Two particles travel along the space curves r(t) and u(t). Do the particles collide? Do their paths intersect?                                                                     r(t) = ti + t2j + t3k,                                                                                           u(t) = (−2t + 3)i + 8tj + (12t + 2)k

Q: Find the domain of the function f(x) = √(x - 1) + √(x + 5)

A: Given function f(x) - f(x) = x-1 + x+5For a square root function to be defined, the value inside the...

Q: The population of a small town was 500 in 2018 and is growing at a rate of 24 people per year. Find ...

A: Let the population of a town as a function of time 't' be Pt Where t is the number of years after 20...

Q: Express the function G in the form fo g. (There is more than one correct answer. The function is of ...

A: We have  Gx=1x+9   Gx=fgx If gx=x+3 and fx=1x then  fgx=fx+3Gx=1x+3

Q: Evaluate in terms of the constant a.

A: Given :            limx→ax-ax-a

Q: Find both first partial derivatives. z = ln√xy

A: Given  z = ln√xy. We need to find both the first partial derivatives of the above function.

Q: Evaluate the integrals

A:

Q: If u ≠ 0 and if u * v = u * w andu # v = u . w, then does v = w? Give reasons for your answer.

A: Given, u≠0, u·v=u·w  &  u×v=u×w Consider, u·v=u·w⇒u·v-u·w=0⇒u·v-w=0⇒v-w=0           ∵u≠0⇒v=w &am...

Q: Suppose that f(a) = f(b) = 0 and the second derivatives of f exist on the closed interval [a, b]. Pr...

A: Prove that ∫abx-ax-bf"xdx=2∫abfxdx.

Q: Evaluate dx x²V9 – x²

A: The given integral is ∫dxx29-x2. Evaluate the integral ∫dxx29-x2 as follows

Q: Use properties of logarithms to expand the logarithmic expression logb(x2y/z2) as much as possible. ...

A: Given :           logbx2yz2

Q: How Separable Equations are Applied ?

A: A first order differential equation y′=f(x,y) is called a separable equation if the function f(x,y) ...

Q: Find the derivative of f(x) = x ^2at x = 3.

A:

Q: Partial derivatives Find the first partial derivatives of the following functions.

A: the given function is: gx,y,z=xyzx+y we have to find the first partial derivatives of the given func...

Q: If f(x) = x2 and g(x) = x + 2, find (f ∘ g)(x) and (g ∘ f)(x).

A: Find f∘gx as shown below: f∘gx=fgx              =fx+2              =x+22              =x2+4x+4

Q: Divide (4x4 + 6x3 + 3x - 1) ÷ (2x2 + 1)using long division.

A: 4x4+6x3+3x-1÷2x2+1

Q: Find the domain of the function : g(x) = √(x-2) + √(x+3)

A: Find the domain of the function : g(x) = √(x-2) + √(x+3)

Q: Calculate the area bounded by y=x³, x=0 and y=3.

A:

Q: the graph of ƒ' is given. Determine x-values corresponding to inflection points for the graph of ƒ.

A: Given       

Q: Evaluate the integral.

A: Given: The integral, ∫12x5lnx dx.

Q: Suppose that f(x) = x*, x 2 5, and that g(x) = Vx + 3, x > 5. Find a formula for (f • g)(x) and find...

A: fx=x4, x≥5  &  gx=x+3, x≥5 Now, f∘gx=fgx=fx+3=x+34=x+32

Q: Write the equation of the parabola in standard form, whose vertex is vertex is (-3, -1) and the grap...

A: Given Write the equation of the parabola in standard form, whose vertex is vertex is (-3, -1) and th...

Q: Find the arc length of the graph of y=Ln(cosx) from x=0 x=π/4

A: Given,          The arc length of the graph of y=lncosx from x=0 to  x=π4.We know that,          The...

Q: Fill in the blanks so that the resulting statement is true.The y-coordinate of a point where a graph...

A: Given- The y-coordinate of a point where a graph crosses the y-axis is called a/an_______ . The x-co...

Q: Use a calculator to solve the equation 7 sin2 x - 1 = 0 correct to four decimal places, on the inter...

A:

Q: Find the derivative of the function y=5x3+3−2x3+3√x5.

A: Given function is y=5x3+3-2x3+3x5. Power rule of differentiation states that: ddxn=nxn-1 Differentia...

Q: Determine all the number(s) c which satisfy the conclusion of the Mean Value Theorem for the given f...

A: i. Given, hx=x3-x2+x+8  on  -3, 4 Differentiating with respect to x, we get h'x=3x2-2x+1+0=3x2-2x+1 ...

Q: A function is given by a table of values, a graph, aformula, or a verbal description. Determine whet...

A: Graph: Vertical line test the vertical line test is a visual way to determine if a curve is a graph ...

Q: Explore the convergence of the series:

A:

Q: Find f + g, f - g, fg, and f/g and determine the domain for the given function : f(x) = √(x - 2), g(...

A:

Q: Solve the polynomial inequality (x - 1)(x - 2)(x - 3) ≥ 0 and graph the solution set on a real numbe...

A: Given that x-1x-2x-3≥ 0.

Q: gives an integral over a region in a Cartesian coordinate plane. Sketch the region and evaluate the ...

A: Let the given integral be I - I = ∫-π3π3∫0sect3cost dudtIntegrating with respect to u,I = ∫-π3π33cos...

Q: Find all the asymptotes of the following functions. 2x2 + 6 f1x2 + 3x - 2

A: Consider the following function: fx=2x2+62x2+3x-2 Vertical asymptotes of function: A rational functi...

Q: Use a change of variables or Table toevaluate the following definite integrals.

A: Given, The definite integral ∫1249x2+6x+1dx.   To solve: Using change of variables or using table gi...

Q: 27. Select the correct one SEF (x)dx S% \f(x)\dx, | - Sif(x)dx < S%\f(x) |dx,

A: Given- Consider the all options given below, To find-  The correct one in the above options.  

Q: lim(e2* +1)

A:

Q: A helicopter hovers 1000 feet above a small island. The given figure shows that the angle of depress...

A:

Q: Find the general solution of the differential equation.

A: Given: dydx=4x+4x16-x2

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.