prove the following claims 3. Prove the following claims. (a) The function f(x) = sin x...
19-20-21 19. [5pts.Each] Find the derivative of the functions: sin x a. y = b. g(x)=e* cosh(x²) 20. [5pts.] ALSO Q3 f(x)= x - cos x, 0<x< 277. Find the intervals of increasing & decreasing and intervals of concave up & concave down. Do not graph but find all relative/local maximum & minimum and inflection points if any. 21. [5pts.] Find the dimension of the rectangle of the largest area that can be inscribed in a circle of radius r....
A rectangle is inscribed in a circle of radius 1 (see the figure). Let P = (x,y) be the point in quadrant that is a vertex of the rectangle and is on the circle. Psy) Answer the following questions (a) Express the area A of the rectangle as a function of X A(x) = ax/1-x² (b) Express the perimeter p of the rectangle as a function of x. p(x) = 2(2** 2/7-x) (c) Graph AA(X). For what value of A...
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
Problem 5. Prove that parametric equations: x a-cosh(s) (a > 0) or back half(a < 0) of hyperboloid of one sheet: Χ t), y b-sinh(s) cos (t) zc-sinh(s) sin( t), (x,y,z) lies on the front half L" a2 b2 c2 Problem 6 What graph of this Compute the arc length : rit)- < sin t, cos t, 2Vt', when 0<t < function: a) Compute the arc length : re)-3cos(9) and 0 < θ < π/2 b) Problem 7. Find parametric...
3. 8p] Show that the force field F(x,y, z) sin y, x cos y + cos z, -y sin z) is conservative and use this fact to evaluate the work done by F in moving a particle with unit mass along the curve C with parametrization r(t (sin t, t, 2t), 0 <t<T/2. 4. 8p] A thin wire has the shape of a helix x = sin t, 0 < t < 27r. If the t, y = cos t,...
8. a. Prove that f(x) = cos x is continuous on R. b. If ECR and f: E R is continuous on E, prove that 8(x) = cos (f(x)) is continuous on E. 9. For each of the following equations, determine the largest subset E of R such that the given equation
F(x, y) = (3x2 + sin y)i + (x cos y + 2 sin y)j. Question 1 (8 points) Find a potential function for the vector field F. Enter this function in the answer box. - Format B I U , . A X Question 2 (6 points) Use the potential function you found in problem 1 to evaluate F. dr, where Cis given by r(t) = (2-t)i + (ret/2), 0 st < 1.
(f) If a and b are unit vectors and θ is the angle between them. prove that sin -b Prove that v2·“2 + 2,6 , where the symbol's h) have their usual meanings 12. Find the following dx dx tan'xdx (i) 1+cos x 13 Find the area enclosed between the circle 2y25 and the straight line x +y- s 14. Solve the following equations dy dy dy dx (f) If a and b are unit vectors and θ is the...
On the back, prove the identity: tan^3(x)csc^2(x)cot^2(x)cos(x)sin(x)=1 Use only the left side and try changing everything to sine and cosine. Original Question Image: On the back, prove the identity: tan'(r)csc(r)cot'(x)cos(x)sin(r)-1 Use only the left side and try changing everything to sine and cosine.
Is the following equation an identity? If so, prove it. sin- x - cos x cos x = sin’x - cos x The given equation is an identity. sinx - cos*x = (sin’x)2 – (cos?x)2 = (sinºx – cos x) (sinºx + cos²x) = (sin x – cos2x) (1) = sinºx - cos x. The given equation is not an identity. The given equation is an identity. sin- x - cos*x = (sin?x)2 – (cos2x)2 (sinºx – cos x) (sinºx...