(e) f,g : R-R given by f(x)and g(x) 1+2r ro0 (f) f, g : (0, oo) → R given by f(r) = In(2) and g(z) = (z-1)3 + In(z): zo = 1. (g) f.g: (0, oo) -R given by f(x)-In(x) and g(x)-(x 1)200 +ln(x); ro 1 x-1)200
Homework Answers
We have used the definition of k th order contact of two curves to solve the problems.
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*9. For each of the following pairs of functions, determine the highest order of contact between the two functions at the indicated point xo: (e) f,g : R-R given by f(x)and g(x) 1+2r ro0 (f) f, g : (...
9. For each of the following pairs of functions, determine the highest order of contact between...
9. For each of the following pairs of functions, determine the highest order of contact between the two functions at the indicated point ro: (a) f,g :R-R given by f(x)-cos a and g(xo0 (b) f,g RR given by f(x) coS and g(x) 1 - j; ro0. (c) f,g:R-R given by f(x)-sinr and g(x0 (d) f,g:R-R given by f(x) sinz and ga)0. (e) f, g : R → R given by f(x)-e*2 and g(x)-1+z?: zo = 0. (f) f, g :...
Co are 5. Suppose that the functions f :R3 R, g:R R, and h:RR ously differentiable and let (xo. o...
co are 5. Suppose that the functions f :R3 R, g:R R, and h:RR ously differentiable and let (xo. o, zo) be a point in R3 at which f(xo, yo, zo-g(xo, yo, zo)sh(xo, yo, zo)s0 and By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a neighborhood of the point (xo, yo, Zo) the system of equations f(x, y, z) g(x, y, 2)0 hCx, y,...
(B)(C)(D)(E)(G)(I)(J) 39 Write the Taylor expansion of function f order n at to given below. 1...
(B)(C)(D)(E)(G)(I)(J)
39 Write the Taylor expansion of function f order n at to given below. 1 (a) (g)2+v1+ I, n 2,ro - 0 n = 7, xo = 0 1 -2-3 (b) sin z cos(2x), (c) z In(2+3z), VI+I n= 3, To = 0 1 +e-1/ (h) 2+x n =5,xo= +o0 n= 3, ro = 1 T (i) cos (2r), n 4, xo= 6 (d) n = 7,xo = 0 COSI i) V+-VI3-, n 4, 1o =+00 (e) In(1+ arcsin(2r)),...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) =...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
(1 point) Use the given functions to evaluate the statements. f(g(-2))= (f.g)(3) = f(g(-3)) = 3...
(1 point) Use the given functions to evaluate the statements. f(g(-2))= (f.g)(3) = f(g(-3)) = 3 2 -2 -1 1 -3 -4 X 31 -4 -1 f(x) -2 0 4 -3 2 е(к)
4. Given a function f(x), use Taylor approximations to derive a second order one-sided ap- proximation to f'(ro...
4. Given a function f(x), use Taylor approximations to derive a second order one-sided ap- proximation to f'(ro) is given by f(zo + h) + cf (zo + 21) + 0(h2). f' (zo) = af(xo) + What is the precise form of the error term? Using the formula approximate f' (1) where r) = e* for h 1/(2p) for p = 1 : 15, Form a table with columns giving h, the approximation, absolute error and absolute error divided by...
(7) Compute the derivatives of order 4 ands of the following functions at the point Xo=0:1...
(7) Compute the derivatives of order 4 ands of the following functions at the point Xo=0:1 (1) f(x) = sin D F(x) = Sinh (x²+28inta) I to
1. (10 pts) For each of the following pairs of functions, indicate whether f = 0(g),...
1. (10 pts) For each of the following pairs of functions, indicate whether f = 0(g), f = Ω(g), or both (in which case f-6(1). You do not need to explain your answer. f(n) (n) a) n (b) n-1n+1 (c) 1000n 0.01n2 (d) 10n2 n (lg n)2 21 е) n (f) 3" (g) 4" rl. 72 i-0 2. (12 pts) Sort the following functions by increasing order of growth. For every pair of consecutive functions f(n) and g(n) in the...
1. Find lim f(x) and lim f(x) for each of the following. Assign oo or -...
1. Find lim f(x) and lim f(x) for each of the following. Assign oo or - where appropriate: (a) f(z)=42 -2 (b) f(x)= 3r-2 (e) f(1)--5rt6 (x - 2)2 2. Find each of the f limits, assigning oo or-o where appropriate x+2z-1 (b)lm 3r-2 3 r-2 (c) li 42+1 (k) im (2r V4r2-8r 3) 15r-2 3. Find the horizontal and vertical asymptotes of each of the following functions 4. Sketch the graph of each of the following functions and determine,...
(1 point) Suppose f, g: R² + R2 are continuous functions, where g is surjective. Determine...
(1 point) Suppose f, g: R² + R2 are continuous functions, where g is surjective. Determine if the following sets are open, closed, neither, both or if it can't be determined. 1.9-1 (R) 2. (f • g)-+ ({(1, 2)}) 3. (f+9) (B(0; 1)) 4. (f+g)-1({(x, y) : x > 0}) 5.9-1 (B(0; 1))
9. For each of the following pairs of functions, determine the highest order of contact between the two functions at the indicated point ro: (a) f,g :R-R given by f(x)-cos a and g(xo0 (b) f,g RR given by f(x) coS and g(x) 1 - j; ro0. (c) f,g:R-R given by f(x)-sinr and g(x0 (d) f,g:R-R given by f(x) sinz and ga)0. (e) f, g : R → R given by f(x)-e*2 and g(x)-1+z?: zo = 0. (f) f, g :...
co are 5. Suppose that the functions f :R3 R, g:R R, and h:RR ously differentiable and let (xo. o, zo) be a point in R3 at which f(xo, yo, zo-g(xo, yo, zo)sh(xo, yo, zo)s0 and By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a neighborhood of the point (xo, yo, Zo) the system of equations f(x, y, z) g(x, y, 2)0 hCx, y,...
(B)(C)(D)(E)(G)(I)(J)
39 Write the Taylor expansion of function f order n at to given below. 1 (a) (g)2+v1+ I, n 2,ro - 0 n = 7, xo = 0 1 -2-3 (b) sin z cos(2x), (c) z In(2+3z), VI+I n= 3, To = 0 1 +e-1/ (h) 2+x n =5,xo= +o0 n= 3, ro = 1 T (i) cos (2r), n 4, xo= 6 (d) n = 7,xo = 0 COSI i) V+-VI3-, n 4, 1o =+00 (e) In(1+ arcsin(2r)),...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
(1 point) Use the given functions to evaluate the statements. f(g(-2))= (f.g)(3) = f(g(-3)) = 3 2 -2 -1 1 -3 -4 X 31 -4 -1 f(x) -2 0 4 -3 2 е(к)
4. Given a function f(x), use Taylor approximations to derive a second order one-sided ap- proximation to f'(ro) is given by f(zo + h) + cf (zo + 21) + 0(h2). f' (zo) = af(xo) + What is the precise form of the error term? Using the formula approximate f' (1) where r) = e* for h 1/(2p) for p = 1 : 15, Form a table with columns giving h, the approximation, absolute error and absolute error divided by...
(7) Compute the derivatives of order 4 ands of the following functions at the point Xo=0:1 (1) f(x) = sin D F(x) = Sinh (x²+28inta) I to
1. (10 pts) For each of the following pairs of functions, indicate whether f = 0(g), f = Ω(g), or both (in which case f-6(1). You do not need to explain your answer. f(n) (n) a) n (b) n-1n+1 (c) 1000n 0.01n2 (d) 10n2 n (lg n)2 21 е) n (f) 3" (g) 4" rl. 72 i-0 2. (12 pts) Sort the following functions by increasing order of growth. For every pair of consecutive functions f(n) and g(n) in the...
1. Find lim f(x) and lim f(x) for each of the following. Assign oo or - where appropriate: (a) f(z)=42 -2 (b) f(x)= 3r-2 (e) f(1)--5rt6 (x - 2)2 2. Find each of the f limits, assigning oo or-o where appropriate x+2z-1 (b)lm 3r-2 3 r-2 (c) li 42+1 (k) im (2r V4r2-8r 3) 15r-2 3. Find the horizontal and vertical asymptotes of each of the following functions 4. Sketch the graph of each of the following functions and determine,...
(1 point) Suppose f, g: R² + R2 are continuous functions, where g is surjective. Determine if the following sets are open, closed, neither, both or if it can't be determined. 1.9-1 (R) 2. (f • g)-+ ({(1, 2)}) 3. (f+9) (B(0; 1)) 4. (f+g)-1({(x, y) : x > 0}) 5.9-1 (B(0; 1))
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