Quantitatively prove the following: If f: R3 → R  has continuous second derivatives, then  ∇ × (∇f) = 0. (In order words, “curl ◦ grad = 0”)

Q: 5 Find the center and radius of the circle. Write the standard form of the equation. The center of…

A: We know that the centre is the mid point of diameter.

Q: A tank contains 100 L of water. A solution with salt concentration of 0.4 kg/L is added at a rate of…

A:

Q: Problem 2 Given the following differential equation: (y³+kxy4 - 2x)dx + (3xy² + 20x²y³-ky)dy a)…

A:

Q: Solve the equation: 5-2 = 32+1

A: See this solution in below.

Q: The graphs of the function F (left, in blue) and G (right, in red) are below. Let P(x)=F(x)G(x) and…

A:

Q: Evaluate the definite integral. π/2 x² sin x [12²232² -π/2 1 + x8 - dx

A:

Q: Suppose we have two interesting functions f and g. We have limited knowledge of them as follows: •…

A:

Q: If f(4) = 11, f'is continuous, and f(6) = Submit Answer fºt f'(x) dx = 19, what is the value of…

A:

Q: 1+t² e. y' = tye-t

A: We need to determine D.E. is separable or not.

Q: 1 2 ső ső (x + y) dydx 0 O none is correct O-3 02 0 3

A:

Q: 1. (sin³1y) (cos ydy 5 8 0 2 2

A:

Q: -5-4-3-2-1 123 1 2 3 4 5 The function graphed above is decreasing on the interval The inflection…

A: Given query is to find the interval where functions is decreasing.

Q: Concept Check: Use integration by parts to find fln(x) dx. (Hint available for u and dv?) Use…

A:

Q: Parameterize boundary of triangle (0,0),(5,0),(3,9)

A: We can parameterize the triangle in the below steps.

Q: Question 6 Determine the derivative of - sec z cosz-sin r sec" tan z sec² z 1-sin z sec² z O cos²x -…

A: Take y= Sin(x)/Sec(x)

Q: Use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. (Write your…

A:

Q: Solve for the particular solution: 4e-3x-2 y' e -6x +3e-3x+2 ; y (0) = 1

A:

Q: r12 Find ſ² ſ¹²(x + ln y) dydx 3 8

A:

Q: Consider the following functions: f(x) = 16-3x - x² 22 g(x) = 5-2x Evaluate (fog)(x). O-4x² +26x -…

A:

Q: Rewrite the quadratic function f(x) f(x)= ■ Vertex: = 2x² + 4x + 7 in standard form, f(x) = a(x -…

A:

Q: Hi can you type out the answer cause the answer is kinda messy

A: We need to write convert the given non linear differential equation into linear differential…

Q: 10 S3 X (xy+8) + dydx Now try the integral $10 S3 order of integration is important. = X (xy+8)¹…

A: Consider:…

Q: one has a Tess man its polynomial function. The volume of a cone is V = ²h for radius r and height…

A: Given query is to find the volume of the function.

Q: List the first 4 terms of the sequence starting from n=1 n {(-1)^ ^ +1} ₁ n 2

A:

Q: INTEGRAL CALCULUS (UPVOTE WILL BE GIVEN. PLEASE

A:

Q: The third derivative of the second derivative of y = f(x) is also O y(4) Oy (3) O f(5) (x) O D²y

A:

Q: Concept check: Evaluate the following integrals and use the appropriate transformations when needed.…

A: Need to find out the integration.

Q: DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t≥ 0. Then the integral £{f(t)} =…

A: Given function is ft=9,    0≤t<30,    t≥3 We need to find laplace tranform of the given function.

Q: simplify the expression:  tan^2theta/sec^2theta+cot^2theta/csc^2theta

A:

Q: Evaluate the following integral using trigonometric substitution. S dt 1²√36-1² OC. t-6 tan 0…

A:

Q: Evaluate the following integral. e2. 1 In ³ (x²) X dx

A: To evaluate the following integral.∫1e2ln3x2x dx

Q: Find the equation of the plane tangent to the surface G(x,y,z)=2x^2+y^2+3z^2=25 at P=(−3,−2,1).

A: Topic = Derivative

Q: Determine the derivative of ln(csc) CSC I POMOSOME CSC I O - cot a Ocot z

A:

Q: Find ff f(x, y) dA where f(x, y) = 3x + 1 and R= [0, 3] x [1,3]. R SSR f(x, y) dA =

A:

Q: A rectangular poster is to contain 162 square centimeters of print. The margins at the top and…

A:

Q: Let f(x) = √28 - 4x and g(x) = 4x² — 2. • You must combine all like terms before yo your answer. •…

A:

Q: 8. A rectangular box with a square base is to be made where the material for the sides and the lid…

A:

Q: How would you evaluate S tan 7x sec ³x dx? Choose the correct answer below. OA. Rewrite the power of…

A: I am attaching image so that you understand each and every step.

Q: Solve each of the following: (x − y − 2)dx + (3x + y - 10)dx = 0

A:

Q: 0 = XP(^vx-2)^x + ^p (Tvx-tv^) differential equations.

A: A differential equation of the form Mdx+Ndy=0 is exact if ∂M∂y=∂N∂x, and general solution is…

Q: Question 23 arctan y + ln(1 + y²) =a²x³: Find y': Gaz (1+²) 1+2y O Jamani No answer text provided. y…

A:

Q: I need the P prime and Q prime of 1 and 6 P'(1)= Q'(1)= P'(6)= Q'(6)=

A:

Q: Give the substitution that converts the given nonlinear equation into a linear equation then find…

A:

Q: tanh(lnx) = x² - 1 x² + 1 22

A: I am attaching image so that you understand each and every step.

Q: An operator found a 1.3 meters-tall dead man at the bottom of an inverted conical tank that has…

A: The height of the dead man is 1.3 meters, so, if the man died due to drowning then, the water level…

Q: Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. || y=…

A:

Q: Find the derivative of the function: y=xz²

A: We need to find out the derivative of the  function.

Q: Find the indefinite integral. (6x + 1)-9dx

A:

Q: 2x^2-4x+1/4x^2+1 determine the limit of n----> infinity an

A: Given query is to find the limit of the expression.

Q: Part C. Find the √√x-5 8). x-15 1 9). +3+ + √x + 10

A: Topic = Function