2. Let f,g: R → R³ be differentiable functions. Show that d dt (f(t) × g(t)) = d d f(t) × g(t) + f(t) × 7+g(t) dt dt

Q: Show that the function u(z, t) : satisfies ua = %3D %3D a2t2-2

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Q: 2. Use the First Fundamental Theorem of Calculus to differentiate the function f(r) = VI+ tª dt.

A: We will use the first fundamental theorem of calculus to evaluate the derivative.   The detailed…

Q: 2 =1+ %3D V1+f(t) dt, x 21

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Q: Differentiate the function. = In({ (4t+ 1)³ (4t-1)5 F'(t) = F(t) = In

A: The derivative of the function in the division form is given by the quotient rule which is given by-…

Q: Show that the function u(x,t) = t/2e-*/4 (t > 0) satisfies u, = uxx.

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Q: he linear approximation to approximate a suitable function f(x, y) and thereby estimate…

A: To approximate a funtion f(x,y) from R^2 to R at some point(xo,yo)

Q: me please: Suppose X1, X2 ∼ Unif(0, 1) and are independent. Let

A: We have given that X1, X2 ∼ Unif(0, 1)

Q: Let f(x, y) = xy+ + 2x²y + 4y5 (a) Find fr- (b) Find fy-

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Q: Use the Fundamental Theorem of Calculus Part I to find dh/dx given h(x) = cost dt (Hìnt: need…

A: Answer:  By fundamental theorem of calculus  Let fx=∫axbx gt. dt   dfdx=gbx.b'x-gax.a'x

Q: Compute Re f(i) when: f(z)= sqrt(z-1) and f(0)=-i

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Q: Let F(x, y) =. Evaluate (31) F dr (1,0,0) O a. 9 + 3e3 8+ 3e3 С. 8— Зе3 O d. 10 – 3e3

A: We will use basic knowledge of vector calculus and integral calculus to answer this question.

Q: Where did F(x, y, z) = Em(1,4,5) come from

A: The steps to follow to write a function in SOP form:    1. Write AND terms for each input…

Q: Let f(x, y) = vax?y? + by. Compute f,(c, d) for the following values: a = 9, b = 3, c =3, d = 2.…

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Q: Evaluate O A. OB. O C. O D. -2 (2) (1) -² (+-) 2 - dt at t = for the function w(x,y,z)=- 4 9 xy ()…

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Q: Q5. Find the directional derivative of the function [f(x, y, z) = xy + y.z + z.x] at the point 1-z…

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Q: Let f(x, y) = xty¬4. Estimate the change Af = f(4.03, 2.9) – f(4, 3) using the equation below.…

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Q: te r'"(t) and r'(t) for the fo

A: Given that : The function is r(t) =t+4 i + tt +7 j - e-t2 k .

Q: f'(t) = tf(t). HINT: Write f'(t) – tf(t) = 0 and multiply the equation by e-2/12.

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Q: = 4+ J₁ Find F'(x). 2t² + 1 t² + 3 dt.

A: