Sketching a Graph Sketch the graph of a differentiable function f such that f(2) = 0, f' < 0 for -o < x < 2, and f'> 0 for 2 < x < 0. Explain how you found your answer. %3D
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- Proof of the Quotient Rule Let F = ƒ/g be the quotient of twofunctions that are differentiable at x.Finding a Pattern Develop a general rule for the nthderivative of xf (x), where f is a differentiable function of x.A right triangle has one vertex on the graph of y = x3, x > 0, at (x, y) another at the origin, and the third on the positive y-axis at (0, y). Express the area A of the triangle as a function of x.
- Identifying homogenous or non homogenous functions. Show your complete solution. when x=0, y=3 xy3dx+ex^2dy=0Curve sketching show all work f(x) = sin(x)cos(x) on [-pi, pi]Fundamental Theorem of Calculus. Suppose that g(x) is a differentiable function on [a, b]. Express g(b) − g(a) in terms of a function on the interior of [a, b].
- Derivative information Suppose a continuous function ƒ is concaveup on (- ∞, 0) and (0, ∞). Assume ƒ has a local maximumat x = 0. What, if anything, do you know about ƒ'(0)? Explainwith an illustration.Analyzing critical points Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points.Analyzing critical points Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points. ƒ(x, y) = yex - ey
- Determining Differentiability. Describe the x-values at which f is differentiable.Continuous functionsChange of variables Consider the parameterized curvesr(t) = ⟨ƒ(t), g(t), h(t)⟩ and R(t) = ⟨ƒ(u(t)0, g(u(t)), h(u(t))⟩, where ƒ, g, h, and u are continuously differentiable functions and u has an inverse on [a, b].a. Show that the curve generated by r on the intervala ≤ t ≤ b is the same as the curve generated by R onu-1(a) ≤ t ≤ u-1(b) (or u-1(b) ≤ t ≤ u-1(a)).b. Show that the lengths of the two curves are equal.(Hint: Use the Chain Rule and a change of variables in the arc length integral for the curve generated by R.)