Calculus Archive: Questions from July 30, 2022
-
Evaluate the following sums using summation formulas. a. \( \sum_{k=1}^{17} k \quad \) b. \( \sum_{k=1}^{17} k^{2} \quad \) c. \( \sum_{k=1}^{17} k^{3} \) a. \( \sum_{k=1}^{17} k= \) b. \( \sum_{k=1}^3 answers -
La variable aleatoria \( X \) tiene la siguiente función de densidad \[ f(x)=\left\{\begin{array}{ll} \frac{1}{6}, & \text { en } 11 answer -
6. If function \( f(x) \) satisfies \( \int_{0}^{3 x} f\left(\frac{t}{3}\right) d t+e^{2 x} \), show \( f(x) \). Answer: \( \quad f(x)=3 e^{3 x}-2 e^{2 x} \).1 answer -
3. If \( f \) is differentiable and \( y=e^{\sin x} \sin e^{x}+f\left(\arctan \frac{1}{x}\right) \), show \( y^{\prime} \). (20 scores)1 answer -
Obtain the general solution of the following DEs: a. \( (x+2 y+1)-(2 x+y-1) y^{\prime}=0 \) b. \( \left(x^{3}+y^{2} \sqrt{x^{2}+y^{2}}\right) d x-x y \sqrt{x^{2}+y^{2}} d y=0 \) c. \( \left(y \sec ^{21 answer -
1 answer
-
Express tue integrand as a Sum of por pootial fraction and evaluate the integral. \[ \int \frac{6 x-12}{x^{2}-4 x-5} d x \] \( 3 \ln |x+5|+3 \ln |x-1|+c \) \( 4 \ln |x-5|-3 \ln |x+1|+c \) In \( 13(x-53 answers -
\( \left\{\begin{array}{l}x^{\prime}=-15 x-6 y-2 z \\ y^{\prime}=32 x+14 y+2 z \\ z^{\prime}=16 x+6 y+3 z\end{array}\right. \)1 answer -
Do any four. (a) Find \( d y / d x \) \( x=\sin ^{2}(\pi y-2) . \) (b) Find \( d r / d \theta \) \[ r=\frac{\sin \sqrt{2 \theta}}{\sqrt{2 \theta}} . \] (c) Find \( d y / d x \) \[ y=\int_{1}^{\sin x}1 answer -
(d) Find \( d y / d x \) \[ y=\sqrt[10]{\frac{1-x}{1+x}} \] (e) Find \( d y / d t \) \[ y=\sqrt{t+\sqrt{2 t+\sqrt{3 t-1}}} \] Evaluate the following limit \[ \lim _{n \rightarrow \infty}\left(\sqrt{\1 answer -
Evaluate the following limit \[ \lim _{n \rightarrow \infty}\left(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\sqrt{\frac{3}{n}}+\cdots+\sqrt{\frac{n}{n}}\right) \frac{1}{n} \] \( \frac{-1}{5(-x+1)^{\frac{1 answer -
1 answer