The quantities 2,y,z and t satisfy z= f (2,y), 1 = g(t) et y=h(t). Given g(3)...
If z = f(x,y), where f is differentiable, and x = g(t) y = hết) g(3) = 2 h(3) = 7 g'(3) = 5 h'(3) = -4 fx(2,7) = 6 fy(2,7) = -8 Find dz/dt when t = 3.
57. Find the total derivative dz/dt, given (a) z = x^2− 8xy − y^3 , where x = 3t and y = 1 − t. (b) z = f(x, y, t), where x = a + bt, and y = c + k
[3 marks] d) Suppose f(x, y,z) x3yzxy +z 3; Given: x 3 cos t; y 3 sint; z=2; i. Finds ii. Evaluate it when t -0 for f(3,1,2) iii. Evaluate it when t for f (1,1,2) dt 13 marks] 3 marks]
Find dz d given: z = xeyy, x = = to, y= – 2 + 2t dz dt Your answer should only involve the variable t. Let z(x, y) = xºy where x = tº & y = +8. Calculate dz by first finding dt dx -& dt dy and using the chain rule. dt dx d = dy dt Now use the chain rule to calculate the following: dz dt
(6). The quantities x(t) and y(t) satisfy the simultaneous equations dt dt dx dt where x(0)-y(0)-ay (0)-0, and ax (0)-λ. Here n, μ, and λ are all positive real numbers. This problem involves Laplace transforms, has three parts, and is continued on the next page. You must use Laplace transforms where instructed to receive credit for your solution (a). Define the Laplace Transforms X(s) -|e"x(t)dt and Y(s) -e-"y(t)dt Laplace Transform the differential equations for x(t) and y(t) above, and incorporate...
Consider z-f(x,y)-1-xy cos(xy) at (2,-1/2) variations in x and y respectively. and let ΔΧ and ây represent small a) (i) Compute ΔΖ, given that ΔΧ_ 0.028 and Δy_-0.039. 1 1 6DP Az 5DP ii) Write out an expression for dz in terms of x,y and d, dy. dz= 2 (iii) Compute dz assuming dr_Δι and dy_ây dz- 5DP b) Use the equation of the tangent plane to z at (2,-1/2) to approximate Approximate value = 1 5DP Consider z-f(x,y)-1-xy cos(xy)...
(The integral of a Gaussian/Bell curve) Let Exercise 34: e~t2(1+z2) -dz 12 da f(t) and g(t) = e and h(t) f(t2 g(t) 1 Problem sheet 9 Homework 29. Mai 2019 a) Compute h(0). b) Compute h'(t) for all t > 0 Remark: You have to argue why you can interchange differentiation and integration c) Compute lim4-,00 h(t) d) Use a) c) to show that 1 d 1 VT JR da and 2 Remark: The elegant proof of the integral of...
Problem 4: Evaluation of the convolution integral too y(t) = (f * h)(t) = f(t)h(t – 7)dt is greatly simplified when either the input f(t) or impulse response h(t) is the sum of weighted impulse functions. This fact will be used later in the semester when we study the operation of communication systems using Fourier analysis methods. a) Use the convolution integral to prove that f(t) *8(t – T) = f(t – T) and 8(t – T) *h(t) = h(t...
5 Consider the functions f and g whose graphs are given below. z y = f(x) -4 A3 -2 -1 1 2 3 4 y = 9(2) -4 -3 -2 -1 1 2 3 4 1 + f. Find (3) a. Find f'(-3). b. Find f'(1). g. Suppose p(x) = f(x)g(2). Find p'(-3). c. Find f'(3). h. Suppose q(z) = 5(). Find g(3). d. Find t'(-3). g(2) e. Find g'(1). i. Suppose r(x) = x2 f(x). Find r'(1).
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...