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Calculus Archive: Questions from July 25, 2022

• question: Find the slope of the line tangent to the function y = (tan x) x at the point where x = 𝜋 /4 image:
  Encuentre la pendiente de la recta tangente a la función \( y=(\tan x)^{x} \) en el punto donde \( x=\frac{\pi}{4} \) (12 puntos)
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  Solve initial value Problem \[ \sqrt{y} d x+(4+x) d y=0, y(-3)=1 \]
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• both a&b
  Problem 3: Let \( f(x, y)=x y^{3}-5 x^{2}+3 x-x y+y^{4}+15 \) (a) Find \( f_{x}(x, y) \) and \( f_{y}(x, y) \). (b) Find \( f_{x x}(x, y), f_{x y}(x, y), f_{y x}(x, y) \) and \( f_{y y}(x, y) \).
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  10) \( \int \frac{\cos \theta}{\sin \theta \sqrt{36+\sin ^{2} \theta}} d \theta \) A) \( -\frac{1}{6} \ln \left|\frac{6+\sqrt{36+\sin ^{2} \theta}}{\sin \theta}\right|+C \) B) \( -\frac{1}{6} \ln \lef
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• Given the function f(x)=y°-9x2+15x-5, sketch the graph of / using the obtaining the following: the relative extremes of f, the inflection points of the graph of f. the intervals at which fes increase
  8. Dada la función \( f(x)=x^{3}-9 x^{2}+15 x-5 \), trace el bosquejo de la gráfica de \( f \) por medio de la obtención de lo siguiente: los extremos relativos de \( f \) los puntos de inflexión
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• y = x3 - 3x2 Determine: a. The equation of the tangent line parallel to the line y= -2x+1 b. The equation of the normal line to the graph of the function at its point of inflection
  Dada la siguiente función \( 7 . y=x^{3}-3 x^{2} \) Determine: Valor 12 ptos a. La ecuación de la recta tangente paralela a la recta \( y=-2 x+1 \) b. La ecuación de la recta nomal a la gráfica de
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  Determine the gradient of the following functions: (i) \( f: \mathbb{R}^{2} \rightarrow \mathbb{R}:(x, y) \mapsto 2 x^{3}+(4-3 x) y^{2} \), (ii) \( f: \mathbb{R}^{3} \rightarrow \mathbb{R}:(x, y, z) \
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• a)find dz/dt b)evaluate for s=1, & t=0
  \( x=s^{2}+t^{2}, y=s t, \quad z=e^{x} \tan y \)
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  The solution of the differential equation \[ \frac{d y}{d x}-\frac{3 x^{2}}{y}=0, \quad y(1)=-2 \] \[ y=\sqrt{2}\left(2 x^{3}+1\right)^{\frac{1}{2}} \] \[ y=\sqrt{2}\left(x^{3}+1\right) \] \[ y=\left(
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  Solve \[ \frac{d y}{d x}-\frac{2 y}{x^{2}}=0 \] \( y=C e^{2 x} \) \( y=2 e^{\frac{-2}{x}}+C \) \( y=C e^{\frac{-2}{x}} \) \( y=e^{\frac{2}{x}}+2 \) None of the above
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• differentiate the function: y= ln(3squareroot/(2x+1))
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• If f(x)=2x2+5x+5√xf(x)=2x2+5x+5x, then: f'(x)f′(x) = ​
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  Solve for \( y \). \( y^{\prime}=6 x+x y ; y=-1 \) when \( x=0 \) \[ y= \]
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  Let \( f(x, y, z)=2 x y+2 y z+2 x z \). Find \( f_{x}(x, y, z), f_{y}(x, y, z) \), and \( f_{z}(x, y, z) \). (Use symbolic notation and fractions where needed.) \[ f_{x}(x, y \] \[ f_{y}(x, y, z) \] \
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• If f(x)=4x2+2x+3√xf(x)=4x2+2x+3x, then: f'(x)=
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  Question 4 (1 point) Find the derivative of \( y=e^{3 x+2} \) a) \( y^{\prime}=e^{3 x+2} \) b) \( y^{\prime}=3 e^{3 x+2} \) c) \( y^{\prime}=(3 x+2) e^{3 x+2} \) d) \( y^{\prime}=(3 x) e^{3 x+2} \) h
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  6. Given the formula \( \int u^{\prime} e^{u} d x=e^{u}+c \), find three different \( f(x) \). So we can apply the formula to \( \int f(x) e^{x^{a}} d x \). (a is an integer) (15 points)
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  1. Find the following for the function \( f(x, y)=3 x^{3} y^{2}-5 x+26 y-9 \). a. \( f(1,2)= \) b. \( f_{y}(x, y)= \) c. \( f_{x}(x, y)= \) d. \( f_{x}(1,2)= \) e. \( f_{y y}(x, y)= \) f. \( f_{x x}(x
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• Thank you!
  \( \int \sqrt{\sin x} \cos ^{3} x d x \) \( \int \tan x \sec ^{3} x d x \)
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• the area please
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  Given \( =f(x, y)=x y e^{-8 y} \) What Are \( \begin{array}{ll}f_{x}(x, y)= & f_{y}(x, y)= \\ f_{x y}(x, y)= & f_{y z}(x, y)=\end{array} \) \( f_{x y}(x, y)=\quad f_{y z}(x, y)= \)
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