show that the Punctionc Sinh t V3 and ye T5 Sinh t+ CoshE 13 taken together satisfy thediff erential equations. dx dt dy +y=0

Q: et Evaluate the derivative: da 4 S (3t2 – 2t + 4) dæ.

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Q: dy Find where y = sin²(V3x²). dx

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Q: t4+9 Evaluate Jo 74+9t² dt - Evaluate S7 x3. Cot-1x dx

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Q: CSCZ dz Where y is the circle 12- il =

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Q: dy in each part. dr Find and simplify a) Use implicit differentiation 2x5 – y + 4ry³ = -8

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Q: Evaluate: d dxJ (21² + 5)² dt.

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Q: Find w, where w3 = 1 + i. i? = -1.

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Q: Calculate the derivative using Part 2 of the Fundamental Theorem of Calculus. X 21 d 1/² (316-1) ²¹…

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Q: dy Find , for the implicit function 7 x²y³ + 2 x² = 3xy. dx

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Q: dy Part II: Find the derivative ) implicitly. Show all work. dx 11. e" – 4x³ + 5y? = 10

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Q: dz Use implicit differentiation to find 3x2 1+2y 33y+z x+2yz 323y 1+2y %3D 1+2y

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Q: Calculate the derivative d da 34 dt using Part 2 of the Fundamental Theorem of Calculus. 34 d dx…

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Q: –1 dx

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Q: ) Use implicit differentiation: + x²y +4y? = 6 %3D

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Q: - If y = u – 4u and u = = 3x + r, then dy с. dr 1 dy а. dx 12x3 + 1 12x3 + 1 3u? 4 dy (3u² – 4)(12r³…

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Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.