreasing: e) Find the intervals on which f is concave up and concave down. Concave up: Concave down: f) answers as (x, y) points. Find all local maxima, local minima, and inflection points of f. Be sure to write your Local max: Local min: Inflection point(s): -4 -3 -1 g) Sketch the graph of f.arrow_forward
- In Basic Calculus. Thank youarrow_forwardA Find the value of JF; G · dL if G = x'a, - xyza,, given points P,(1,2,3) and P;(5,0,4) and a path consisting of: (a) the straight line segments connecting P, to (5,2,3) to (5,0,3) 10 (5,0,4); (b) the straight line which is the intersection of the planes x = 5 - 2y and y = 8 – 22.arrow_forwardy=t^2+4t See question in imagearrow_forward
- Show the solutionarrow_forwarda) Describe the curve obtained when we make y= 2 and z = √2 a f b) Represent on this curve the partial derivative Click here to Download Image (GIF) at the point P(,1,√2)arrow_forwardFind y in terms of x. dy = x 4(x 5 - 3) 2, curve passes through (1, 5) dx 17 A y = - (x5 -3) 3 + 15 1 B) y = x 15-3 + 15 27 с) у3 - (x 5 - 3) 1+ 83 D y = (x 5 - 3) 3 + 15 15arrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
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