Calculus

Problem 21801

Evaluate the integral from -2 to 5 of the function $- |x|$ .

See Solution

Problem 21802

Find the equation of the tangent line to $f(t)=\left(e^{-t}+\sqrt{t}\right)^{2}$ at $(1/2, f(1/2))$ , for $t>0.1$ .

See Solution

Problem 21803

Find where the curve $3x + 3y^2 - y = 10$ has vertical and horizontal tangent lines.

See Solution

Problem 21804

Find the limit as $\Delta x \rightarrow 0$ of $\sum_{k=1}^{n}(x_{k}^{*})^{6} \Delta x_{k}$ over the interval $[3,9]$ .

See Solution

Problem 21805

Calculate the integral from 0 to 2 of $\sqrt{4-x^{2}}$ with respect to $x$ .

See Solution

Problem 21806

Calculate the line integral $\int_{C} \vec{F} \cdot d \vec{r}$ for $\vec{F} = <x^{2}, y^{2}>$ and $C: \vec{r}(t)=\langle\sin(t), \cos(t), t^{2}\rangle$ for $0 \leq t \leq \frac{\pi}{2}$ . Set up the integral but do not evaluate it.

See Solution

Problem 21807

Set up the integral $\int_{C} \vec{F} \cdot d \vec{r}$ with $\vec{F}=\langle x^{2}, y^{2}, z^{2} \rangle$ and $C: \vec{r}(t)=\langle \sin(t), \cos(t), t^{2} \rangle$ , $0 \leq t \leq \frac{\pi}{2}$ .

See Solution

Problem 21808

Calculate $\int_{0}^{5} f(x) dx$ for the piecewise function $f(x)=\begin{cases}4 & \text{if } x \leq 4 \\ 3x - 8 & \text{if } x > 4\end{cases}$ .

See Solution

Problem 21809

Calculate the indefinite integral using substitution: $\int \frac{x}{6-x^{2}} \, dx$ . Don't forget absolute values!

See Solution

Problem 21810

Find the derivative of $g(t) = 3 \tan(t^{3})$ .

See Solution

Problem 21811

Evaluate $\int_{7}^{2} g(x) d x$ given $\int_{2}^{7} g(x) d x=4$ .

See Solution

Problem 21812

Calculate the average rate of change of $g(x)=-5+3(x-1)$ from $x=-2$ to $x=0$ .

See Solution

Problem 21813

Given $\int_{2}^{4} f(x) d x=5$ , $\int_{2}^{7} f(x) d x=-3$ , and $\int_{2}^{7} g(x) d x=4$ , find:
1. $\int_{7}^{2} g(x) d x$
2. $\int_{2}^{7} 9 g(x) d x$
3. $\int_{2}^{7}[g(x)-f(x)] d x=\square$

See Solution

Problem 21814

Given $\int_{2}^{4} f(x) d x=5$ , $\int_{2}^{7} f(x) d x=-3$ , and $\int_{2}^{7} g(x) d x=4$ , find $\int_{2}^{7} 9 g(x) d x=\square$ .

See Solution

Problem 21815

Given $\int_{2}^{7} f(x) d x=5$ , $\int_{2}^{6} f(x) d x=-3$ , and $\int_{2}^{7} g(x) d x=4$ , find:
1. $\int_{7}^{2} g(x) d x$
2. $\int_{2}^{7} 9 g(x) d x$
3. $\int_{2}^{7}[g(x)-f(x)] d x=\square$

See Solution

Problem 21816

Find the biomass at $t=13$ hours if the average rate of change from $t=3$ to $t=13$ is $\frac{1}{5} \mathrm{mg}/\mathrm{hour}$ and at $t=3$ is 13.

See Solution

Problem 21817

Find $\int_{2}^{6} 7 f(x) d x$ given $\int_{2}^{7} f(x) dx=11$ , $\int_{2}^{7} g(x) dx=7$ , $\int_{6}^{7} f(x) dx=6$ , and $\int_{2}^{6} g(x) dx=3$ .

See Solution

Problem 21818

Evaluate the following integrals given $\int_{2}^{4} f(x) d x=5$ , $\int_{2}^{7} f(x) d x=-3$ , and $\int_{2}^{7} g(x) d x=4$ :
1. $\int_{7}^{2} g(x) d x$
2. $\int_{2}^{7} 9 g(x) d x$
3. $\int_{2}^{7}[g(x)-f(x)] d x$
4. $\int_{2}^{7}[6 g(x)-f(x)] d x$

See Solution

Problem 21819

Evaluate $\int_{s} \int \vec{F} \cdot \vec{N} d S$ for $\vec{F}(x, y, z)=\langle x, y, z\rangle$ on $S: x^{2}+y^{2}+z^{2}=9$ in the first octant.

See Solution

Problem 21820

Given functions $f$ and $g$ on $[2,7]$ with $\int_{2}^{7} f(x) dx=11$ , $\int_{2}^{7} g(x) dx=7$ , $\int_{6}^{7} f(x) dx=6$ , and $\int_{2}^{6} g(x) dx=3$ :
a. Find $\int_{2}^{6} 7 f(x) dx$ .
b. Find $\int_{2}^{7}(f(x)-g(x)) dx=\square$ .

See Solution

Problem 21821

Find the derivative of the function $f(x) = \sqrt{x + \sqrt{x}}$ .

See Solution

Problem 21822

Find the value of $\int_{2}^{6} (f(x) - g(x)) d x$ given $\int_{2}^{6} 7 f(x) d x=35$ and $\int_{2}^{7}(f(x)-g(x)) d x=4$ .

See Solution

Problem 21823

Given functions $f$ and $g$ on $[2,7]$ , find:
a. $\int_{2}^{6} 7 f(x) d x$ b. $\int_{2}^{7}(f(x)-g(x)) d x$ c. $\int_{2}^{6}(f(x)-g(x)) d x=\square$

See Solution

Problem 21824

Find the derivative $f^{\prime}(x)$ of the function $f(x)=x^{2}-2x$ and evaluate it at $x=3$ .

See Solution

Problem 21825

Find the limit $\lim _{x \rightarrow 0} f(x)$ for the piecewise function:
$f(x)=\begin{cases} \lfloor x+3\rfloor & x<0 \\ 4 e^{-2 x} & 0 \leq x<6 \\ 2 \ln (x-2) & x \geq 6 \end{cases}$

See Solution

Problem 21826

Find the derivative $f^{\prime}(2)$ for the function $f(x)=|x-2|$ .

See Solution

Problem 21827

Calculate the indefinite integral of $e^{-x^{2}} x \, dx$ .

See Solution

Problem 21828

Calculate the integral using substitution: $\int\left(x^{6}-9\right)^{7} x^{5} dx$ .

See Solution

Problem 21829

Find the tangent and normal line equations to $x^{3}+2xy-y^{2}=11$ at the point $(2,3)$ .

See Solution

Problem 21830

Find the area $A$ between the curves $x=y^{2}+2y$ and $x=y+12$ . Provide the answer in exact form.

See Solution

Problem 21831

Calculate the integral of $e^{x^{6}} x^{6}$ with respect to $x$ : $\int e^{x^{6}} x^{6} d x$ .

See Solution

Problem 21832

Find the time $x$ when the drug concentration $K(x)=\frac{12x}{x^2+25}$ is at its maximum.

See Solution

Problem 21833

Approximate the change in drug concentration $C(x)=\frac{10 x}{9+x^{2}}$ from $x=0.5$ to $x=10$ .

See Solution

Problem 21834

Bestimme den Grenzwert $\varlimsup_{n \to \infty} \sqrt{n^{2}-n}$ für die Folge $c_{n}$ .

See Solution

Problem 21835

Find the instantaneous rate of change of height $h$ at $t=20 \mathrm{~s}$ from $h(t)=15 \cos \left(\frac{\pi t}{120}\right)+18$ . What does this rate represent?

See Solution

Problem 21836

Find the monkey's acceleration at $t=3$ for the distance $S(t)=t \sin (2 t)+t^{3}$ . Round up the result.

See Solution

Problem 21837

Find the critical points of the function $y=f(x)=4 x^{3}-6 x^{2}-72 x+3$ .

See Solution

Problem 21838

Given a conservative vector field $\vec{F}(x, y)$ with potential function $f(x, y)$ , find the integrals along paths $C_1$ , $C_2$ , $C_3$ , and $C_4$ from $\left(x_1, y_1\right)$ to $\left(x_2, y_2\right)$ , where $f\left(x_1, y_1\right)=1$ , $f\left(x_2, y_2\right)=4$ .

See Solution

Problem 21839

Calculate the integral of $(x^{2}+3)^{9} 5 x \, dx$ .

See Solution

Problem 21840

Calculate the indefinite integral: $\int \sqrt[3]{z^{3}+8} z^{2} \, dz$ .

See Solution

Problem 21841

Compare the integrals $\int_{S_{1}} \int(\operatorname{curl} \overrightarrow{\mathbf{F}}) \cdot \overrightarrow{\mathbf{N}} d S$ and $\int_{S_{2}} \int(\operatorname{curl} \overrightarrow{\mathbf{F}}) \cdot \overrightarrow{\mathbf{N}} d S$ . Which is greater?

See Solution

Problem 21842

Compare the values of the integrals over two surfaces: $\int_{S_{1}} \operatorname{curl} \overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathbf{N}} d S$ and $\int_{S_{2}} \operatorname{curl} \overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathbf{N}} d S$ to determine which is greater.

See Solution

Problem 21843

Find the tangent line equation for the function $f(t)=\left(e^{-2 t}+\sqrt{t}\right)^{2}$ at $(3, f(3))$ , $t>0.1$ .

See Solution

Problem 21844

Find the monkey's acceleration at $t=3$ for $S(t)=t \sin (2 t)+t^{3}$ . Round up your answer. a) 15 b) 20 c) 26 d) 27

See Solution

Problem 21845

Calculate the indefinite integral using substitution: $\int \sqrt[8]{x^{8}+256} x^{6} d x$ .

See Solution

Problem 21846

Find the tangent line equation in terms of $x$ and $y$ for $r=3 \sin 4 \theta$ at $\theta=\pi / 3$ . $y=$

See Solution

Problem 21847

Calculate $14 \int_{0}^{2} 15 \sqrt{4 x+1} \, dx$ .

See Solution

Problem 21848

Approximate the integral $\int_{1}^{10}\left(x^{2}+4\right) d x$ using a Riemann sum with 3 subintervals of length 3.

See Solution

Problem 21849

Approximate the integral $\int_{2}^{11}(x^{3}+4) dx$ using a Riemann sum with 3 subintervals of length 3.

See Solution

Problem 21850

Differentiate and optimize the equation $5 x^{2}+4 x-16$ . Find $x=$ and state max or min value. Show workings to 1.d.p.

See Solution

Problem 21851

Differentiate and optimize $5 x^{2}+4 x-16$ . Find $x=$ and state max or min value, correct to 1.d.p.

See Solution

Problem 21852

Find the total reaction from $t=1$ to $t=10$ for $R^{\prime}(t)=\frac{6}{t}+\frac{4}{t^{3}}+e^{-2t}$ . Round up.

See Solution

Problem 21853

Evaluate the integral: $\int(\sin (x))^{20}(\cos (x)) d x$ .

See Solution

Problem 21854

Find the derivative of the function $f(x)=(x^{8}+6 x)^{5}$ . What is $\frac{d f(x)}{d x}$ ?

See Solution

Problem 21855

Find $\frac{dy}{dt}$ if $y = \left( \tan \left( x^{8} \right) + 6 \right)^{6}$ .

See Solution

Problem 21856

Evaluate the integral $\int_{0}^{1} x^{4} x^{x^{5}} d x$ .

See Solution

Problem 21857

Calculate the integral $\int_{0}^{1} x^{4} e^{x^{5}} d x$ .

See Solution

Problem 21858

Find $\frac{d y}{d t}$ for $y=\left(\tan \left(x^{8}\right)+6\right)^{6}$ . Choose the correct option.

See Solution

Problem 21859

Find the integral of the function: $\int(e^{3 t}+t+5) dt$ . Choose the correct answer from the options.

See Solution

Problem 21860

If $f$ is even and $\int_{0}^{2} f(x) dx=1$ , $\int_{0}^{6} f(x) dx=3$ , find $\int_{-2}^{6} f(x) dx$ . a) 6 b) 8 c) 10 d) 4

See Solution

Problem 21861

Find the total reaction from $t=1$ to $t=10$ for $R^{\prime}(t)=\frac{6}{t}+\frac{4}{t^{3}}+e^{-2t}$ . Round up.

See Solution

Problem 21862

Find the partial derivative $\frac{\partial}{\partial y} 200 x^{\frac{1}{5}} y^{\frac{4}{5}}$ .

See Solution

Problem 21863

Bestimmen Sie $f^{\prime}(x)$ für: a) $f(x)=(2 x-1)^{5}$ , b) $f(x)=x \cdot(7 x+3)^{6}$ , c) $f(x)=\sqrt{x} \cdot(x^{2}-1)$ , d) $f(x)=x \cdot \cos (2 x+1)$ , e) $f(x)=\sqrt{5-3 x}$ , f) $f(x)=x^{2} \cdot \cos (3 x)$ .

See Solution

Problem 21864

Find the derivative of the function $\left(\frac{2 x^{3}-12 x+1}{1+x^{2}}\right)$ .

See Solution

Problem 21865

Find the area under the curve $y = 2x - x^2$ from $x = 0$ to $x = 2$ : $\int_{0}^{2} (2x - x^2) \, dx$ .

See Solution

Problem 21866

Evaluate the integral $\int_{1}^{e^{3}} \frac{(\ln x)^{5}}{x} dx$ .

See Solution

Problem 21867

Calculate the integral from -1 to 1 of $\frac{1}{4} x^{3}-\frac{2}{3} x$ .

See Solution

Problem 21868

Berechne die Ableitung $f^{\prime}(5.94)$ für die Funktion $f(x)=3.3 x^{4}-64 x^{2}+9.2 x+30$ und runde auf zwei Nachkommastellen.

See Solution

Problem 21869

Evaluate the integral $\int_{0}^{\pi / 3} \frac{5 \sin (5 t)}{4-\cos (5 t)} d t$ .

See Solution

Problem 21870

Find the area under the curve $y = \frac{3}{x^3}$ from $x=1$ to $x=3$ using the integral $\int_{1}^{3} \frac{3}{x^3} \, dx$ .

See Solution

Problem 21871

Gegeben ist die Funktion $f(x)=\frac{1}{3} x^{3}-2 x+3$ . Finde die Tangentengleichung bei $B(0 \mid f(0))$ und zeige, dass $f$ für $x>0$ linksgekrümmt ist.

See Solution

Problem 21872

Untersuchen Sie das Krümmungsverhalten der Funktion $f(x)=(2 x-3)^{3}-12 x^{2}$ .

See Solution

Problem 21873

Bestimme die Ableitung von $f(x)=3\left(x^{3}-1\right) \sqrt{x^{7}}$ bei $x_{0}=4.3$ . Ergebnis: $f^{\prime}(4.3)=$

See Solution

Problem 21874

Evaluate the integral $\int_{C} 2xyz \, ds$ for the curve $C: \vec{r}(t)=12t\vec{i}+5t\vec{j}+84t\vec{k}$ , $0 \leq t \leq 1$ .

See Solution

Problem 21875

Leiten Sie die Funktionen ab: a) $f(x)=x^{a}$ b) $f(x)=x^{t+1}$

See Solution

Problem 21876

Find the derivative of $\sin ^{2} x + \sqrt{\pi} + \cos ^{2} x + 5^{7}$ with respect to $x$ .

See Solution

Problem 21877

Bestimme die richtigen Ableitungen der Funktion $f(x)=\left(\ln \frac{x^{2}+1}{e^{x}}\right)^{20}$ .

See Solution

Problem 21878

Welche Ableitung ist richtig für $f(x)=\left(\ln \frac{x^{2}+1}{e^{x}}\right)^{20}$ ?

See Solution

Problem 21879

Bestimme die Ableitung von $f(x)=2\left(x^{2}-1\right) \sqrt{x^{5}}$ bei $x_{0}=4.2$ . Ergebnis: $f^{\prime}(4.2)=$

See Solution

Problem 21880

Find the derivative of the function $f(x)=-3 x^{8} \ln x$ and evaluate it at $x=e^{2}$ .

See Solution

Problem 21881

Find the velocity of a particle given by $s(t)=4 t^{3}-78 t^{2}+360 t$ at time $t$ , where $t$ is in seconds.

See Solution

Problem 21882

Find the tangent line equation for $y=\cos^{2}(2^{x})$ at $x=\frac{\pi}{3}$ . Round to two decimal places.

See Solution

Problem 21883

Approximate the displacement of an object with velocity $v=\frac{1}{2} t^{2}+5(f t / s)$ from $t=0$ to $t=12$ using $n=6$ .

See Solution

Problem 21884

Bestimme die Extrem- und Wendepunkte der Funktion $f(x)=x \cdot(x-3)^{2}$ .

See Solution

Problem 21885

Berechne die Grenzwerte von $f(x)$ bei $x_{0}=2$ : links für $0<x<2$ und rechts für $x>2$ . Ergebnisse auf zwei Dezimalstellen.

See Solution

Problem 21886

Find the indefinite integral: $\int\left(4 y^{2}+8 y\right)^{3}(y+1) d y$

See Solution

Problem 21887

Find the derivative of the function $f(x)=\ln \left(\sqrt{\frac{8 x-6}{5 x+4}}\right)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 21888

Calculate the indefinite integral using substitution:
$\int \frac{x^{3}+x^{2}}{3 x^{4}+4 x^{3}} d x$

See Solution

Problem 21889

Differentiate the function $f(x) = 7^{-9x + 4}$ . What is $f^{\prime}(x) =$ ?

See Solution

Problem 21890

Bestimmen Sie die Ableitung von $f(x)=2\left(x^{3}-1\right) \sqrt{x^{7}}$ bei $x_{0}=1$ und runden Sie auf zwei Nachkommastellen.

See Solution

Problem 21891

Find the derivative of the function $f(x)=20-3^{x}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 21892

Find the derivative of $f(x)=x^{5} e^{3.5 x}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 21893

Find the derivative of $f(x) = x^{7} e^{5.5 x}$ . What is $f'(x)$ ?

See Solution

Problem 21894

Find the derivative of $f(x)=5 e^{-8 x^{10}+3 x^{5}}$ using the chain rule. What is $f^{\prime}(x)=$ ?

See Solution

Problem 21895

Compute $F(0)$ , $F^{\prime}(x)$ , and $F^{\prime}(2)$ for $F(x)=\int_{0}^{x} \sqrt{t^{3}+1} dt$ .

See Solution

Problem 21896

Find the tangent line to $y=e^{x}$ parallel to $2x-y=5$ and the intervals of increase/decrease for $f(x)=x^{3} e^{x}$ .

See Solution

Problem 21897

Find the derivative of $g(x)=\frac{e^{x}}{1-3 x}$ . What is $g^{\prime}(x)$ ?

See Solution

Problem 21898

Find the tangent line to $y=e^{x}$ parallel to $2x-y=5$ .

See Solution

Problem 21899

Welche Ableitung ist richtig für $f(x)=2\left(x^{3}-1\right) \sqrt{x^{7}}$ ? Optionen sind: $f^{\prime}(x)=13 \sqrt[5]{x^{11}}-7 \sqrt{x^{5}}$ , $f^{\prime}(x)=6 x^{\frac{11}{2}}+7 x^{\frac{5}{2}} \cdot\left(x^{3}-1\right)$ , $f^{\prime}(x)=\sqrt{x^{5}} \cdot\left(13 \sqrt[5]{x^{11}}-7\right)$ , $f^{\prime}(x)=13 \sqrt{x^{11}}-7 \sqrt{x^{5}}$ , oder keine der Antworten.

See Solution

Problem 21900

Find the derivative of the function $f(x)=7 e^{x}+x^{e}$ . What is $f^{\prime}(x)$ ?

See Solution

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Calculus

Problem 21801

Evaluate the integral from -2 to 5 of the function −∣x∣- |x|−∣x∣.

Problem 21802

Find the equation of the tangent line to f(t)=(e−t+t)2f(t)=\left(e^{-t}+\sqrt{t}\right)^{2}f(t)=(e−t+t​)2 at (1/2,f(1/2))(1/2, f(1/2))(1/2,f(1/2)), for t>0.1t>0.1t>0.1.

Problem 21803

Find where the curve 3x+3y2−y=103x + 3y^2 - y = 103x+3y2−y=10 has vertical and horizontal tangent lines.

Problem 21804

Find the limit as Δx→0\Delta x \rightarrow 0Δx→0 of ∑k=1n(xk∗)6Δxk\sum_{k=1}^{n}(x_{k}^{*})^{6} \Delta x_{k}∑k=1n​(xk∗​)6Δxk​ over the interval [3,9][3,9][3,9].

Problem 21805

Calculate the integral from 0 to 2 of 4−x2\sqrt{4-x^{2}}4−x2​ with respect to xxx.

Problem 21806

Problem 21807

Problem 21808

Calculate ∫05f(x)dx\int_{0}^{5} f(x) dx∫05​f(x)dx for the piecewise function f(x)={4if x≤43x−8if x>4f(x)=\begin{cases}4 & \text{if } x \leq 4 \\ 3x - 8 & \text{if } x > 4\end{cases}f(x)={43x−8​if x≤4if x>4​.

Problem 21809

Calculate the indefinite integral using substitution: ∫x6−x2 dx\int \frac{x}{6-x^{2}} \, dx∫6−x2x​dx. Don't forget absolute values!

Problem 21810

Find the derivative of g(t)=3tan⁡(t3)g(t) = 3 \tan(t^{3})g(t)=3tan(t3).

Problem 21811

Evaluate ∫72g(x)dx\int_{7}^{2} g(x) d x∫72​g(x)dx given ∫27g(x)dx=4\int_{2}^{7} g(x) d x=4∫27​g(x)dx=4.

Problem 21812

Calculate the average rate of change of g(x)=−5+3(x−1)g(x)=-5+3(x-1)g(x)=−5+3(x−1) from x=−2x=-2x=−2 to x=0x=0x=0.

Problem 21813

Problem 21814

Given ∫24f(x)dx=5\int_{2}^{4} f(x) d x=5∫24​f(x)dx=5, ∫27f(x)dx=−3\int_{2}^{7} f(x) d x=-3∫27​f(x)dx=−3, and ∫27g(x)dx=4\int_{2}^{7} g(x) d x=4∫27​g(x)dx=4, find ∫279g(x)dx=□\int_{2}^{7} 9 g(x) d x=\square∫27​9g(x)dx=□.

Problem 21815

Problem 21816

Find the biomass at t=13t=13t=13 hours if the average rate of change from t=3t=3t=3 to t=13t=13t=13 is 15mg/hour\frac{1}{5} \mathrm{mg}/\mathrm{hour}51​mg/hour and at t=3t=3t=3 is 13.

Problem 21817

Find ∫267f(x)dx\int_{2}^{6} 7 f(x) d x∫26​7f(x)dx given ∫27f(x)dx=11\int_{2}^{7} f(x) dx=11∫27​f(x)dx=11, ∫27g(x)dx=7\int_{2}^{7} g(x) dx=7∫27​g(x)dx=7, ∫67f(x)dx=6\int_{6}^{7} f(x) dx=6∫67​f(x)dx=6, and ∫26g(x)dx=3\int_{2}^{6} g(x) dx=3∫26​g(x)dx=3.

Problem 21818

Problem 21819

Evaluate ∫s∫F⃗⋅N⃗dS\int_{s} \int \vec{F} \cdot \vec{N} d S∫s​∫F⋅NdS for F⃗(x,y,z)=⟨x,y,z⟩\vec{F}(x, y, z)=\langle x, y, z\rangleF(x,y,z)=⟨x,y,z⟩ on S:x2+y2+z2=9S: x^{2}+y^{2}+z^{2}=9S:x2+y2+z2=9 in the first octant.

Problem 21820

Problem 21821

Find the derivative of the function f(x)=x+xf(x) = \sqrt{x + \sqrt{x}}f(x)=x+x​​.

Problem 21822

Find the value of ∫26(f(x)−g(x))dx\int_{2}^{6} (f(x) - g(x)) d x∫26​(f(x)−g(x))dx given ∫267f(x)dx=35\int_{2}^{6} 7 f(x) d x=35∫26​7f(x)dx=35 and ∫27(f(x)−g(x))dx=4\int_{2}^{7}(f(x)-g(x)) d x=4∫27​(f(x)−g(x))dx=4.

Problem 21823

Given functions fff and ggg on [2,7][2,7][2,7], find:a. ∫267f(x)dx\int_{2}^{6} 7 f(x) d x∫26​7f(x)dx b. ∫27(f(x)−g(x))dx\int_{2}^{7}(f(x)-g(x)) d x∫27​(f(x)−g(x))dx c. ∫26(f(x)−g(x))dx=□\int_{2}^{6}(f(x)-g(x)) d x=\square∫26​(f(x)−g(x))dx=□

Problem 21824

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=x2−2xf(x)=x^{2}-2xf(x)=x2−2x and evaluate it at x=3x=3x=3.

Problem 21825

Problem 21826

Find the derivative f′(2)f^{\prime}(2)f′(2) for the function f(x)=∣x−2∣f(x)=|x-2|f(x)=∣x−2∣.

Problem 21827

Calculate the indefinite integral of e−x2x dxe^{-x^{2}} x \, dxe−x2xdx.

Problem 21828

Calculate the integral using substitution: ∫(x6−9)7x5dx\int\left(x^{6}-9\right)^{7} x^{5} dx∫(x6−9)7x5dx.

Problem 21829

Find the tangent and normal line equations to x3+2xy−y2=11x^{3}+2xy-y^{2}=11x3+2xy−y2=11 at the point (2,3)(2,3)(2,3).

Problem 21830

Find the area AAA between the curves x=y2+2yx=y^{2}+2yx=y2+2y and x=y+12x=y+12x=y+12. Provide the answer in exact form.

Problem 21831

Calculate the integral of ex6x6e^{x^{6}} x^{6}ex6x6 with respect to xxx: ∫ex6x6dx\int e^{x^{6}} x^{6} d x∫ex6x6dx.

Problem 21832

Find the time xxx when the drug concentration K(x)=12xx2+25K(x)=\frac{12x}{x^2+25}K(x)=x2+2512x​ is at its maximum.

Problem 21833

Approximate the change in drug concentration C(x)=10x9+x2C(x)=\frac{10 x}{9+x^{2}}C(x)=9+x210x​ from x=0.5x=0.5x=0.5 to x=10x=10x=10.

Problem 21834

Bestimme den Grenzwert lim‾⁡n→∞n2−n\varlimsup_{n \to \infty} \sqrt{n^{2}-n}limn→∞​n2−n​ für die Folge cnc_{n}cn​.

Problem 21835

Find the instantaneous rate of change of height hhh at t=20 st=20 \mathrm{~s}t=20 s from h(t)=15cos⁡(πt120)+18h(t)=15 \cos \left(\frac{\pi t}{120}\right)+18h(t)=15cos(120πt​)+18. What does this rate represent?

Problem 21836

Find the monkey's acceleration at t=3t=3t=3 for the distance S(t)=tsin⁡(2t)+t3S(t)=t \sin (2 t)+t^{3}S(t)=tsin(2t)+t3. Round up the result.

Problem 21837

Find the critical points of the function y=f(x)=4x3−6x2−72x+3y=f(x)=4 x^{3}-6 x^{2}-72 x+3y=f(x)=4x3−6x2−72x+3.

Problem 21838

Problem 21839

Calculate the integral of (x2+3)95x dx(x^{2}+3)^{9} 5 x \, dx(x2+3)95xdx.

Problem 21840

Calculate the indefinite integral: ∫z3+83z2 dz\int \sqrt[3]{z^{3}+8} z^{2} \, dz∫3z3+8​z2dz.

Problem 21841

Problem 21842

Problem 21843

Find the tangent line equation for the function f(t)=(e−2t+t)2f(t)=\left(e^{-2 t}+\sqrt{t}\right)^{2}f(t)=(e−2t+t​)2 at (3,f(3))(3, f(3))(3,f(3)), t>0.1t>0.1t>0.1.

Problem 21844

Find the monkey's acceleration at t=3t=3t=3 for S(t)=tsin⁡(2t)+t3S(t)=t \sin (2 t)+t^{3}S(t)=tsin(2t)+t3. Round up your answer. a) 15 b) 20 c) 26 d) 27

Problem 21845

Evaluate the integral from -2 to 5 of the function $- |x|$ .

Find the equation of the tangent line to $f(t)=\left(e^{-t}+\sqrt{t}\right)^{2}$ at $(1/2, f(1/2))$ , for $t>0.1$ .

Find where the curve $3x + 3y^2 - y = 10$ has vertical and horizontal tangent lines.

Find the limit as $\Delta x \rightarrow 0$ of $\sum_{k=1}^{n}(x_{k}^{*})^{6} \Delta x_{k}$ over the interval $[3,9]$ .

Calculate the integral from 0 to 2 of $\sqrt{4-x^{2}}$ with respect to $x$ .

Calculate $\int_{0}^{5} f(x) dx$ for the piecewise function $f(x)=\begin{cases}4 & \text{if } x \leq 4 \\ 3x - 8 & \text{if } x > 4\end{cases}$ .

Calculate the indefinite integral using substitution: $\int \frac{x}{6-x^{2}} \, dx$ . Don't forget absolute values!

Find the derivative of $g(t) = 3 \tan(t^{3})$ .

Evaluate $\int_{7}^{2} g(x) d x$ given $\int_{2}^{7} g(x) d x=4$ .

Calculate the average rate of change of $g(x)=-5+3(x-1)$ from $x=-2$ to $x=0$ .

Given $\int_{2}^{4} f(x) d x=5$ , $\int_{2}^{7} f(x) d x=-3$ , and $\int_{2}^{7} g(x) d x=4$ , find $\int_{2}^{7} 9 g(x) d x=\square$ .

Find the biomass at $t=13$ hours if the average rate of change from $t=3$ to $t=13$ is $\frac{1}{5} \mathrm{mg}/\mathrm{hour}$ and at $t=3$ is 13.

Find $\int_{2}^{6} 7 f(x) d x$ given $\int_{2}^{7} f(x) dx=11$ , $\int_{2}^{7} g(x) dx=7$ , $\int_{6}^{7} f(x) dx=6$ , and $\int_{2}^{6} g(x) dx=3$ .

Evaluate $\int_{s} \int \vec{F} \cdot \vec{N} d S$ for $\vec{F}(x, y, z)=\langle x, y, z\rangle$ on $S: x^{2}+y^{2}+z^{2}=9$ in the first octant.

Find the derivative of the function $f(x) = \sqrt{x + \sqrt{x}}$ .

Find the value of $\int_{2}^{6} (f(x) - g(x)) d x$ given $\int_{2}^{6} 7 f(x) d x=35$ and $\int_{2}^{7}(f(x)-g(x)) d x=4$ .

Given functions $f$ and $g$ on $[2,7]$ , find:
a. $\int_{2}^{6} 7 f(x) d x$ b. $\int_{2}^{7}(f(x)-g(x)) d x$ c. $\int_{2}^{6}(f(x)-g(x)) d x=\square$

Find the derivative $f^{\prime}(x)$ of the function $f(x)=x^{2}-2x$ and evaluate it at $x=3$ .

Find the derivative $f^{\prime}(2)$ for the function $f(x)=|x-2|$ .

Calculate the indefinite integral of $e^{-x^{2}} x \, dx$ .

Calculate the integral using substitution: $\int\left(x^{6}-9\right)^{7} x^{5} dx$ .

Find the tangent and normal line equations to $x^{3}+2xy-y^{2}=11$ at the point $(2,3)$ .

Find the area $A$ between the curves $x=y^{2}+2y$ and $x=y+12$ . Provide the answer in exact form.

Calculate the integral of $e^{x^{6}} x^{6}$ with respect to $x$ : $\int e^{x^{6}} x^{6} d x$ .

Find the time $x$ when the drug concentration $K(x)=\frac{12x}{x^2+25}$ is at its maximum.

Approximate the change in drug concentration $C(x)=\frac{10 x}{9+x^{2}}$ from $x=0.5$ to $x=10$ .

Bestimme den Grenzwert $\varlimsup_{n \to \infty} \sqrt{n^{2}-n}$ für die Folge $c_{n}$ .

Find the instantaneous rate of change of height $h$ at $t=20 \mathrm{~s}$ from $h(t)=15 \cos \left(\frac{\pi t}{120}\right)+18$ . What does this rate represent?

Find the monkey's acceleration at $t=3$ for the distance $S(t)=t \sin (2 t)+t^{3}$ . Round up the result.

Find the critical points of the function $y=f(x)=4 x^{3}-6 x^{2}-72 x+3$ .

Calculate the integral of $(x^{2}+3)^{9} 5 x \, dx$ .

Calculate the indefinite integral: $\int \sqrt[3]{z^{3}+8} z^{2} \, dz$ .

Find the tangent line equation for the function $f(t)=\left(e^{-2 t}+\sqrt{t}\right)^{2}$ at $(3, f(3))$ , $t>0.1$ .

Find the monkey's acceleration at $t=3$ for $S(t)=t \sin (2 t)+t^{3}$ . Round up your answer. a) 15 b) 20 c) 26 d) 27

Calculate the indefinite integral using substitution: $\int \sqrt[8]{x^{8}+256} x^{6} d x$ .

Find the tangent line equation in terms of $x$ and $y$ for $r=3 \sin 4 \theta$ at $\theta=\pi / 3$ . $y=$

Calculate $14 \int_{0}^{2} 15 \sqrt{4 x+1} \, dx$ .

Approximate the integral $\int_{1}^{10}\left(x^{2}+4\right) d x$ using a Riemann sum with 3 subintervals of length 3.

Approximate the integral $\int_{2}^{11}(x^{3}+4) dx$ using a Riemann sum with 3 subintervals of length 3.

Differentiate and optimize the equation $5 x^{2}+4 x-16$ . Find $x=$ and state max or min value. Show workings to 1.d.p.

Differentiate and optimize $5 x^{2}+4 x-16$ . Find $x=$ and state max or min value, correct to 1.d.p.

Find the total reaction from $t=1$ to $t=10$ for $R^{\prime}(t)=\frac{6}{t}+\frac{4}{t^{3}}+e^{-2t}$ . Round up.

Evaluate the integral: $\int(\sin (x))^{20}(\cos (x)) d x$ .

Find the derivative of the function $f(x)=(x^{8}+6 x)^{5}$ . What is $\frac{d f(x)}{d x}$ ?

Find $\frac{dy}{dt}$ if $y = \left( \tan \left( x^{8} \right) + 6 \right)^{6}$ .

Evaluate the integral $\int_{0}^{1} x^{4} x^{x^{5}} d x$ .

Calculate the integral $\int_{0}^{1} x^{4} e^{x^{5}} d x$ .

Find $\frac{d y}{d t}$ for $y=\left(\tan \left(x^{8}\right)+6\right)^{6}$ . Choose the correct option.

Find the integral of the function: $\int(e^{3 t}+t+5) dt$ . Choose the correct answer from the options.

If $f$ is even and $\int_{0}^{2} f(x) dx=1$ , $\int_{0}^{6} f(x) dx=3$ , find $\int_{-2}^{6} f(x) dx$ . a) 6 b) 8 c) 10 d) 4

Find the total reaction from $t=1$ to $t=10$ for $R^{\prime}(t)=\frac{6}{t}+\frac{4}{t^{3}}+e^{-2t}$ . Round up.

Find the partial derivative $\frac{\partial}{\partial y} 200 x^{\frac{1}{5}} y^{\frac{4}{5}}$ .

Find the derivative of the function $\left(\frac{2 x^{3}-12 x+1}{1+x^{2}}\right)$ .

Find the area under the curve $y = 2x - x^2$ from $x = 0$ to $x = 2$ : $\int_{0}^{2} (2x - x^2) \, dx$ .

Evaluate the integral $\int_{1}^{e^{3}} \frac{(\ln x)^{5}}{x} dx$ .

Calculate the integral from -1 to 1 of $\frac{1}{4} x^{3}-\frac{2}{3} x$ .

Berechne die Ableitung $f^{\prime}(5.94)$ für die Funktion $f(x)=3.3 x^{4}-64 x^{2}+9.2 x+30$ und runde auf zwei Nachkommastellen.

Evaluate the integral $\int_{0}^{\pi / 3} \frac{5 \sin (5 t)}{4-\cos (5 t)} d t$ .

Find the area under the curve $y = \frac{3}{x^3}$ from $x=1$ to $x=3$ using the integral $\int_{1}^{3} \frac{3}{x^3} \, dx$ .

Gegeben ist die Funktion $f(x)=\frac{1}{3} x^{3}-2 x+3$ . Finde die Tangentengleichung bei $B(0 \mid f(0))$ und zeige, dass $f$ für $x>0$ linksgekrümmt ist.

Untersuchen Sie das Krümmungsverhalten der Funktion $f(x)=(2 x-3)^{3}-12 x^{2}$ .

Bestimme die Ableitung von $f(x)=3\left(x^{3}-1\right) \sqrt{x^{7}}$ bei $x_{0}=4.3$ . Ergebnis: $f^{\prime}(4.3)=$

Evaluate the integral $\int_{C} 2xyz \, ds$ for the curve $C: \vec{r}(t)=12t\vec{i}+5t\vec{j}+84t\vec{k}$ , $0 \leq t \leq 1$ .