Vector-valued functions and continuity: Given:  r(t) = <4t+2, sin(pi t), 1/t3> what is the lim r(t) as t --> -1

Q: degree Taylor Polynomial at a= a.) Find the of f(x)=cos (Zx), centered

A: Since you have posted multiple questions, we will solve only first question for you. To getremaining…

Q: lim sin Zy Fan3y ア

A: Limit y tends to 0.

Q: sinx Prove that lim 1|

A: Solution: Consider the given limit function and we have to show that  limx→0sin xx = 1 So, limx→0sin…

Q: sin z Find the derivative of the function g(x) = V1+ 2t dt using the Fundamental Theorem of…

A:

Q: sin z et f(z) = 23(z – 3i)² sin (a) Find all the singularities of the function. Are they isolated?…

A: see attachments below

Q: Tangent lines Find an equation of the line tangent to the cycloidx = t - sin t, y = 1 - cos t at the…

A:

Q: z+y <=0. (a) Using - definition, show that lim (a)+(0,0) 2²+ y² +1 [2²sin +3² sin, (2, 3) (0,0) (b)…

A:

Q: series bind the limi t use to Siny limit tary y?cosy

A:

Q: Vector-Valued Function r(t) = (2 sin t, 6t, 2 cos t) O O NACE anand -15 -10 Find its length over the…

A:

Q: 8. Gradient orthogonal to tangent Suppose that a differentiable function f(x, y) has the constant…

A:

Q: lim cot(z) %3D - 10 -1 0 3 0

A:

Q: tagrange interpolation polynomial for " the" qunction y = Sinł e, Construct the taking the points…

A:

Q: Find the limit

A: We have to evaluate

Q: Estimate `sin(-0.01)` using an appropriate linear approximation L(x) = f(a) + f'(a)(x – a).…

A:

Q: Tangent line given Determine the constants b and c such that the line tangent to f(x) = x² + bx + c…

A: Here we have to determine the constant a and b. The given function is f(x)=x2+bx+c…

Q: sin(z) lim Z+0 II

A:

Q: 8. Define a) the derivative of f(x) with respect to r (in terms of a limit). b) the function f(z) is…

A: According to guidelines we solve only first three parts.

Q: lim (x,y)→(0,0) r sin² y + y sin² a x² + 3y²

A:

Q: sin z Prove that lim Cosz - = 1 and lim = 0. Z→0 Z Z→0 Z Hint: use the definition of the derivative,…

A:

Q: sin√√x +. +5 ; (x, y) = (0,0) Set f(x,y)= A ; (x,y)=(0,0) Find the value of A continuation f(x,y) at…

A:

Q: ition of cont s continuous at the given point. a). f(x) = , a = 3 Use the 2z– -32

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: sin Show that the set of functions. 3πX 5TX 2L 2L TCX 2L sin sin is orthogonal over (0, L).

A:

Q: [cost- sin t Q2) Determine whether x, = ,X1 = |cost| and x, = Lcos t] Co t sin t| are linearly…

A: For linearly independent , determinant of matrix should be non zero.

Q: sin(z) - lim

A: The given limit is limz→0sinzz. Evaluate the limit of the given function as follows.…

Q: sinar lim,-1 r-1

A: Given limit is  limr→1sinπrr-1 Evaluate the limit. limr→1sinπrr-1=00 indeterminate form

Q: Theoretical assignment Consider hyperbolic tangent function exp(x) – exp(-æ) exp(x)+ exp(-æ) tanh(x)…

A: Linear Spline Intepolation: A linear spline is a sequence of straight lines. So, in between the data…

Q: convolution Theorem to Poove that =Lie-sint-cost) Use 2

A: Convolution of function: The convolution of function f(t) and g(t) is denoted by f*g and defined as,…

Q: Topic: Differentiability, Differentials and Local Linear Approximation Question: Show that fx (0,…

A:

Q: sin x + sin y 1. Sine limits Verify that lim (x, y)(0, 0) x + y

A:

Q: sin 5z-sin 3z 14. lim,» +0 sin z

A:

Q: sin -,x 0 Q: Let f(x) =- 0 ,x = 0 Find the set of values of m for which (i)f(x) is continuous at x =…

A:

Q: 4) V9-z zt2-6

A:

Q: uestion nd the first four non-zero terms of the Maclaurin ries for the function f(x) = sin(x10), and…

A: Solution of the problem as follows

Q: Determine the open t-intervals on which the curve is concave downward or concave upward in interval…

A:

Q: (sin(x² +y²) ²+y² if (x,y) # (0,0) (ii) Check continuity of the function at (0,0), if (x,y) = (0, 0

A:

Q: sin (z – 1) z <1 (z – 1)3 – 3 1<z Let F (2) Then lim F(x) =

A:

Q: Using Cauchy Principal Value and Residue Theorem evaluate the definite ntegral sin? dx n x2 1 : 1

A:

Q: lim sin"(2 – t) cot“(t“ - t+2 - t) -t- 2)

A:

Q: Derivative rules Suppose u and v are differentiable functions at t = 0 with u(0) = ⟨0, 1, 1⟩, u'(0)…

A: Given: u and v are differentiable function at t=0, withu0=0,1,1, u'0=0,7,1v0=0,1,1, v'0=1,1,2 To…

Q: Using L'hopital's Rule evaluate the following derivatives: 5x – 2x lim x→0 7x3 + 3 sin t2 lim 8x2…

A: If limx→x0 f(x)g(x) is of the 00 form then By L'Hospital rule limx→x0 f(x)g(x)=limx→x0 f'(x)g'(x)If…

Q: Vector-Valued Function r(t) = - + 3tj + 4tk 5 x Z Find its length over the given interval. 26…

A: 1st we will draw the curve using its initial and final point .  To find arc length of the curve r(t)…

Q: Bessel fundion Satis fies veyi fy that the Sinx the Bessel equation of

A: We have to verify that the Bessel function J12x=2πxsinx satisfies the Bessel equation of 12.

Q: · divergence of F that F is con ser vati ve at the point (0, 3, 1 ) R° with out finding its…

A: Div (F) = i∧∂∂x+j∧∂∂y+k∧∂∂zF1 i∧+ F2j∧+F3k∧ F is conservative if curl F =0 i.e,…

Q: lim f(z) = #t * NE

A:

Q: sin(z) lim z+0 (1)

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Use Mean-Value theorem to find c X +2x-1 ; [o,i] xj[3,5] Use L'Hôspital vule to find the wmit lime…

A: We will solve the Problem on limits and mean value theorem.

Q: sin 3x . [T] lim S.M; x→0 ; ±0.1, ±0.01, ±0.001, ± 0.0001 sin 3x sin 3x X -0.1 а. 0.1 е. -0.01 b.…

A: Evaluate the limit for x→±0.1limx→-0.1sin 3xx=Sin 3x-0.1-0.1                    =-Sin0.3-0.1…

Q: sin(t) Suppose F(x) = dt. Use the Fundamental of Theorem of Calculus to calculate F'(x). 3 +t 6.…

A:

Q: sin z show that the function is complete(valued function) with the help of f(2) = 1 a series z =0

A: given function is                                           we know that…

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.