Exercícios de Cálculo

4 Calcule um valor aproximado para a variação ΔA na área de um retângulo quando os lados variam de x 2 m e y 3 m para x 201 m e y 297 m 5 Uma caixa de forma cilíndrica é feita com um material de espessura 003 m As medidas internas são altura 2 m e raio da base 1 m A caixa é sem tampa Calcule um valor aproximado para o volume do material utilizado na caixa 6 A energia consumida num resistor elétrico é dada por P V²R watts Se V 100 volts e R 10 ohms calcule um valor aproximado para a variação ΔP em P quando V decresce 02 volt e R aumenta de 001 ohm 7 A altura de um cone é h 20 cm e o raio da base r 12 cm Calcule um valor aproximado para a variação ΔV no volume quando h aumenta 2 mm e r decresce 1 mm 1 Calcule fu x₀ y₀ sendo dados a fx y x² 3y² x₀ y₀ 1 2 e u o versor de 2 i j b fx y ex² y² x₀ y₀ 1 1 e u o versor de 3 4 c fx y arctg xy x₀ y₀ 3 3 e u 12 12 d fx y xy x₀ y₀ 1 1 e u o versor de i j 3 Seja fx y x arctg xy Calcule fu 1 1 onde u aponta na direção e sentido de máximo crescimento de f no ponto 11 4 Calcule a derivada direcional de fx y 1 x² y² no ponto 22 e na direção a v 1 2 b w i 2 j 5 Calcule a derivada direcional de fx y 2x² y² no ponto 1 1 e na direção 2 i 3 j 4 Axy xy Ax xy y dA y dx x dy Ay xy x NO PONTO 23 dA 3 dx 2 dy x 201 m y 297 m dA dA ΔA 3001 2003 ΔA dA ΔA 003 m² dx 001 dy 003 5 Vhr πhr² dv vh hr dh vr hr dr vh hr πr² vr hr 2πhr ESPESSURA 003 m dh 003 m dr 003 m Δv π 003 4 π 003 Δv 015 π m³ Δv 047124 m³ dv πr² dh 2πhr dr COM AS DIMENSOES DA CAIXA 6 dP Pv vR dv PR vR dR Pv vR 2vR PR vR v²R² dP 2vR dv v²R² dR dP 20 dv 200 dR V 100 VOLTZ R 10 Ω V DECRESCE 02V E R AUMENTA 001 Ω dv 02 dR 001 ΔP 2002 200001 ΔP 5 WATTS 7 V πr²h3 dv vr rh dr vh rh dh d v 2 π rh3 dr π r²3 dh vr rh 2πrh3 vh rh πr²3 h 20 cm r 12 cm dv 160 π dr 48 π dh h AUMENTA 2 mm dh 01 r DECRESCE 1 mm dh 02 Δv 160 π 01 48 π 02 Δv 64 π 1 a fxy x2 3y2 x0y0 12 and vector u versor of 2i j partial derivative of f respect vector u x0y0 gradient of fx0y0 dot u gradient fxy 2x 6y gradient f12 212 partial derivative of partial u partial of f with respect to u 12 212 dot 1sqrt5 21 1sqrt5 4 12 8sqrt5 8sqrt5 b ex2 y2 x0y0 11 and vector u versor of 34 partial derivative of f respect x 2x ex2 y2 partial derivative respect y 2y ex2 y2 gradient f11 22 vector u 15 34 partial derivative with respect to u 11 22 dot 15 34 15 6 8 25 25 c fxy arctan xy x0y0 33 and vector u 1sqrt2 1sqrt2 partial derivative respect x 1 1 x2y2 1y y x2 y2 partial derivative respect y 1 1 x2y2 xy2 x x2 y2 gradient f 33 318 318 16 11 partial derivative with respect to u 33 16 11 dot 1sqrt2 11 0 0 d fxy xy x0y0 11 and vector u versor of i j gradient fxy y x partial derivative respect u 11 1sqrt2 11 dot 11 1sqrt2 1 1 2 sqrt2 sqrt2 gradient f11 11 3 partial derivative of f respect u 11 normgradient f11 partial derivative respect x arctan xy x 1 1 xy2 1y arctan xy xy x2 y2 partial derivative respect x 11 arctan 1 1 1 1 12 π4 partial derivative respect y 11 1 1 1 12 partial derivative respect y x 1 1 xy2 x y x2 x2 y2 gradient f11 π4 12 12 partial derivative respect u 11 gradient f11 sqrtπ4 122 122 sqrtπ216 π4 12 4 fxy sqrt1 x2 y2 on 22 gradient fxy y 2 sqrt1 x2 y2 2x sqrt10 2y 1 sqrt1 x2 y2 xy partial derivative respect x x0y0 gradient f x0y0 dot u gradient f22 13 22 a vector n 12 norm is sqrt1 4 sqrt5 vector n hat 1sqrt5 12 partial derivative respect n hat 22 13 22 dot 1sqrt5 12 1 3 sqrt5 6 2 sqrt5 b vector w 1 2 partial derivative respect w 22 13 22 dot 1sqrt5 1 2 1 3 sqrt5 2 2 3 sqrt5 5 fxy 2 x2 y2 on point 11 direction 2i 3j norm sqrt22 32 sqrt13 gradient f partial derivative respect x vector i partial derivative respect y vector j 4x x2 y22 vector i 4y x2 y22 vector j vector u 2 sqrt13 vector i 3 sqrt13 vector j gradient f11 vector i vector j directional derivative of f11 gradient f11 dot u vector i vector j dot 2 sqrt13 vector i 3 sqrt13 vector j 2 sqrt13 3 sqrt13 1 sqrt13 directional derivative of f11 1 sqrt13