CALCULO 2 SEMANA 4PERGUNTA 1Seja f colon straight real numbers...

CALCULO 2 SEMANA 4

PERGUNTA 1

Seja f colon straight real numbers squared rightwards arrow straight real numbers dada por f left parenthesis x comma y right parenthesis equals 2 x cubed plus y cubed plus x y cubed plus 2. O plano tangente ao gráfico de f no ponto left parenthesis 1 comma 1 comma f left parenthesis 1 comma 1 right parenthesis right parenthesis é dado pela equação vetorial:

PERGUNTA 2

Seja f colon straight real numbers squared rightwards arrow straight real numbers dada por f left parenthesis x comma y right parenthesis equals e to the power of 2 x end exponent s e n left parenthesis y cubed right parenthesis. Denotando por P subscript 1 left parenthesis x comma y right parenthesis e P subscript 2 left parenthesis x comma y right parenthesis os polinômios de Taylor de f desenvolvidos em left parenthesis 0 comma 0 right parenthesis de ordem 1 e 2, respectivamente, podemos afirmar que.

PERGUNTA 3

Sejam f colon straight real numbers squared rightwards arrow straight real numbers dada por f left parenthesis x comma y right parenthesis equals x y e capital psi colon straight real numbers squared rightwards arrow straight real numbers squared dada por capital psi left parenthesis s comma t right parenthesis equals left parenthesis s squared plus t squared comma s t right parenthesis. Se a função g colon straight real numbers squared rightwards arrow straight real numbers for dada por f ring operator capital psi, temos que nabla g left parenthesis 1 comma 2 right parenthesis é dado por:

PERGUNTA 4

Sejam f colon straight real numbers squared rightwards arrow straight real numbers dada por f left parenthesis x comma y right parenthesis equals x plus y e capital phi colon straight real numbers squared rightwards arrow straight real numbers squared dada por capital phi left parenthesis s comma t right parenthesis equals left parenthesis 2 s plus t squared comma t right parenthesis. Seja g colon straight real numbers squared rightwards arrow straight real numbers dada por g left parenthesis s comma t right parenthesis equals left parenthesis f ring operator capital phi right parenthesis left parenthesis s comma t right parenthesis, temos que nabla g left parenthesis 1 comma 1 right parenthesis é dado por:

PERGUNTA 5

Seja f colon straight real numbers squared rightwards arrow straight real numbers dada por f left parenthesis x comma y right parenthesis equals x squared plus y cubed, usando o polinômio de Taylor de ordem 1 de f desenvolvido em left parenthesis 2 comma 1 right parenthesis, e considerando duas casas decimais, o valor que melhor aproxima f left parenthesis 13 over 6 comma 11 over 10 right parenthesis entre as alternativas é:

PERGUNTA 6

Seja f colon straight real numbers squared rightwards arrow straight real numbers dada por f left parenthesis x comma y right parenthesis equals x squared minus y cubed plus x y minus 1. O plano tangente ao gráfico de f no ponto left parenthesis 1 comma 1 comma f left parenthesis 1 comma 1 right parenthesis right parenthesis é dado pela equação:

PERGUNTA 7

A temperatura em uma placa é modelada pela função z equals f left parenthesis x comma y right parenthesis equals x y squared plus x y, sendo que, a placa é descrita por P equals left curly bracket left parenthesis x comma y element of straight real numbers squared divided by 0 less or equal than x less or equal than 10 comma 0 less or equal than y less or equal than 10 right parenthesis right curly bracket. Uma particula se desloca sobre essa placa pela curva gamma colon left square bracket 0 comma 1 right square bracket rightwards arrow straight real numbers squared dada por gamma left parenthesis t right parenthesis equals left parenthesis t cubed plus 2 t comma 2 t right parenthesis. A taxa de variação de temperatura sofrida por essa particula no instante t equals 1 vale:

PERGUNTA 8

Considere f colon straight real numbers squared rightwards arrow straight real numbers dada por f left parenthesis x comma y right parenthesis equals square root of x squared plus x y end root e gamma colon straight real numbers rightwards arrow straight real numbers squared dada por gamma left parenthesis t right parenthesis equals left parenthesis e to the power of t comma e to the power of negative t end exponent right parenthesis. Temos que fraction numerator d f left parenthesis gamma left parenthesis t right parenthesis right parenthesis over denominator d t end fraction dada por: