1. Consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t > 0. Find an expression for the temperature u(x, t) if the initial temperature distribution in the rod is u(x, 0) = 25 0 < x < 40. Suppose that a² = 1.

Q: A mass is attached to a spring and then the spring is compressed and released. Suppose the…

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Q: 10. (a) State the Fundamental Theorem of Calculus. -x sin t %3D (b) Let J(x) = S*. Evaluate: (i)…

A: Fundamental theorem of calculus

Q: MCQ 2_ 0.25e-2 + 3 grams 0.25e-1 grams O -8e-1+ -1 + 10 grams -8e-2 + 10 grams

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Q: Sara draws the 4 of hearts from a standard deck of 52 cards. Without replacing the first card, she…

A: Given, Sara draws the 4 of hearts from a standard deck of 52 cards without replacing. a) we know…

Q: 1. determine the derivative with respect to time t of the following: a) 4x + 8y = 12 b) xy = 15 c)…

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Q: Consider the function f (x) ax³ + bæ where a and b are constants. Suppose f' ( 1) = 12 and f' (0) =…

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Q: Find the total derivative of the following functions. (a) u = In (1+se²t). (b) z= r coshy+ysinh r. w…

A: Since you have asked multiple questions, we will solve the first question for you.If you want any…

Q: Find f(x), where f'(x) = sin(x) + cos(x) + 1 and f(#)=2. %3D %3D

A: Using anti derivative formula.

Q: If y = xsin then y is equal to = X" at x =

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Q: If f(-0.1) = 5, f(O) = 7, and f(0.1) = 8 then f'(0) is approximately equal to: a. 0 Ob. 15 Ос. 30…

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Q: Let f(x) = (x" + 10x) e 6x-5, where n is an unspecified constant. Find the value of n such that the…

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Q: LET F(X) =2 x SIN(X), AND EPSILON = 3 × 10-7. FIND OPTIMUM STEP VALUE THAT CAN BE USED WITH…

A: We are given f(x)=2 sin(x) and ε=3×10-7. The formula for central difference is…

Q: 1 If f(t) cos(6t), find f'(t). Use exact values. f'(t) =

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Q: 1. Calculate an expression for = y if V+b°y = tan-(x"y)

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Q: 6. A sensor is attached to a spring and its velocity is given by the function v(t) = cos(2t) feet…

A: Derivative of velocity function gives displacement function of a body.

Q: 8. X is a voltage that is uniform in [-2,2]. X is passed through a device with the output as 0, X 0…

A: The random variable represents the voltage.and X follows the uniform distribution(-2,2) I.e.…

Q: Let f(r) = 2 sec(2r). Determine the following %3D f'(x) = 4 sec(2x) tan(2x) %3D f'(4) =

A: As we know that; The chain rule: fgx'=f'gx·g'x And ddxsecx=secxtanx From the given function:…

Q: Let f(x) and g(x) be differentiable functions such that • f(3) = 1, f(0) = f'(3) = 2. • g(1) = 0,…

A: Given, fx and gx be differentiable functions such thatf3=1f0=2f'3=2g1=0g'1=2 Used differentiation…

Q: 6. df Let f(x) = -x+ 6. Find f'(x). Graph f(x) and f'(x) together. Evaluate 1 df 1 at x= 1 and dx at…

A: Givenf(x)=-67x+6 We need to find f-1(x) and evaluate dfdx at x=1and df-1dx at x=f(1)

Q: Suppose that g is a function that is differentiable at x = 8 and that g(8) = -4 and g' (8) = 3. Find…

A: Given,            Suppose that g is a function that is differentiable at x=8 and that g8=-4…

Q: 17. If P'(t) is the rate a deer population grows per year, write a sentence explaining what S P'(t)…

A: 17. ∫06p'(t)dt =p(6)-p(0) this implies ∫06p'(t)dt represents the total increment in the population…

Q: 4. Let f(x)= 5Vx and g(x)= 4 + sin x. (a) Find (f ° (x) and (f ° g'(x). (b) Find (g ºf)(x) and…

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Q: Let y = g(x) cos (In x). Compute y'(1), if g(1) = 1 and g'(1) = -2.

A: We have to find the derivative of y at x=1