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Home Question Bank Math Calculus Archive 2025 January 18

Calculus Questions from Jan 18,2025

Browse the Calculus Q&A Archive for Jan 18,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Consider a region bounded by the x-axis and the line \( y = 2x \) from \( x = 0 \) to \( x = 1 \). If this shape is revolved around the x-axis, what is the volume of the resulting solid using triangular cross-sections perpendicular to the x-axis? The function \( f \) is defined by \( f(x)=x^{2}-2+3 \cos (2 x) \). Find all values of \( x \) that satisfy the conclusion of the Mean Value Theorem on the interval \( [-2,3] \). You may use a calculator and round to the nearest thousandth. The function \( f \) is defined by \( f(x)=x^{2}-2+3 \cos (2 x) \). Find all values of \( x \) that satisfy the conclusion of the Mean Value Theorem on the interval \( [-2,3] \). You may use a calculator and round to the nearest thousandth. Sia \( f \) una funzione derivabile su \( \mathbb{R} \), tale che \( \lim _{x \rightarrow+\infty} f^{\prime}(x)=3 \). È vero che: Scegli un'alternativa: \( f \) è limitata su una semiretta \( (a,+\infty) \) \( \lim _{x \rightarrow+\infty} f(x)=3 \) \( f \) ha asintoto obliquo destro \( f \) è un infinito di ordine 1 rispetto a \( x \), per \( x \rightarrow+\infty \) © non è necessariamente crescente su una semiretta \( (a,+\infty) \approx \) Data la funzione \( f(x)=\frac{x-1}{x^{3}+x^{2}-x-1} \), è vero che Scegli un'altemativa: \( f \) è limitata \( x \) \( f \) è un infinito di ordine 2 rispetto a \( \frac{1}{x+1} \), per \( x \rightarrow-1 \) \( \lim _{x \rightarrow-1} f(x) \) non esiste \( \lim _{x \rightarrow-1} f(x) \) esiste finito \( f \) è un infinito di ordine 1 rispetto a \( \frac{1}{x+1} \), per \( x \rightarrow-1 \) \( \int _ { - 1 } ^ { 0 } \sqrt { 2 + x ^ { 3 } } d x \quad n = 8 \) \( \int _ { 1 } ^ { 3 } e ^ { \sqrt { x } } d x \quad n = 5 \) (b) Use an appropriate method of differentiation to determine the derivative of the following functions (sim- plify your answers as far as possible): (i) \( f(x)=\cos ^{2}(x) e^{\tan (x)} \) (ii) \( g(x)=\frac{\tan ^{-1}(2 x)}{4 x^{2}+1} \) (iii) \( h(x)=\left(x^{3}-9\right)^{50} e^{x} \) (iv) \( p(x)=x \ln \left\{\frac{1-\cos x}{1+\sin x}\right\} \) \( \int _ { 0 } ^ { n } \frac { d x } { 1 + \sin x } \quad n = 6 \) Compute the average value of \( f(t) = \sin(t) + \cos(t) \) over the interval \([0, \pi]\).
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