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Home Question Bank Math Calculus Archive 2024 November 23

Calculus Questions from Nov 23,2024

Browse the Calculus Q&A Archive for Nov 23,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Recall the Chain Rule in terms of the functions \( h(x) \) and \( g(x) \), which states that if \( g \) is differentiable at \( x \) and \( h \) is differentiable at \( g(x) \), then the composite function \( F=h 。 g \) defined by \( F(x)=h(g(x)) \) is differentiable at \( x \) and \( F^{\prime} \) is given by the following. \[ F^{\prime}(x)=h^{\prime}(g(x)) \cdot g^{\prime}(x) \] We are given the function \( f(x)=\sin (8 \ln (x)) \) To write this is the form \( F(x)=h(g(x)) \) we can let \( h(x)= \) Submit (a) Find \( y^{\prime} \) by implicit differentiation. \( y^{\prime}=\frac{7-27 x^{2}}{5 y^{4}} \) (b) Solve the equation explicitly for \( y \) and differentiate to get \( y^{\prime} \) in terms of \( x \). \( y^{\prime}=\frac{7-27 x^{2}}{5\left(7 x-9 x^{3}\right)\left(\frac{4}{5}\right)} \) (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for \( y \) into your solution for part (a) \( y^{\prime}=\square \) Determine el volumen de la región acotada arriba por el paraboloide \( z=x^{2}+y^{2} \), \( y \) abajo por el cuadrado \( R:-1 \leq x \leq 1,-1 \leq y \leq 1 \) Write \( \cosh (\ln (x)) \) as a rational function of \( x \). Write \( 4 e^{2 x}+8 e^{-2 x} \) in terms of \( \sinh (2 x) \) and \( \cosh (2 x) \) \( (\square) \cosh (2 x) \) What is the value of the definite integral \( \int_{0}^{\frac{\pi}{2}} \sin(x) \, dx \)? \( \int \frac { 1 } { \sqrt { 4 x ^ { 2 } + 7 } } d x \) Determine whether the infinite geometric series converges or diverges. If it converges, find its si \( \sum_{k=1}^{\infty} \frac{1}{5} 9^{k-1} \) Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. \( 8+4+2+\cdots \) descrolla Longitud de curva colculer la longitud de la cuova de la siguiente funcion en el intervalo dado \( f(x) \) \( -2+x^{3 / 2} \) en el intervalo dado \( [0,1] \)
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