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Home Question Bank Math Calculus Archive 2025 January 21

Calculus Questions from Jan 21,2025

Browse the Calculus Q&A Archive for Jan 21,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Determine el volumen resultante de rotar la función \( f(x) = \frac{1}{x} \), desde \( x = 1 \) hasta \( x = 2 \), alrededor del eje y usando el método de lavadoras. \( f(x)=\frac{1}{2} x^{2}-5 x-2 \) Berken het differentiequalient van \( f(x) \) op \( [-3,1] \) Beceken de gemiddelde verandering van y op \( [2,5] \) van de functie \( f(x)=2 x^{3}-4 x+1 \) Fund the area of the region bounded by \( y=\sqrt{x} \quad y=18-x^{2} \quad \) and the \( x \)-axis in the 1st Quadvant. (1) Hallar la derivada \( y=\left[\operatorname{arcsen}\left(\mathbb{e}^{2 x}\right)\right] \mathbb{C}^{2} \) Where the functina is increasing and decreasing, \[ f(x)=3 x^{2}+5 \] a. The minimum value is 5 . The domain is all real numbers and the range is \( y \geq 5 \). The frnction is decreasing to the left of \( x=0 \) and increasing to the right of \( x=0 \). b. The maxin value is 5 . The domain is all real rubers and the mige it \( 1 \leq 5 \). The function is increasing to the left of \( x=0 \) and decreasing to the right of \( x=0 \). c. The minimum value is -5 . The domain is all real numbers and the range is \( y \geq-5 \). The fumction is decreasing to the left of \( x=0 \) and increasing to the right of \( x=0 \). d. The maxim walue is -5 . The domain is all real numbers and the range is \( y \leq-5 \). Th function is increasing to the left of \( x=0 \) and decreasing to the right of \( x=0 \). \( \int _ { 0 } ^ { \pi } e ^ { \cos ( t ) } \sin ( 2 t ) d t \) Name: 14. For this problem, \( f(x)=x^{2} \). Because \( f \) is differentiable everywhere, it satisfies all hypotheses of the MVT on any interval \( [a, b] \). a) What does the MVT say about \( f \) on \( [\mathrm{a}, \mathrm{b}]=[-1,1] \) ? Find all suitable values of \( c \). How many are there? b) What does the MVT say about \( f \) on \( [\mathrm{a}, \mathrm{b}]=[1,2] \) ? Find all suitable values of \( c \). How many are there? c) Show that for \( f(x)=x^{2} \) and any interval \( [a, b] \), the MVT's number \( c \) is the midpoint of \( [\mathrm{a}, \mathrm{b}] \). \( y ^ { \prime \prime } + 4 y ^ { \prime } + 5 y = 0 \quad \operatorname { con } y ( 0 ) = 1 \quad y ^ { \prime } ( 0 ) \Rightarrow 6 \) 13. Verify that the hypotheses of the Mean Value Theorem are satisfied on the given interval, and find all values of \( c \) in that interval that satisfy the conclusion of the theorem. a) \( f(x)=x^{2}+x ;[-4,6] \) d) \( f(x)=\sqrt{2}-x-x^{2} \); b) \( \left.\left.f(x)=x^{3}+x-4 ;-1,2\right]\right] \) c) \( f(x)=x+\frac{1}{x} \); 3,4\( ] \) Use \( z \)-substitution, i.e \( z=\tan \frac{x}{2} \) to determine the integral \[ \int \frac{2 \sin x}{1-\cos 2 x} d x \]
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