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Calculus Archive: Questions from February 27, 2023

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  (b) Given \( y=\arcsin \frac{1}{2 x} \), find \( \cos y \).
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  If \( f(x)=\cos x-5 \tan x \), the \[ f^{\prime}(x)= \] \[ f^{\prime}(4)= \]
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  Pars esta evaluacion, el estudiante deberí desarrollar y tesolver los siguientes problemas: Problem 1. a) Determine la energia de la siguiente sellal: \[ x(t)=2 \operatorname{rect}\left(\frac{t}{4}\r
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• venty the identity. (cos^(2)t)/(sint)=csct-sint
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• Verify: (sinx)/(cot^(2)x+1)+(cosx)/(tan^(2)x+1)=sin^(3)x+cos^(3)x
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• tion and simplify if possible. Assume all cancel (3s^(3)-2s^(2)t+12s-8t)/(9s^(3)+36s)-:(36s^(2)-16t^(2))/(48s^(4))
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• rify the identity. (1+cosx)/(sinx)+(sinx)/(1+cosx)=2cscx
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• plify each expressio (sinx)/(1+cosx)+(1+cosx)/(sinx)
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• en the domain. Show all work. csc^(2)(x)-2=0
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  b) Una sehal periddica de tiempo continuo con periodo fundamental \( T=12 \), e descrita sobte un periodo de la siguiente maneta: \[ x(t)=3 \operatorname{rect}\left(\frac{t+3}{4}\right)-4 \operatornam
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  6. Find \( y^{\prime} \) and \( y^{\prime \prime} \) if: \[ y=\frac{(x-1)(x-4)}{(x-2)(x-3)} \]
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• Find f_{x}; f_{y}; f xy , and f_{y}; f(x, y) = y ^ 2 * cos(2xy) + x ^ 4 * y ^ 2 + 7x + y + 1
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• Dividir. (3x^(2)+16x+18)-:(3x+4) La respuesta debe rendir el cociente y el residuo.
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• tion for all values of x by completi 3x^(2)-60=24x
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• r the given function, fin f(x)=(2x^(2)-7x-9)/(9x^(2)-3x+5), hori
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• Subtract. Simplify the result if possible. (5x^(2)+8x)/(x-8)-(37x+88)/(x-8)
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• List all possible zeros, and f(x)=-4x^(3)+15x^(2)-8x-3
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• For the function whose graph is illustrated, provide the value of the given quantity, if it exists, if not, write does not exist
  Para la función cuya gráfica se ilustra, proporcione el valor de la cantidad dada, si existe, si no la hay escriba no existe. a) \( \lim _{x \rightarrow 1} g(x) \) b) \( \lim _{x \rightarrow 2} g(x)
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  9. Halle \( f^{\prime}(x) \) y también \( f^{\prime}(x+3) \) en los siguientes casos. Hay que ser muy metódico para no cometer un error en algún paso. Consulte las respuestas (después de resolver
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  7. Suponga que \( f(x)=x^{3} \). (a) ¿Cuál es el valor de \( f^{\prime}(9), f^{\prime}(25), f^{\prime}(36) \) ? (c) Calcule \( f^{\prime}\left(a^{2}\right), f^{\prime}\left(x^{2}\right) \). Si no en
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  13-22. Critical points Find all critical points of the following functions. 13. \( f(x, y)=3 x^{2}-4 y^{2} \) 14. \( f(x, y)=x^{2}-6 x+y^{2}+8 y \) 15. \( f(x, y)=3 x^{2}+3 y-y^{3} \) 16. \( f(x, y)=x
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• Use the graph of f provided to determine the value of the quantity, if it exists. If it does not exist write it does not exist
  Use la gráfica de \( f \) que se proporciona para determinar el valor de la cantidad, si existe. Si no existe escriba no existe. a) \( \lim _{t \rightarrow-2} f(t) \) b) \( \lim f(t) \) \( t \rightar
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• Use the graph of f provided to determine the value of the quantity, if it exists. If it does not exist write it does not exist
  Use la grảfica de f que se proporciona para determinar el valor de la cantidad, si existe. Si no existe escriba no existe. a) \( \lim f(x) \) \[ x \rightarrow 0^{+} \] b) \( \lim f(x) \) \[ x \right
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• suppose that
  Suponga que: \( \lim _{x \rightarrow 1} h(x)=5 \) que \( \lim _{x \rightarrow 1} p(x)=1 \lim _{x \rightarrow 1} r(x)=2 \) calcule el límite \( \lim _{x \rightarrow 1} \frac{\sqrt{5 h(x)}}{p(x)(4-r(x)
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• suppose that
  Suponga que \( \lim _{x \rightarrow 0} f(x)=1 \) y que \( \lim _{x \rightarrow 0} g(x)=-5 \) calcucle: \( \lim _{x \rightarrow 0} \frac{2 f(x)-g(x)}{(f(x)+7)^{2 / 3}} \)
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• Find the limits that exist. If the limit does not exist, write does not exist. Write your answer with three decimal places.
  encuentre los limites que existan. Si el limite no existe, escriba no existe. Escriba su respuesta hasta con tres decimales. 1) \( \lim _{x \rightarrow-1^{+}} f(x) \) 2) \( \lim f(x) \) \[ x^{-1}-1^{-
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  \( \int \frac{3 \sin (\theta)}{\cos ^{3}(\theta)+4 \cos (\theta)} d \theta= \)
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  Given \( F(r, s, t)=-r\left(9 s^{2}+9 t^{2}\right) \), compute: \[ F_{r s t}= \] Use the Product Rule to comput \( f_{x} \) and \( f_{y} \) for \( f(x, y)=3 x e^{6 x} y \sin (y) \) (Use symbolic nota
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  (1 point) Let \( f(x, y, z)=\frac{x^{2}-5 y^{2}}{y^{2}+3 z^{2}} \). Then \( f_{x}(x, y, z)= \) \( f_{y}(x, y, z)= \) \( f_{z}(x, y, z)= \)
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  Find all the second partial derivatives. \[ f(x, y)=x^{4} y^{9}+4 x^{5} y \]
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  Find the first partial derivatives of the function. \[ \begin{array}{l} f(x, y, z)=x z-6 x^{5} y^{9} z^{3} \\ f_{x}(x, y, z)=z-30 x y^{9} z^{3} \\ f_{y}(x, y, z)=-54 x^{5} y^{8} z^{2} \\ f_{z}(x, y, z
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• Era el mediod ́ıa en un fr ́ıo d ́ıa de diciembre en la CDMX: 16◦C. El detective Leo, de la unidad de victimas espe- ciales, llego ́ a la escena del crimen para hallar al sargento Cris junto
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• In each exercise, express the given inequality in interval notation and draw the graph of the interval.
  Problema 1 En cada ejercicio expresar la desigualdad dada en notación de intervalos y trazar la gráfica del intervalo. i) \( 2
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• Express each of the intervals as an inequality in the variable x
  Problema 2 Expresar cada uno de los intervalos como una deslgualdad en la varlable \( x \) i) \( (-2,7] \) ii) \( [7,17] \) iii) \( [5,9) \) iv) \( (7,+\infty) \) v) \( (-\infty, 8] \)
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• The number of miles M(v) that a certain compact car can go on one gallon of gasoline is related to its speed v in mi/h (miles/hours) by: For what speeds are at least 45 miles traveled?
  Problema 6 El número de millas \( M(v) \) que clerto auto compacto puede recorrer con un galón de gasolina está relacionado con su velocldad \( v \) en \( \mathrm{mi} / \mathrm{h} \) (millas/horas)
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  21. Match the functions (a)-(f) with their graphs (A)-(F) in Figure 20. a. \( f(x, y)=|x|+|y| \) b. \( f(x, y)=\cos (x-y) \) c. \( f(x, y)=\frac{-1}{1+9 x^{2}+y^{2}} \) d. \( f(x, y)=\cos \left(y^{2}\
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  Find the intersection. \[ \text { 19) } x+y+z=1, x+y=10 \]
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  7. Calculate \( f_{z}(2,3,1) \), where \( f(x, y, z)=x y z \).
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• Encuentra el área limitada por las gráficas siguientes, estos tipos de figuras aparecen en los ciclos termodinámicos. ​​​​​​​
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• Find dy/dx by implicit differentiation. tan(x − y) = y 3 + x2 y' =
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• Solve 2cosx-sin^(2)x+2=0 for all values of x
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  Use the Product Rule to comput \( f_{x} \) and \( f_{y} \) for \( f(x, y)=3 x e^{6 x} y \sin (y) \) (Use symbolic notation and fractions where needed.) \[ f_{x}(x, y)= \] \[ f_{y}(x, y)=3 e^{6} x^{2}(
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  Evaluate the integral. \[ \int_{1}^{4} \int_{y}^{y^{2}} \sqrt{\frac{y}{x}} d x d y \] \[ \int_{1}^{4} \int_{y}^{y^{2}} \sqrt{\frac{y}{x}} d x d y= \] (Simplify your answer.)
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  \( f(x)=\left(x^{2}+5\right)^{5} \) \( y=x^{\frac{4}{9}}+4 x \)
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  \[ \frac{d}{d x} e^{x} \cos (2 x) \] (a) \( e^{x}(\sin (2 x)+\cos (2 x)) \) (b) \( e^{x} \sin (2 x) \) (c) \( e^{x}(-2 \sin (2 x)+\cos (2 x)) \) (d) \( -2 x^{x} \sin (2 x) \) \[ \frac{d}{d x} \sqrt{x}
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  Find the derivative of the function. \[ y=\frac{2}{5} \sec ^{2} x \quad y=\frac{2}{5} \sec ^{2} x \] A) \( y^{\prime}=\frac{2}{5} \sec ^{2} x \tan x \quad y^{\prime}= \) B) \( y^{\prime}=\frac{4}{5} \
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  Solve the initial value problem \[ \begin{array}{l} \frac{\mathrm{d}}{\mathrm{d} x} y(x)=(-10 x-6)(5 y(x)-25) ; \quad y(0)=0 \\ y(x)= \end{array} \]
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  Find the intersection. 19) \( x+y+z=1, x+y=10 \)
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• Find the absolute maximum and minimum values of f on the set D. f(x, y) = 4x + 6y − x2 − y2 + 1, D = {(x, y) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 5}
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  \( \begin{array}{l}\ln x^{y}=\ln \ln ^{x} y^{x} \text { if } x^{y}=y^{x} \\ \frac{d}{d x} y \ln x=\frac{d}{d x} x \ln y\end{array} \)
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  1. Find \( y^{\prime}, y^{\prime \prime} \), and \( y^{\prime \prime \prime} \) for each of the following functions a. \( y=x^{3} \) b. \( y=x^{-1} \) c. \( y=x^{\frac{1}{2}} \) d. \( y=x^{-\frac{1}{2
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  \( y=\left(1+\sin ^{8}\left(x^{9}\right)\right)^{3} \)
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• Please show all work.
  Exercise 2.5.9:* Solve \( y^{\prime \prime}+2 y^{\prime}+y=x^{2}, y(0)=1, y^{\prime}(0)=2 \).
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• 1) If f(x) = 2x^2- 4x+8, find f' (1) 2) If f(x) = 2x^2- 5x + 6, solve f’(x) = 20 3) If f(x) = 4x^2- 6x + 8, solve f’(x) = 0
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  Solve the initial value problem \[ \begin{array}{l} \frac{\mathrm{d}}{\mathrm{d} x} y(x)=(-10 x-6)(5 y(x)-25) ; \quad y(0)=0 \\ y(x)= \end{array} \]
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  \( \begin{array}{l}x y^{\prime}=\frac{1}{x \sqrt{9-x^{2}}}-2 y, y(3)=0 \\ \frac{d y}{d x}-\frac{1}{x} y=\frac{7 x^{2}+14 x}{x^{2}+3 x-10}, y(3)=3\end{array} \) \( y^{\prime}+y=\sin \left(e^{x}\right),
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  \( \begin{array}{l}h(x, y)=g(f(x, y)) \\ \quad g(t)=t^{2}+\sqrt{t}, \quad f(x, y)=7 x+3 y-21\end{array} \)
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• If g(2)=6, g'(2)=4, f(2)=8, and f'(2)=-4, find h'(2) when h(x)=f(x)/g(x)
  2. If \( g(2)=6, g^{\prime}(2)=4, f(2)=8 \), and \( f^{\prime}(2)=-4 \), find \( h^{\prime}(2) \) when \( h(x)=\frac{f(x)}{g(x)} \) \[ h^{\prime}(2)= \]
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  \( \begin{array}{l}\text { (8 pts.) } \\ \int \cos ^{2} \theta d \theta\end{array} \) \( \int \tan ^{2} x \sec ^{4} x d x \)
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• r the expression, if possible. Factor out ible, enter the original expression.
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• Resuelve la ecuacion. Encuentra las soluciones imaginarlas cuando sea posible. x^(4)-3x^(2)=4
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  2. (3 pts) Find the derivative, \( y^{\prime} \). \[ y=\left(5 x^{2}-7 x\right)^{-5}\left(x^{3}+1\right)^{3} \]
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• Verify the identity. (1+cosx)/(1-cosx)=(cscx+cotx)^(2)
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