Calculus Archive: Questions from September 30, 2022
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Problem 1. Find all second-order partial derivatives of the following functions. (a) \( f(x, y)=x^{2} y+\cos y+y \sin x \) (b) \( g(x, y)=\ln (x+y) \) (c) \( h(\vec{x}, y)=x^{2} \tan (x y) \)2 answers -
2. Find \( \frac{d y}{d x} \) and \( \frac{d^{2} y}{d x^{2}} \) for the following functions a. \( y=3 e^{\tan x} \) b. \( y=\sqrt{1+\sin x} \) c. \( y=5 \csc \left(x^{2}\right) \)2 answers -
1 answer
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0 answers
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\( \lim _{x \rightarrow 0}\left(\frac{6 x e^{x}}{4 x-e^{8 x}+1}\right) \) \( \lim _{x \rightarrow 0^{+}}\left((\cos (3 x))^{\frac{5}{x}}\right) \)2 answers -
number 44 please
part (b)? 39-48 Find the limit. 39. \( \lim _{x \rightarrow 0} \frac{\sin 3 x}{x} \) 40. \( \lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 6 x} \) 41. \( \lim _{t \rightarrow 0} \frac{\tan 6 t}{\sin 2 t2 answers -
3.)(2pts)Solve the IVP \( \quad(x+1)^{2} \frac{d y}{d x}+y=\frac{1}{x+1}, \quad y(0)=0 \). 4.)(2pt) Solve the exact DE \( \quad\left(x+e^{-y}\right)\left(y^{2}-x^{2}\right) d x+\left[y x\left[x+(2-y)2 answers -
10 and 16 please
1-16 Differentiate. 1. \( f(x)=3 x^{2}-2 \cos x \) 2. \( f(x)=\sqrt{x} \sin x \) 3. \( f(x)=\sin x+\frac{1}{2} \cot x \) 4. \( y=2 \sec x-\csc x \) 5. \( y=\sec \theta \tan \theta \) 6. \( g(\theta)=e2 answers -
2 answers
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2 answers
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\[ y^{\prime}=2(y-70) \] \[ y=70 e^{2 x}+c \] none of these \[ y=c e^{x^{2}}+70 \] \[ y=c e^{70 x}+2 \]2 answers -
11. \[ y^{\prime}+\frac{y}{x}=\frac{e^{x}}{x} \] \[ y=\frac{e^{x}}{x}+\frac{c}{x} \] \[ y=\frac{e^{x}}{x}+c \] \[ y=\frac{e^{x}}{x}+c x \] none of these2 answers -
10. \( y^{\prime}=2 x y, \quad y(0)=3 \) \( y=e^{x^{2}}+2 \) \( y=3 e^{x} \) \( y=2 e^{x}+1 \) \[ y=3 e^{x^{2}} \]2 answers -
\( x y^{\prime}=3 y, \quad y(1)=5 \) \( y=5 x^{2} \) \( y=3 x^{3}+2 \) \( y=5 x^{3} \) \( y=3 x^{2}+2 \)2 answers -
\( x y^{\prime}=3 y, \quad y(1)=5 \) \( y=5 x^{2} \) \( y=3 x^{3}+2 \) \( y=5 x^{3} \) \( y=3 x^{2}+2 \)2 answers -
\( y^{\prime}=-x \div y, \quad y(4)=3 \) \( x^{2}+y^{2}=25 \) \( x^{2}-y^{2}=7 \) \( x+y=7 \) \( x-y=1 \)2 answers -
Find the derivative of the function. \[ y=3 \tan ^{-1}\left(x+\sqrt{1+x^{2}}\right) \] \[ y^{\prime}= \]2 answers -
2 answers
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2 answers
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6. \( (x-y) d x+x d y=0 \) \( y=c x+x \ln x \) \( y=c x^{2}+x \ln x \) \( y=c x-x \ln x \) \( y=x+x \ln x+c \)2 answers -
3. \( y^{\prime}+2 y \div x=x \) \[ y=x^{2} \div 4+c \div x^{2} \] \[ y=x^{2} \div 2+c \div x^{2} \] \[ y=x^{4} \div 4+c \div x^{4} \] \[ y=x^{2} \div 2+c \div x^{3} \]2 answers -
\( y^{\prime}=2 x y^{2} \) \( y=1 \div x^{2}+c \) \( y=x^{2}+c \) \( y=1 \div\left(x^{2}+c\right) \) \( y=1 \div\left(c-x^{2}\right) \)2 answers -
2 answers
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Dada la función \( f(x)=\frac{11 x+3}{14 x-1} \), encontrar \( f^{\prime}(8) \), usando la regla del cociente. En el numerador de la derivada tenemos: \[ u v^{\prime}-v u^{\prime}=\left(\quad \sigma^2 answers -
Translation: Find the critical points of the functions below and determine whether it is a relative maximum, relative minimum, or a saddle point
Resuelva: Halle los puntos críticos de las funciones que se presentan a continuación y determine si es un máximo relativo, mínimo relativo o un punto de silla. 1. \( f(x, y)=80 x+80 y-x^{2}-y^{2}2 answers -
1. If \( f(x)=\frac{3 x+5}{7 x+2} \), find; a. \( f^{\prime}(x) \) b. \( f^{\prime}(2) \) 2. Differentiate \( y=\frac{2-\sec x}{\tan x} \)2 answers -
2 answers
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Solve the given initial-value problem. \[ 5 y^{\prime \prime}+y^{\prime}=-8 x, y(0)=0, y^{\prime}(0)=-10 \]2 answers -
Which function is differentiable at \( (0,0) \) \[ f(x, y)=\left\{\begin{array}{ll} \frac{x^{3}}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0) \end{array}\right. \] \[ f(2 answers -
2 answers
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1. Considere la siguiente información. \[ A=\left[\begin{array}{ccc} \frac{1}{8} & 0 & 0 \\ 0 & \frac{1}{8} & 0 \\ 0 & 0 & \frac{1}{8} \end{array}\right] \] Teniendo en cuenta los datos de arriba, in2 answers -
Q3. Let \( \mathbf{x}=\left(x_{1}, x_{2}\right), \mathbf{y}=\left(y_{1}, y_{2}\right) \in \mathbb{R}^{2} \). We define \[ \langle\mathbf{x}, \mathbf{y}\rangle:=x_{1} y_{1}+2 x_{1} y_{2}+3 x_{2} y_{1}+2 answers -
Find the general solution of the equation. \[ \begin{aligned} y^{\prime}-\frac{y}{10} &=-8 \\ y &\left.=C e^{10 x}+\frac{4}{5}\right) \\ y &=C e^{x / 10}-80 e^{x} \\ y &=C e^{x / 10}+80 \\ y &=C e^{102 answers -
Find the general solution of the equation. \[ \begin{aligned} y^{\prime}-\frac{y}{10} &=-8 \\ y &\left.=C e^{10 x}+\frac{4}{5}\right) \\ y &=C e^{x / 10}-80 e^{x} \\ y &=C e^{x / 10}+80 \\ y &=C e^{102 answers -
P 1 (cont.) b. \( \left(2 y-\frac{1}{x}+\cos 3 x\right) \frac{d y}{d x}+\frac{y}{x^{2}}-4 x^{3}-3 y \sin 3 x=0 \)2 answers -
Find \( d y / d t \). \[ \begin{aligned} y=& \cos ^{4}(\pi t-19) \\ &-4 \cos ^{3}(\pi t-19) \sin (\pi t-19) \\ & 4 \cos ^{3}(\pi t-19) \\ &-4 \pi \sin ^{3}(\pi t-19) \\ &-4 \pi \cos ^{3}(\pi t-19) \si2 answers -
Which function is differentiable at \( (0,0) \) \[ \begin{array}{r} f(x, y)=\left\{\begin{array}{ll} \frac{x}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0) \end{array}\ri1 answer -
Which function is differentiable at \( (0,0) \) \[ \begin{array}{c} f(x, y)=\left\{\begin{array}{ll} \frac{x^{2}}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0) \end{array2 answers -
Find \( D_{x} y \). \[ \begin{aligned} y=& 3 x(2 x+4)^{4} \\ & 3(2 x+4)^{4}(6 x+4) \\ & 3(10 x+4)^{3} \\ & 3(2 x+4)^{3} \\ & 3(2 x+4)^{3}(10 x+4) \end{aligned} \]2 answers