Calculus

Problem 3201

Find the integral of $\frac{x}{x^{4}+9}$ with respect to $x$ .

See Solution

Problem 3202

Evaluate the integral: $\int \frac{\ln x}{x \sqrt{1+(\ln x)^{2}}} d x$

See Solution

Problem 3203

Evaluate the integral $\int(1+\tan x)^{2} \sec x \, dx$ .

See Solution

Problem 3204

Monopoly profit function: $\Pi=-Q^{3}+48 \times Q^{2}-180 \times Q-800$ . Which statement is true about it? A, B, C, D, or E?

See Solution

Problem 3205

Find the derivative $\frac{d y}{d x}$ of the function $y=\frac{3 x^{2}-9 x+13}{3 x+4}$ .

See Solution

Problem 3206

Find the derivative $\frac{d y}{d x}$ for $y = \frac{x^{2}+6 x-1}{x^{2}+3 x-1}$ . No need to expand.

See Solution

Problem 3207

Find the derivative of the function $s(x)=\frac{2}{x}$ without and with the product/quotient rule.

See Solution

Problem 3208

Evaluate the integral from 1 to 2 of $\frac{1}{4 x} \, dx$ .

See Solution

Problem 3209

Find the derivatives of $u(x)=\sin(x)$ , $v(x)=x^{11}$ , and $f(x)=\frac{u(x)}{v(x)}$ .

See Solution

Problem 3210

Find the derivative $f'(x)$ of $f(x)=\frac{3 \sin x}{1+\cos x}$ and calculate $f'(2)$ .

See Solution

Problem 3211

Find the derivative $f^{\prime}(\pi)$ for the function $f(x)=\frac{-8 x}{\sin x+\cos x}$ .

See Solution

Problem 3212

Find $h$ and the secant slope for points $(1.5,2)$ and $(1.5+h, f(1.5+h))$ using given values.

See Solution

Problem 3213

Calculate the Riemann sum for $f(x)=x^{2}$ on $[1,3]$ with 4 equal subdivisions.

See Solution

Problem 3214

Find the tangent line $y=m x+b$ to $f(x)=\frac{2 \sin x}{2 \sin x+6 \cos x}$ at $a=\frac{\pi}{3}$ . Determine $m$ and $b$ .

See Solution

Problem 3215

Find the average velocity of the particle on $[2, 2+h]$ using $\frac{s(2+h)-s(2)}{h}$ for $s(t)=6 t^{2}-10 t+1$ .

See Solution

Problem 3216

Find the integral: $\int \frac{1}{x-\sqrt{2 x}} d x$

See Solution

Problem 3217

Calculate the integral: $\int x \sqrt[3]{x-4} \, dx$

See Solution

Problem 3218

Find the integral: $\int x \sqrt[3]{x-4} \, dx$

See Solution

Problem 3219

Find the limit of the series $\sum_{r=0}^{\infty} \frac{1}{r !}$ . Show your calculations.

See Solution

Problem 3220

Calculate the balance after 1, 5, and 20 years for \$8000 at an APR of 3.75\%. Find the APY using continuous compounding.

See Solution

Problem 3221

Bestimme die Wendetangente am Wendepunkt $(0, 9)$ von $f(x)=\frac{1}{3} x^{4}-\frac{4}{3} x^{3}+9$ .

See Solution

Problem 3222

Finde die erste und zweite Ableitung von $f(x)=\frac{1}{6} x^{4}-9 x^{2}+2 x-3$ .

See Solution

Problem 3223

Find the limit: $\lim _{x \rightarrow \infty} \sqrt{x^{2}+x+3}-\sqrt{x^{2}-3 x+1}$ .

See Solution

Problem 3224

Ein Rollkörper schwingt auf zwei Bahnen.
a) Ordnen Sie die Graphen den Bahnen zu.
b) Erklären Sie die Diagrammänderung bei Reibung.
c) Berechnen Sie die Gesamtenergie und maximale Geschwindigkeit für $m=50 \mathrm{~kg}$ und Höhe $h=2 \mathrm{~m}$ . (Kontrollergebnis: $E_{\text {Gesamt }}=981$ )

See Solution

Problem 3225

Finde die erste und zweite Ableitung von $f(x) = \frac{1}{4} x^{4}-\frac{1}{2} x^{3}-x^{2}-17$ .

See Solution

Problem 3226

Find the roots of the derivative $f^{\prime}(x)=8 x^{3}-2 x$ .

See Solution

Problem 3227

Bestimmen Sie die Ableitung von $f(x) = 2x^4 - x^2 + 5$ .

See Solution

Problem 3228

Find the derivatives of these functions:
a) $f(x)=\frac{1}{3}(x+2)^{-2}$
b) $f(t)=\sqrt{\frac{2}{t}}$
c) $f(t)=\frac{1}{\sin (t)}$

See Solution

Problem 3229

Find a function $f(x)$ such that $\int_{-3}^{1} f(x) \, dx = 2$ .

See Solution

Problem 3230

Lösen Sie diese inhomogenen linearen Differentialgleichungen: a) $y^{\prime}+x \cdot y=4 x$ b) $x \cdot y^{\prime}+y=x \cdot \sin x$

See Solution

Problem 3231

Finden Sie eine nicht konstante Funktion $f(x)$ , für die gilt: $\int_{1}^{-1} f(x) dx = 4$ .

See Solution

Problem 3232

Für welchen Wert von $a$ hat die Funktion $f(x)=x^{3}$ im Intervall $[0 ; a]$ eine Änderungsrate von 16?

See Solution

Problem 3233

Bestimmen Sie die Steigung von $f_{a}$ an den Stellen $x_{0}$ für die Funktionen a) bis d).

See Solution

Problem 3234

Find the one-sided limit: $\lim _{x \rightarrow(1 / 2)^{+}} 13 x^{2} \tan \pi x$ .

See Solution

Problem 3235

Find the limit as $x$ approaches -1 for $\frac{2x^2 - 4x - 6}{x + 1}$ . If DNE, state that. Also, define $g(x)$ that matches it except at one point.

See Solution

Problem 3236

Find the value of $x$ where the functions are discontinuous and if it's a removable discontinuity: $7 f(x)=\frac{\sin 2 x}{x}$ , $8 f(x)=\frac{e^{x}-2}{x-1}$ .

See Solution

Problem 3237

Find the limit: $\lim _{x \rightarrow 6^{+}} \frac{|x-6|}{x-6}$ . If it doesn't exist, write DNE.

See Solution

Problem 3238

Find the limit $L = \lim_{x \rightarrow 2} (x + 5)$ and prove it using the $\varepsilon-\delta$ definition.

See Solution

Problem 3239

Find the derivative of $\cos^{3}(2x)$ and use it to evaluate $\int_{0}^{\frac{\pi}{4}} 2 \sin^{3}(2x) \, dx$ .

See Solution

Problem 3240

Leiten Sie die folgenden Funktionen ab: a) $f(x)=0,5 \cdot(6-x)$ , b) $f(x)=x \cdot(2-x)$ , c) $f(x)=(1-x)^{2}$ , d) $f(x)=(x-1)^{2}$ , e) $f(x)=(x+1) \cdot(x-3)$ , f) $f(x)=x \cdot(x+2) \cdot(x-2)$ .

See Solution

Problem 3241

Bestimme die Nullstellen von $f$ und den Gesamtinhalt der Fläche zwischen $f$ und der $x$ -Achse über den Intervallen: a) $f(x)=x^{3}+2 x^{2}-3 x, \quad I=[-2 ; 2,5]$ b) $f(x)=(x+2)(x-1)^{2}, I=[-2 ; 2]$ c) $f(x)=(x-1)(x+2)(x-3), I=[-1 ; 2]$ d) $f(x)=x^{4}+x^{2}-2, \quad I=[-2 ; 3]$

See Solution

Problem 3242

Is the function $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-49}{x-7} & \text { if } x \neq 7 \\ 11 & \text { if } x=7\end{array}\right.$ continuous at $x=7$ ?

See Solution

Problem 3243

A body is thrown up and reaches $48 \mathrm{ft}$ in 1 second. Find its maximum height with gravity at $32 \mathrm{ft/s^2}$ .

See Solution

Problem 3244

Check if the function $f(x)=\frac{x^{2}-49}{x-7}$ for $x \neq 7$ and $f(7)=11$ is continuous at $a=7$ .

See Solution

Problem 3245

Bestimme die Tangentengleichung $t$ an $f(x)$ im Punkt $P(2 \mid f(2))$ und finde die Nullstelle von $t$ . Berechne die Fläche zwischen $t$ und den Achsen.

See Solution

Problem 3246

Is the function $y=\frac{6 x-5}{x^{2}-11 x+30}$ continuous at $a=5$ ?

See Solution

Problem 3247

Check if the Intermediate Value Theorem applies to $f(x)=\frac{x^{2}+x}{x-1}$ on $\left[\frac{5}{2}, 4\right]$ and find $c$ where $f(c)=6$ .

See Solution

Problem 3248

Bestimme den Wert von $a$ für die Funktion $f(x)=a x^{2}$ , so dass: a) $f'(3)=1$ und b) $\frac{f(5)-f(1)}{5-1}=3$ .

See Solution

Problem 3249

Determine if $f$ is continuous, analyze continuity at 4, and find intervals of continuity for $f(x)=\begin{cases} x^{2}+2x & \text{if } x \geq 4 \\ 5x & \text{if } x < 4 \end{cases}$ .

See Solution

Problem 3250

Check if $f$ is continuous, if it's left/right continuous at 4, and find intervals of continuity for: $f(x)=\left\{\begin{array}{ll} x^{2}+2 x & \text { if } x \geq 4 \\ 5 x & \text { if } x<4 \end{array}\right.$

See Solution

Problem 3251

Analyze the continuity of the function $f(x)=\frac{1}{2}[[x]]+x+1$ . Where is $f$ continuous or discontinuous?

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Problem 3252

Untersuchen Sie den Graphen von $f(x)=x^{3}-6 x^{2}+12 x+2$ auf Extrempunkte.

See Solution

Problem 3253

Bestimme die Uhrzeit, zu der die Besucherzahl $f(x)=-\frac{1}{15} x^{3}+\frac{2}{5} x^{2}+\frac{1}{15}$ maximal ist.

See Solution

Problem 3254

Find the limits and roots of the function $f(x)=\frac{x}{2x^{2}+1}$ .

See Solution

Problem 3255

Find the first derivative of $f(x)=\sqrt{3} x^{2}-\frac{3}{x}+7 x+1$ .

See Solution

Problem 3256

Calculate the integral $\int_{1}^{16}(x-5) x^{-1 / 2} d x$ .

See Solution

Problem 3257

Analyze the function $f(x)=\frac{1}{(x-2)^{2}}$ and find the limits as $x$ approaches 2 from the left and right.

See Solution

Problem 3258

Find the position $\vec{r}(t)$ given $\vec{v}(t)=\langle e^{-t}, 0, \sin t\rangle$ and $\vec{r}(0)=\langle 0,2,3\rangle$ . Calculate distance from $t=0$ to $t=5$ .

See Solution

Problem 3259

Find the limit of the function $f(x)=4 x^{2}-8 x$ as $\Delta x$ approaches 0: $\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$

See Solution

Problem 3260

Find the limit: $\lim _{x \rightarrow 1} f(x)$ for the piecewise function $f(x)=\left\{\begin{array}{ll} x^{2}-1 & \text { if } x \neq 1 \\ -2 & \text { if } x=1 \end{array}\right.$ .

See Solution

Problem 3261

Zeichnen Sie den Graphen einer Funktion $f$ mit negativer Steigung. Bestimmen Sie die Ableitung von $f(x) = x$ .

See Solution

Problem 3262

Find the limit: $\lim _{\theta \rightarrow 0} \frac{\cos (7 \theta) \tan (7 \theta)}{\theta}$ . If none, enter DNE.

See Solution

Problem 3263

Solve the differential equation: $(x+2) dx + (x+y-1) dy = 0$ .

See Solution

Problem 3264

Solve the differential equation $(x+2) dx+(x+y-1) dy=0$ for the general solution.

See Solution

Problem 3265

Solve the differential equation: $(x+2) dx + (x+y-1) dy = 0$ .

See Solution

Problem 3266

Find the limit $L$ and use the $\varepsilon-\delta$ definition to prove it for $\lim _{x \rightarrow 25} \sqrt{x}$ .

See Solution

Problem 3267

Find the limit $\lim _{x \rightarrow 2} \frac{|x-2|}{x-2}$ using a graph of two lines at (2,1) and (2,-1). If DNE, state that.

See Solution

Problem 3268

How many days for a 60g sample of ${ }^{65} \mathrm{Ni}$ (half-life 2.5 days) to decay to under 1g? Report whole number.

See Solution

Problem 3269

Bestimme die Ableitung von $g(x)=\frac{x^{2}+2x}{3x-4}$ und finde die Tangente bei $P(3, g(3))$ . Untersuche auch auf Asymptoten.

See Solution

Problem 3270

Bestimmen Sie die erste Ableitung von $g(x)=\frac{x^{2}+2x}{3x-4}$ und die Tangenten am Punkt $\mathrm{P}(3, g(3))$ . Untersuchen Sie auch Asymptoten.

See Solution

Problem 3271

Find the limit: $\lim _{x \rightarrow \pi / 2} 2 \tan x$ . If it doesn't exist, write DNE.

See Solution

Problem 3272

Bestimme die Zeit-Ort-Funktion $s(t)$ für die Geschwindigkeit $v(t)$ und den Abstand $a$ zum Start. a) $v(t)=10 t ; a=12$ b) $v(t)=0,4 t^{2}+3 t ; a=15$ c) $v(t)=0,7 t^{2} ; a=5$

See Solution

Problem 3273

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

See Solution

Problem 3274

Calculate the integral from 2 to 5 of $\frac{1}{x^{2}}$ with respect to $x$ .

See Solution

Problem 3275

Calculate the integral $\int_{2}^{3}\left(4 x^{3}-x+2\right) d x$ .

See Solution

Problem 3276

Calculate the integral: $\int_{1}^{4}\left(x^{2}+\frac{1}{x^{3}}\right) dx$

See Solution

Problem 3277

Evaluate the integral: $\int_{1}^{2}\left(\frac{1}{\sqrt{x}}-x\right) d x$

See Solution

Problem 3278

Finde die Stellen, an denen $f(x)=\frac{1}{4} x^{4}-6 x$ die Steigung $m=2$ hat.

See Solution

Problem 3279

Solve the initial value problem using Laplace transforms: $y'' - 4y' + 20y = 85e^{3t}$ , $y(0)=5$ , $y'(0)=19$ . Find $y(t)=$ (exact answer in terms of $e$ ).

See Solution

Problem 3280

Bestimme die Stellen, an denen $f$ die Steigung $m$ hat: a) $f(x)=\frac{1}{4} x^{4}-6 x, m=2$ ; b) $f(x)=-\frac{1}{6} x^{3}+x^{2}, m=-2,5$ .

See Solution

Problem 3281

Solve the initial value problem using Laplace transforms: $y'' - 4y' + 20y = 85e^{3t}$ , $y(0)=5$ , $y'(0)=19$ . Find $y(t)=$ (exact answer in terms of $e$ ).

See Solution

Problem 3282

Bestimme die Stellen, an denen die Funktion $f$ die gegebene Steigung $m$ hat. a) $f(x)=\frac{1}{4} x^{4}-6 x, m=2$ b) $f(x)=-\frac{1}{6} x^{3}+x^{2}, m=-2,5$ c) $f(x)=\frac{2}{x}-x, m=-3$ d) $f(x)=3 \sqrt{x}, m=3$

See Solution

Problem 3283

Find the derivative of $2 e^{-5 x^{3}+6 x^{9}}$ .

See Solution

Problem 3284

Find the derivative of $8 \sqrt{6 x^{6}+10 x^{3}}$ using the chain rule, without fractions or negatives.

See Solution

Problem 3285

Find the derivative of $f(x)=sqrt(\frac{x^{2}+100}{x^{2}+64})$ using the chain rule and simplify your answer. $f^{\prime}(x)=$

See Solution

Problem 3286

Find the derivative of $f(x) = (4x^{2} - 2)^{4}(5x^{2} + 8)^{15}$ . Use the product and chain rules to compute $f'(x)$ .

See Solution

Problem 3287

Bestimme die höchste und niedrigste Temperatur zwischen 7 und 18 Uhr für die Funktion $f(t)=-0,04 t^{3}+1,31 t^{2}-12,3 t+38,4$ .

See Solution

Problem 3288

Find the derivative of $f(x)=4 e^{4 x^{8}-8 x^{9}}$ using the chain rule. What is $f^{\prime}(x)=?$

See Solution

Problem 3289

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

See Solution

Problem 3290

Bestimme die Tangentengleichung $t$ an die Exponentialfunktion im Punkt $P(2 \mid f(2))$ und finde die Nullstelle. Berechne die Fläche zwischen $t$ und den Achsen.

See Solution

Problem 3291

Determine the long run behavior of the functions:
1. $\frac{x^{3}+1}{x^{2}+2}$
2. $\frac{x^{2}+1}{x^{2}+2}$
3. $\frac{x^{2}+1}{x^{3}+2}$

See Solution

Problem 3292

Find $\frac{d y}{d x}$ for the equation $4 x^{3}+4 y^{3}-2 x y=0$ .

See Solution

Problem 3293

Find $\frac{d y}{d x}$ for the equation $(2 x-y)^{2}-5 y=2$ .

See Solution

Problem 3294

Find $\frac{d^{2} y}{d x^{2}}$ at (4,5) for the curve defined by $x^{2}+y^{2}=41$ .

See Solution

Problem 3295

Find the slope of the tangent line to $x^{2}+9xy^{2}+3y^{3}=25$ at the point $(2,1)$ .

See Solution

Problem 3296

Find points where the tangent line to $4 x^{2}+6 y^{2}=5$ is horizontal using implicit differentiation.

See Solution

Problem 3297

Explain why the limit does not exist for $\lim _{x \rightarrow 0} \frac{x}{|x|}$ . Choose A or B and fill in the boxes.

See Solution

Problem 3298

Find the limit as $x$ approaches 6 for the expression $-x^{2}+8x-7$ . What is the value? A. Answer = $\square$ , B. Limit does not exist.

See Solution

Problem 3299

Find points where the tangent line to $xy + 5y^2 = -9$ is vertical using implicit differentiation.

See Solution

Problem 3300

Find the limit as $x$ approaches 1 for $3x^3 - 4x^2 + 5x + 6$ . What is the result? A. $\square$ B. Limit does not exist.

See Solution

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Calculus

Problem 3201

Find the integral of xx4+9\frac{x}{x^{4}+9}x4+9x​ with respect to xxx.

Problem 3202

Evaluate the integral: ∫ln⁡xx1+(ln⁡x)2dx\int \frac{\ln x}{x \sqrt{1+(\ln x)^{2}}} d x∫x1+(lnx)2​lnx​dx

Problem 3203

Evaluate the integral ∫(1+tan⁡x)2sec⁡x dx\int(1+\tan x)^{2} \sec x \, dx∫(1+tanx)2secxdx.

Problem 3204

Monopoly profit function: Π=−Q3+48×Q2−180×Q−800\Pi=-Q^{3}+48 \times Q^{2}-180 \times Q-800Π=−Q3+48×Q2−180×Q−800. Which statement is true about it? A, B, C, D, or E?

Problem 3205

Find the derivative dydx\frac{d y}{d x}dxdy​ of the function y=3x2−9x+133x+4y=\frac{3 x^{2}-9 x+13}{3 x+4}y=3x+43x2−9x+13​.

Problem 3206

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=x2+6x−1x2+3x−1y = \frac{x^{2}+6 x-1}{x^{2}+3 x-1}y=x2+3x−1x2+6x−1​. No need to expand.

Problem 3207

Find the derivative of the function s(x)=2xs(x)=\frac{2}{x}s(x)=x2​ without and with the product/quotient rule.

Problem 3208

Evaluate the integral from 1 to 2 of 14x dx\frac{1}{4 x} \, dx4x1​dx.

Problem 3209

Find the derivatives of u(x)=sin⁡(x)u(x)=\sin(x)u(x)=sin(x), v(x)=x11v(x)=x^{11}v(x)=x11, and f(x)=u(x)v(x)f(x)=\frac{u(x)}{v(x)}f(x)=v(x)u(x)​.

Problem 3210

Find the derivative f′(x)f'(x)f′(x) of f(x)=3sin⁡x1+cos⁡xf(x)=\frac{3 \sin x}{1+\cos x}f(x)=1+cosx3sinx​ and calculate f′(2)f'(2)f′(2).

Problem 3211

Find the derivative f′(π)f^{\prime}(\pi)f′(π) for the function f(x)=−8xsin⁡x+cos⁡xf(x)=\frac{-8 x}{\sin x+\cos x}f(x)=sinx+cosx−8x​.

Problem 3212

Find h h h and the secant slope for points (1.5,2) (1.5,2) (1.5,2) and (1.5+h,f(1.5+h)) (1.5+h, f(1.5+h)) (1.5+h,f(1.5+h)) using given values.

Problem 3213

Calculate the Riemann sum for f(x)=x2f(x)=x^{2}f(x)=x2 on [1,3][1,3][1,3] with 4 equal subdivisions.

Problem 3214

Find the tangent line y=mx+by=m x+by=mx+b to f(x)=2sin⁡x2sin⁡x+6cos⁡xf(x)=\frac{2 \sin x}{2 \sin x+6 \cos x}f(x)=2sinx+6cosx2sinx​ at a=π3a=\frac{\pi}{3}a=3π​. Determine mmm and bbb.

Problem 3215

Find the average velocity of the particle on [2,2+h][2, 2+h][2,2+h] using s(2+h)−s(2)h\frac{s(2+h)-s(2)}{h}hs(2+h)−s(2)​ for s(t)=6t2−10t+1s(t)=6 t^{2}-10 t+1s(t)=6t2−10t+1.

Problem 3216

Find the integral: ∫1x−2xdx\int \frac{1}{x-\sqrt{2 x}} d x∫x−2x​1​dx

Problem 3217

Calculate the integral: ∫xx−43 dx\int x \sqrt[3]{x-4} \, dx∫x3x−4​dx

Problem 3218

Find the integral: ∫xx−43 dx\int x \sqrt[3]{x-4} \, dx∫x3x−4​dx

Problem 3219

Find the limit of the series ∑r=0∞1r!\sum_{r=0}^{\infty} \frac{1}{r !}∑r=0∞​r!1​. Show your calculations.

Problem 3220

Calculate the balance after 1, 5, and 20 years for \$8000 at an APR of 3.75\%. Find the APY using continuous compounding.

Problem 3221

Bestimme die Wendetangente am Wendepunkt (0,9)(0, 9)(0,9) von f(x)=13x4−43x3+9f(x)=\frac{1}{3} x^{4}-\frac{4}{3} x^{3}+9f(x)=31​x4−34​x3+9.

Problem 3222

Finde die erste und zweite Ableitung von f(x)=16x4−9x2+2x−3f(x)=\frac{1}{6} x^{4}-9 x^{2}+2 x-3f(x)=61​x4−9x2+2x−3.

Problem 3223

Find the limit: lim⁡x→∞x2+x+3−x2−3x+1\lim _{x \rightarrow \infty} \sqrt{x^{2}+x+3}-\sqrt{x^{2}-3 x+1}limx→∞​x2+x+3​−x2−3x+1​.

Problem 3224

Problem 3225

Finde die erste und zweite Ableitung von f(x)=14x4−12x3−x2−17f(x) = \frac{1}{4} x^{4}-\frac{1}{2} x^{3}-x^{2}-17f(x)=41​x4−21​x3−x2−17.

Problem 3226

Find the roots of the derivative f′(x)=8x3−2xf^{\prime}(x)=8 x^{3}-2 xf′(x)=8x3−2x.

Problem 3227

Bestimmen Sie die Ableitung von f(x)=2x4−x2+5f(x) = 2x^4 - x^2 + 5f(x)=2x4−x2+5.

Problem 3228

Find the derivatives of these functions: a) f(x)=13(x+2)−2f(x)=\frac{1}{3}(x+2)^{-2}f(x)=31​(x+2)−2 b) f(t)=2tf(t)=\sqrt{\frac{2}{t}}f(t)=t2​​ c) f(t)=1sin⁡(t)f(t)=\frac{1}{\sin (t)}f(t)=sin(t)1​

Problem 3229

Find a function f(x)f(x)f(x) such that ∫−31f(x) dx=2\int_{-3}^{1} f(x) \, dx = 2∫−31​f(x)dx=2.

Problem 3230

Lösen Sie diese inhomogenen linearen Differentialgleichungen: a) y′+x⋅y=4xy^{\prime}+x \cdot y=4 xy′+x⋅y=4x b) x⋅y′+y=x⋅sin⁡xx \cdot y^{\prime}+y=x \cdot \sin xx⋅y′+y=x⋅sinx

Problem 3231

Finden Sie eine nicht konstante Funktion f(x)f(x)f(x), für die gilt: ∫1−1f(x)dx=4\int_{1}^{-1} f(x) dx = 4∫1−1​f(x)dx=4.

Problem 3232

Für welchen Wert von aaa hat die Funktion f(x)=x3f(x)=x^{3}f(x)=x3 im Intervall [0;a][0 ; a][0;a] eine Änderungsrate von 16?

Problem 3233

Bestimmen Sie die Steigung von faf_{a}fa​ an den Stellen x0x_{0}x0​ für die Funktionen a) bis d).

Problem 3234

Find the one-sided limit: lim⁡x→(1/2)+13x2tan⁡πx\lim _{x \rightarrow(1 / 2)^{+}} 13 x^{2} \tan \pi xlimx→(1/2)+​13x2tanπx.

Problem 3235

Find the limit as xxx approaches -1 for 2x2−4x−6x+1\frac{2x^2 - 4x - 6}{x + 1}x+12x2−4x−6​. If DNE, state that. Also, define g(x)g(x)g(x) that matches it except at one point.

Problem 3236

Find the value of xxx where the functions are discontinuous and if it's a removable discontinuity: 7f(x)=sin⁡2xx7 f(x)=\frac{\sin 2 x}{x}7f(x)=xsin2x​, 8f(x)=ex−2x−18 f(x)=\frac{e^{x}-2}{x-1}8f(x)=x−1ex−2​.

Problem 3237

Find the limit: lim⁡x→6+∣x−6∣x−6\lim _{x \rightarrow 6^{+}} \frac{|x-6|}{x-6}x→6+lim​x−6∣x−6∣​. If it doesn't exist, write DNE.

Problem 3238

Find the limit L=lim⁡x→2(x+5)L = \lim_{x \rightarrow 2} (x + 5)L=limx→2​(x+5) and prove it using the ε−δ\varepsilon-\deltaε−δ definition.

Problem 3239

Find the derivative of cos⁡3(2x)\cos^{3}(2x)cos3(2x) and use it to evaluate ∫0π42sin⁡3(2x) dx\int_{0}^{\frac{\pi}{4}} 2 \sin^{3}(2x) \, dx∫04π​​2sin3(2x)dx.

Problem 3240

Problem 3241

Find the integral of $\frac{x}{x^{4}+9}$ with respect to $x$ .

Evaluate the integral: $\int \frac{\ln x}{x \sqrt{1+(\ln x)^{2}}} d x$

Evaluate the integral $\int(1+\tan x)^{2} \sec x \, dx$ .

Monopoly profit function: $\Pi=-Q^{3}+48 \times Q^{2}-180 \times Q-800$ . Which statement is true about it? A, B, C, D, or E?

Find the derivative $\frac{d y}{d x}$ of the function $y=\frac{3 x^{2}-9 x+13}{3 x+4}$ .

Find the derivative $\frac{d y}{d x}$ for $y = \frac{x^{2}+6 x-1}{x^{2}+3 x-1}$ . No need to expand.

Find the derivative of the function $s(x)=\frac{2}{x}$ without and with the product/quotient rule.

Evaluate the integral from 1 to 2 of $\frac{1}{4 x} \, dx$ .

Find the derivatives of $u(x)=\sin(x)$ , $v(x)=x^{11}$ , and $f(x)=\frac{u(x)}{v(x)}$ .

Find the derivative $f'(x)$ of $f(x)=\frac{3 \sin x}{1+\cos x}$ and calculate $f'(2)$ .

Find the derivative $f^{\prime}(\pi)$ for the function $f(x)=\frac{-8 x}{\sin x+\cos x}$ .

Find $h$ and the secant slope for points $(1.5,2)$ and $(1.5+h, f(1.5+h))$ using given values.

Calculate the Riemann sum for $f(x)=x^{2}$ on $[1,3]$ with 4 equal subdivisions.

Find the tangent line $y=m x+b$ to $f(x)=\frac{2 \sin x}{2 \sin x+6 \cos x}$ at $a=\frac{\pi}{3}$ . Determine $m$ and $b$ .

Find the average velocity of the particle on $[2, 2+h]$ using $\frac{s(2+h)-s(2)}{h}$ for $s(t)=6 t^{2}-10 t+1$ .

Find the integral: $\int \frac{1}{x-\sqrt{2 x}} d x$

Calculate the integral: $\int x \sqrt[3]{x-4} \, dx$

Find the integral: $\int x \sqrt[3]{x-4} \, dx$

Find the limit of the series $\sum_{r=0}^{\infty} \frac{1}{r !}$ . Show your calculations.

Bestimme die Wendetangente am Wendepunkt $(0, 9)$ von $f(x)=\frac{1}{3} x^{4}-\frac{4}{3} x^{3}+9$ .

Finde die erste und zweite Ableitung von $f(x)=\frac{1}{6} x^{4}-9 x^{2}+2 x-3$ .

Find the limit: $\lim _{x \rightarrow \infty} \sqrt{x^{2}+x+3}-\sqrt{x^{2}-3 x+1}$ .

Finde die erste und zweite Ableitung von $f(x) = \frac{1}{4} x^{4}-\frac{1}{2} x^{3}-x^{2}-17$ .

Find the roots of the derivative $f^{\prime}(x)=8 x^{3}-2 x$ .

Bestimmen Sie die Ableitung von $f(x) = 2x^4 - x^2 + 5$ .

Find the derivatives of these functions:
a) $f(x)=\frac{1}{3}(x+2)^{-2}$
b) $f(t)=\sqrt{\frac{2}{t}}$
c) $f(t)=\frac{1}{\sin (t)}$

Find a function $f(x)$ such that $\int_{-3}^{1} f(x) \, dx = 2$ .

Lösen Sie diese inhomogenen linearen Differentialgleichungen: a) $y^{\prime}+x \cdot y=4 x$ b) $x \cdot y^{\prime}+y=x \cdot \sin x$

Finden Sie eine nicht konstante Funktion $f(x)$ , für die gilt: $\int_{1}^{-1} f(x) dx = 4$ .

Für welchen Wert von $a$ hat die Funktion $f(x)=x^{3}$ im Intervall $[0 ; a]$ eine Änderungsrate von 16?

Bestimmen Sie die Steigung von $f_{a}$ an den Stellen $x_{0}$ für die Funktionen a) bis d).

Find the one-sided limit: $\lim _{x \rightarrow(1 / 2)^{+}} 13 x^{2} \tan \pi x$ .

Find the limit as $x$ approaches -1 for $\frac{2x^2 - 4x - 6}{x + 1}$ . If DNE, state that. Also, define $g(x)$ that matches it except at one point.

Find the value of $x$ where the functions are discontinuous and if it's a removable discontinuity: $7 f(x)=\frac{\sin 2 x}{x}$ , $8 f(x)=\frac{e^{x}-2}{x-1}$ .

Find the limit: $\lim _{x \rightarrow 6^{+}} \frac{|x-6|}{x-6}$ . If it doesn't exist, write DNE.

Find the limit $L = \lim_{x \rightarrow 2} (x + 5)$ and prove it using the $\varepsilon-\delta$ definition.

Find the derivative of $\cos^{3}(2x)$ and use it to evaluate $\int_{0}^{\frac{\pi}{4}} 2 \sin^{3}(2x) \, dx$ .

Is the function $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-49}{x-7} & \text { if } x \neq 7 \\ 11 & \text { if } x=7\end{array}\right.$ continuous at $x=7$ ?

A body is thrown up and reaches $48 \mathrm{ft}$ in 1 second. Find its maximum height with gravity at $32 \mathrm{ft/s^2}$ .

Check if the function $f(x)=\frac{x^{2}-49}{x-7}$ for $x \neq 7$ and $f(7)=11$ is continuous at $a=7$ .

Bestimme die Tangentengleichung $t$ an $f(x)$ im Punkt $P(2 \mid f(2))$ und finde die Nullstelle von $t$ . Berechne die Fläche zwischen $t$ und den Achsen.

Is the function $y=\frac{6 x-5}{x^{2}-11 x+30}$ continuous at $a=5$ ?

Check if the Intermediate Value Theorem applies to $f(x)=\frac{x^{2}+x}{x-1}$ on $\left[\frac{5}{2}, 4\right]$ and find $c$ where $f(c)=6$ .

Bestimme den Wert von $a$ für die Funktion $f(x)=a x^{2}$ , so dass: a) $f'(3)=1$ und b) $\frac{f(5)-f(1)}{5-1}=3$ .

Determine if $f$ is continuous, analyze continuity at 4, and find intervals of continuity for $f(x)=\begin{cases} x^{2}+2x & \text{if } x \geq 4 \\ 5x & \text{if } x < 4 \end{cases}$ .

Check if $f$ is continuous, if it's left/right continuous at 4, and find intervals of continuity for: $f(x)=\left\{\begin{array}{ll} x^{2}+2 x & \text { if } x \geq 4 \\ 5 x & \text { if } x<4 \end{array}\right.$

Analyze the continuity of the function $f(x)=\frac{1}{2}[[x]]+x+1$ . Where is $f$ continuous or discontinuous?

Untersuchen Sie den Graphen von $f(x)=x^{3}-6 x^{2}+12 x+2$ auf Extrempunkte.

Bestimme die Uhrzeit, zu der die Besucherzahl $f(x)=-\frac{1}{15} x^{3}+\frac{2}{5} x^{2}+\frac{1}{15}$ maximal ist.

Find the limits and roots of the function $f(x)=\frac{x}{2x^{2}+1}$ .

Find the first derivative of $f(x)=\sqrt{3} x^{2}-\frac{3}{x}+7 x+1$ .

Calculate the integral $\int_{1}^{16}(x-5) x^{-1 / 2} d x$ .

Analyze the function $f(x)=\frac{1}{(x-2)^{2}}$ and find the limits as $x$ approaches 2 from the left and right.

Find the position $\vec{r}(t)$ given $\vec{v}(t)=\langle e^{-t}, 0, \sin t\rangle$ and $\vec{r}(0)=\langle 0,2,3\rangle$ . Calculate distance from $t=0$ to $t=5$ .

Find the limit of the function $f(x)=4 x^{2}-8 x$ as $\Delta x$ approaches 0: $\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$

Find the limit: $\lim _{x \rightarrow 1} f(x)$ for the piecewise function $f(x)=\left\{\begin{array}{ll} x^{2}-1 & \text { if } x \neq 1 \\ -2 & \text { if } x=1 \end{array}\right.$ .

Zeichnen Sie den Graphen einer Funktion $f$ mit negativer Steigung. Bestimmen Sie die Ableitung von $f(x) = x$ .

Find the limit: $\lim _{\theta \rightarrow 0} \frac{\cos (7 \theta) \tan (7 \theta)}{\theta}$ . If none, enter DNE.

Solve the differential equation: $(x+2) dx + (x+y-1) dy = 0$ .

Solve the differential equation $(x+2) dx+(x+y-1) dy=0$ for the general solution.

Solve the differential equation: $(x+2) dx + (x+y-1) dy = 0$ .

Find the limit $L$ and use the $\varepsilon-\delta$ definition to prove it for $\lim _{x \rightarrow 25} \sqrt{x}$ .

Find the limit $\lim _{x \rightarrow 2} \frac{|x-2|}{x-2}$ using a graph of two lines at (2,1) and (2,-1). If DNE, state that.

How many days for a 60g sample of ${ }^{65} \mathrm{Ni}$ (half-life 2.5 days) to decay to under 1g? Report whole number.

Bestimme die Ableitung von $g(x)=\frac{x^{2}+2x}{3x-4}$ und finde die Tangente bei $P(3, g(3))$ . Untersuche auch auf Asymptoten.

Bestimmen Sie die erste Ableitung von $g(x)=\frac{x^{2}+2x}{3x-4}$ und die Tangenten am Punkt $\mathrm{P}(3, g(3))$ . Untersuchen Sie auch Asymptoten.

Find the limit: $\lim _{x \rightarrow \pi / 2} 2 \tan x$ . If it doesn't exist, write DNE.

Bestimme die Zeit-Ort-Funktion $s(t)$ für die Geschwindigkeit $v(t)$ und den Abstand $a$ zum Start. a) $v(t)=10 t ; a=12$ b) $v(t)=0,4 t^{2}+3 t ; a=15$ c) $v(t)=0,7 t^{2} ; a=5$

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

Calculate the integral from 2 to 5 of $\frac{1}{x^{2}}$ with respect to $x$ .