Represent the plane curve by a vector-valued function. y = 3 – x² r(t) = %3D

Q: Find the equation in the xy-plane whose graph includes x In(6t) and y = t².

A: Given  x= ln(6t) y=t2

Q: Derivatives of vector-valued functions Differentiate the following function.

A: We have given the function;

Q: What is the graph of r(t)=2 costi + sintj?

A: Vector-valued function : It is a function of the form  r(t)=f(t)i⏜+g(t)j⏜ or…

Q: Find a unit vector that is tangent to the vector function F(t) = at t = 1. Step 1: take derivative…

A:

Q: Calculate Duf(-1, 1, 1) in the direction of =--23-k for the function f(x, y, z)=-4x² + 5xy + y² + 3x…

A: Given function is f(x, y, z) = -4x2 + 5xy + y2 + 3x + 3yz - z2 - 5xz

Q: Graph the curve traced by the given vector function r(t)=ti +2tj+costk;t>0.

A:

Q: d. In which direction is the instantaneous rate of change the maximum from (1,0)? What is the…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: Consider the function f(x.y) = 2x -y - 3 and the point (2, - 1). a. Find the unit vectors that give…

A: Given function is fx,y=2x2−y2−3

Q: Determine the equation of the tangent plane and a vector equation of the normal line to A. tangent…

A:

Q: Calculate the directional derivative of f(x, y) = x²y in the direction of v = −3i+2j at the point P…

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Q: Why is r(t) = (f(t),g(t).h(t)) called a vector-valued function? %3D

A:

Q: Sketch the curve traced out by the vectorvalued function. Indicate the direction in which the curve…

A: Let F:R2→R, and the parametric equation given is , for t≥0.Since, there are 3 components to F(t),…

Q: A2. Consider the function f(x, y) = x³e²y. In what direction should one move from the point (2,0) to…

A:

Q: For the vector valued function below, 1. Find the domain of each component. 2. Find the domain for…

A: Please see the below answer.

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.