Calculus

Problem 7501

Untersuche die Funktion $f(x)=(x-4) \cdot \sqrt{x}$ für $x \geq 0$ auf Monotonieverhalten und finde die innere Extremstelle.

See Solution

Problem 7502

Find the rate of change of the revenue function $R(Q)=100 Q^{2}+500$ at production level $Q_{0}$ : $\Delta R(Q_{0}) / \Delta Q$ .

See Solution

Problem 7503

Untersuchen Sie das Krümmungsverhalten der folgenden Funktionen: a) $f(x)=-x^{2}+2 x+4$ b) $f(x)=x^{3}-x$ c) $f(x)=x^{3}-3 x^{2}-9 x-5$ d) $f(x)=x^{4}+x^{2}$ e) $f(x)=x^{4}-6 x^{2}$ f) $f(x)=\frac{1}{4} x^{4}+3 x^{2}-2$ g) $f(x)=\frac{1}{3} x^{6}-20 x^{2}$ h) $f(x)=\frac{1}{20} x^{5}+\frac{1}{2} x^{4}+\frac{3}{2} x^{3}$ i) $f(x)=(x+2)^{2} \cdot(x-1)^{2}-3$

See Solution

Problem 7504

Evaluate the integral $\int(3 x^{2}+6 x+5) \tan^{-1} x \, dx$ .

See Solution

Problem 7505

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

See Solution

Problem 7506

Prove that $\int_{1}^{e} \frac{d x}{x(\ln x)^{1 / 3}}=\frac{3}{2}$ .

See Solution

Problem 7507

Evaluate the integral from 0 to 9: $\int_{0}^{9} x^{\frac{1}{2}} d x$ .

See Solution

Problem 7508

Evaluate the integral: $\int_{-\infty}^{\infty} \frac{1}{x^{2}+2 x+5} \, dx$

See Solution

Problem 7509

Calculate the integral $\int_{0}^{1} \sqrt{x} \, dx$ .

See Solution

Problem 7510

Calculate the integral from -1 to 0 of $e^{x}$ , i.e., $\int_{-1}^{0} e^{x} d x$ .

See Solution

Problem 7511

Berechne $\int_{-2}^{3}\left(4 x^{2}-3 x+5\right) d x + \int_{-2}^{3}(3 x-5) d x$ .

See Solution

Problem 7512

Choose negative values for $a$ and $d$ ( $a < d < 0$ ) and observe which axis solution trajectories approach as they stabilize at the origin. Prove your observation.

See Solution

Problem 7513

Show that $\int(3 x^{2}+6 x+5) \tan^{-1} x \, dx = (x^{3}+3 x^{2}+5 x) \tan^{-1} x - \frac{x^{2}}{2}-3 x-2 \ln(x^{2}+1)+3 \tan^{-1} x + C$ using integration by parts.

See Solution

Problem 7514

Berechne die bestimmten Integrale und deute die Ergebnisse am Graphen von $f$ : a) $\int_{2}^{2} x^{4} d x$ , b) $\int_{-2}^{2} 6 x^{5} d x$ , c) $\int_{-3}^{4} d x$ , d) $\int_{-3}^{4} 4 \sqrt{2} d x$ , e) $\int_{-3}^{4} 0 d x$ , f) $\int_{1}^{2} \frac{1}{x^{2}} d x$ , g) $\int_{-2}^{-1} \frac{1}{x^{2}} d x$ , h) $\int_{1}^{2} \frac{1}{x} d x$ , i) $\int_{0.5}^{1} \frac{1}{x} d x$ , k) $\int_{-2}^{-1} \frac{1}{x} d x$ , l) $\int_{0}^{9} \sqrt{x} d x$ , m) $\int_{1}^{9} \frac{1}{\sqrt{x}} d x$ , n) $\int_{0}^{\theta} x \sqrt{x} d x$ , o) $\int_{0}^{\pi / 2} \sin x d x$ , p) $\int_{0}^{\pi / 2} \cos x d x$ , q) $\int_{1}^{2} \ln x d x$ , r) $\int_{0.5}^{1} \ln x d x$ , s) $\int_{0.5}^{2} \ln x d x$ , t) $\int_{-1}^{0} e^{x} d x$ , u) $\int_{-1000}^{0} e^{x} d x$ .

See Solution

Problem 7515

Bestimmen Sie den Wert von $k \in \mathbb{R}^{+}$ für a) $\int_{-k}^{k} 3 x^{2} d x=16$ und b) $\int_{-2}^{4} k x d x=18$ .

See Solution

Problem 7516

Berechne die Integrale und erläutere die Ergebnisse am Graphen von $f$ : a) $\int_{2}^{2} x^{4} d x$ , b) $\int_{-2}^{2} 6 x^{5} d x$ , c) $\int_{-3}^{4} d x$ , d) $\int_{-3}^{4} 4 \sqrt{2} d x$ , e) $\int_{-3}^{4} 0 d x$ , f) $\int_{1}^{2} \frac{1}{x^{2}} d x$ , g) $\int_{-2}^{-1} \frac{1}{x^{2}} d x$ , h) $\int_{1}^{2} \frac{1}{x} d x$ , i) $\int_{0.5}^{1} \frac{1}{x} d x$ , k) $\int_{-2}^{-1} \frac{1}{x} d x$ , l) $\int_{0}^{0} \sqrt{x} d x$ , m) $\int_{1}^{9} \frac{1}{\sqrt{x}} d x$ , n) $\int_{0}^{\theta} x \sqrt{x} d x$ , o) $\int_{0}^{\pi / 2} \sin x d x$ , p) $\int_{0.5}^{1} \ln x d x$ , q) $\int_{1}^{2} \ln x d x$ , r) $\int_{0.5}^{2} \ln x d x$ , s) $\int_{-1}^{0} e^{x} d x$ , t) $\int_{0}^{\pi / 2} \cos x d x$ , u) $\int_{-1000}^{0} e^{x} d x$ .

See Solution

Problem 7517

Calculate the integral $\int \sin^{4} x \cos^{6} x \, dx$ .

See Solution

Problem 7518

Calculate the integrals: 1) $\int_{1}^{2} \frac{x^{3}+x^{2}+2}{x^{2}} dx$ and 2) $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos x dx$ .

See Solution

Problem 7519

Finde den Wert von $k \in \mathbb{R}^{+}$ für: a) $\int^{k} 3 x^{2} d x=16$ und b) $\int_{-2}^{4} k x d x=18$ .

See Solution

Problem 7520

Berechnen Sie die 1. und 2. Ableitung der Funktionen und die Definitionsmengen $D_{t}, D_{f}$ . a) $f(x)=\sqrt[4]{x}-2x$ b) $f(x)=\frac{x^{2}+1}{x^{2}}$

See Solution

Problem 7521

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{\sqrt{x+\Delta x}}-\frac{4}{\sqrt{x}}}{\Delta x}$ .

See Solution

Problem 7522

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{\sqrt{x+\Delta x}}-\frac{4}{\sqrt{x}}}{\Delta x}$ . Options: A) $\frac{2}{x \sqrt{x}}$ , B) $\frac{-2}{x \sqrt{x}}$ , C) $\frac{8}{x \sqrt{x}}$ , D) $\frac{-8}{x \sqrt{x}}$ .

See Solution

Problem 7523

Calculate the integral: $\int \frac{x+x^{2 / 3}+x^{1 / 6}}{x\left(1+x^{1 / 3}\right)} d x$

See Solution

Problem 7524

Find $g^{\prime}(0)$ if $g(x)=f(x)^{3 f(x)}$ and the tangent line at $x=0$ is $y=3x+2$ .

See Solution

Problem 7525

Find values of $a, b \in \mathbb{R}$ for which the function
$f(x)=\left\{\begin{array}{ll} \frac{\sin (3 x)}{a x} & \text { if } x>0 \\ b & \text { if } x=0 \\ \arctan \left(\frac{1}{x}\right) & \text { if } x<0 \end{array}\right.$
is continuous.

See Solution

Problem 7526

Find the tangent line equation using implicit differentiation for $y^{2}(y^{2}-4)=x^{2}(x^{2}-5)$ at $(0,-2)$ .

See Solution

Problem 7527

Find the derivative of the inverse function for $f(x)=\frac{e^{x}-e^{-x}}{2}$ using $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$ .

See Solution

Problem 7528

Trova le equazioni degli asintoti per la curva: $f(x)=x+\sqrt{x^{2}-2 x+5}$ .

See Solution

Problem 7529

Find the limit: $\lim _{x \rightarrow 1}\left(\frac{1}{1-x}-\frac{2}{1-x^{2}}\right)$ .

See Solution

Problem 7530

Find the average value $f_{\text {ave }}$ of $f(x)=2 \frac{b}{a} \sqrt{x^{2}-a^{2}}$ over $[a, 2a]$ .

See Solution

Problem 7531

Calculate the integral of $\sin^{4} x \cos^{6} x$ with respect to $x$ : $\int \sin^{4} x \cos^{6} x \, dx$ .

See Solution

Problem 7532

Find the tangent line equation for $y=\sqrt{1-3x}$ at the point $(-1,2)$ .

See Solution

Problem 7533

Berechnen Sie die Punkte, wo der Graph von $f(x)=2 x^{2}+...$ a) eine Tangente mit Steigung 4 hat, b) dieselbe Steigung wie $g(x)=x^{3}-4 x-1$ .

See Solution

Problem 7534

Find the derivative of $f(t)=\frac{1}{t^{2}+1}$ using the limit definition: $f'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}$ .

See Solution

Problem 7535

Find points $c$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on $[-19,19]$ . Answer: $\mathrm{c}=$ (exact values).

See Solution

Problem 7536

Calculate the average rate of change of $f(x)=5 x^{3}-3 x^{2}+7$ from $x=-3$ to $x=2$ .

See Solution

Problem 7537

Find the derivative of $y=\frac{(x^{3}+4)^{5}}{3 x^{4}-2}$ using the chain rule.

See Solution

Problem 7538

Find the slope and equation of the tangent line for $f(x)=x^{2}-5$ at $x=3$ .

See Solution

Problem 7539

Find the tangent line equation for $g(x)=\sqrt{x}$ at $x=16$ using $g^{\prime}(x)=\frac{1}{2 \sqrt{x}}$ .

See Solution

Problem 7540

Find the tangent line equation for $f(x)=\frac{4}{x}$ at $x=-7$ . Use $f^{\prime}(x)=\frac{-4}{x^{2}}$ .

See Solution

Problem 7541

Find the derivative of $y=((x+5)^{5}-1)^{4}$ using the chain rule.

See Solution

Problem 7542

Check if Rolle's theorem applies to $f(x)=-\sin 8 x$ on $\left[\frac{\pi}{8}, \frac{\pi}{4}\right]$ and find guaranteed point(s).

See Solution

Problem 7543

Find the derivative of $y=\left(5 x^{3}-3\right)^{5} \sqrt[4]{-4 x^{5}-3}$ using the chain rule.

See Solution

Problem 7544

Check if Rolle's theorem applies to $f(x)=-\sin 7 x$ on $\left[\frac{\pi}{7}, \frac{2 \pi}{7}\right]$ and find guaranteed point(s).

See Solution

Problem 7545

Find all functions $y$ such that $y' = 9x^8 - 9$ , including the constant $C$ .

See Solution

Problem 7546

Find points $c$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on $[-18,18]$ . Answer: $\mathrm{c}=$

See Solution

Problem 7547

Find the critical points of $f$ if $f^{\prime}(x)=x+13$ and determine where $f$ is increasing or decreasing.

See Solution

Problem 7548

Find critical points of $f$ where $f^{\prime}(x)=x-11$ . Determine intervals where $f$ is increasing or decreasing.

See Solution

Problem 7549

Determine where the function $f(x)=-4 \sin ^{2} x$ is increasing or decreasing on the interval $[-\pi, \pi]$ . Find $f^{\prime}(x)$ .

See Solution

Problem 7550

Find critical points of $f$ if $f^{\prime}(x)=x+11$ and determine where $f$ is increasing or decreasing.

See Solution

Problem 7551

Ein ICE bremst ab. Gegeben ist die Funktion $s(t)=-\frac{1}{5} t^{2}+80 t$ . Berechne: a) Bremszeit, b) Bremsweg, c) Anfangsgeschwindigkeit, d) Zeitpunkt bei $100 \mathrm{~km/h}$ , e) Geschwindigkeit nach $2000 \mathrm{~m}$ .

See Solution

Problem 7552

Given $g^{\prime \prime}(x)=5-x$ , find intervals where $g$ is concave up/down and identify inflection points.

See Solution

Problem 7553

Calculate the average value $f_{avg}$ of $f(x)=2 \frac{b}{a}-\sqrt{x^{2}-a^{2}}$ over $[a, 2a]$ for positive $a$ and $b$ .

See Solution

Problem 7554

Bestimme die Ableitung $f_{a}^{\prime}$ und berechne $f_{2}^{\prime}(1)$ für die Funktionen a) bis f).

See Solution

Problem 7555

Find functions whose derivative is $y' = 9x^8 - 4$ . Express $y$ using the constant $\mathrm{C}$ .

See Solution

Problem 7556

Determine where the function $f(t)=t^{\frac{1}{5}}(t^{2}-99)$ is increasing and decreasing.

See Solution

Problem 7557

Determine where the function $f(x)=-\cos ^{2} x$ is increasing and decreasing on the interval $[-\pi, \pi]$ .

See Solution

Problem 7558

Warum erscheint der Term $\cos(a x) \cdot a$ in der Ableitung der Funktion $f a(x)=(\sin (a x))^{2}$ ?

See Solution

Problem 7559

Gegeben sind die Funktionen $f(x)=-x^{3}+3 x^{2}-3 x+4$ und $g(x)=\frac{1}{x^{2}}+2 x$ . Bestimme den Winkel des Graphen von $f$ bei $P(1 \mid f(1))$ und prüfe, ob $f$ und $g$ sich dort berühren.

See Solution

Problem 7560

Find the leading term of the Taylor series for $f(x)=\int_{0}^{x} \ln (\cosh (t)) dt$ about zero. Options: $\frac{x^{3}}{6}$ , $\frac{x^{3}}{2}$ , $\frac{x^{2}}{2}$ , $-\frac{x^{5}}{60}$ , $-\frac{x^{3}}{18}$ .

See Solution

Problem 7561

Evaluate the Riemann sum for $I=\int_{0}^{2} x \, dx$ and determine the optimal error $E(n)$ as $n \to \infty$ .

See Solution

Problem 7562

Determine where the function $f(t)=t^{\frac{1}{5}}(t^{2}-99)$ is increasing or decreasing. Provide intervals in notation.

See Solution

Problem 7563

Find the average value of $f(x)=2 \frac{b}{a}-\sqrt{x^{2}-a^{2}}$ over the interval $[a, 2a]$ .

See Solution

Problem 7564

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=3 x^{2}-7$ , with $h \neq 0$ .

See Solution

Problem 7565

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=4 x^{2}-1$ , where $h \neq 0$ .

See Solution

Problem 7566

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=5 x^{2}-3 x+8$ , where $h \neq 0$ .

See Solution

Problem 7567

Evaluate the integral $\int \sqrt{1-7 w^{2}} d w$ using the substitution $w=\frac{1}{\sqrt{7}} \sin (\theta)$ .

See Solution

Problem 7568

Find the relative extrema of the function $f(x)=5x \ln\left(\frac{x}{2}\right)$ .

See Solution

Problem 7569

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=9x-2$ , where $h \neq 0$ .

See Solution

Problem 7570

Calculate the integral $\int_{3}^{6}\left(2 x^{\frac{2}{3}}-\frac{4}{3} x\right) dx$ .

See Solution

Problem 7571

Evaluate the integral from 3 to 6 of the function $2 x^{\frac{2}{3}} - \frac{4}{3 x}$ .

See Solution

Problem 7572

Find the horizontal asymptote of the drug concentration $C(t)=\frac{t}{6 t^{2}+6}$ . $C=\square$ (Simplify your answer.)

See Solution

Problem 7573

Find the horizontal asymptote of $C(t)=\frac{t}{6t^{2}+6}$ and what value $C(t)$ approaches as $t$ increases.

See Solution

Problem 7574

Find the volume of revolution when the area under the curve $y=f(x)$ from $x=0$ to $x=k$ is rotated around the $x$ -axis, given $\frac{d y}{d x}=\frac{-y}{\sqrt{1-y^{2}}}$ , $f(0)=1$ , and $f(k)=a$ .

See Solution

Problem 7575

Evaluate the expression: $\int\left(\frac{d}{d x}\left(e^{-x^{2}}\right)\right) d x-\frac{d}{d x}\left(\int e^{-x^{2}} d x\right)$ .

See Solution

Problem 7576

Bestimme das Integral der Funktion $f(x) = 2x^{\frac{2}{3}} - \frac{4}{3x}.$

See Solution

Problem 7577

Use implicit differentiation to find $\frac{d^{2} y}{d x^{2}}$ for: 13) $4 y^{2}+2=3 x^{2}$ , 14) $5=4 x^{2}+5 y^{2}$ .

See Solution

Problem 7578

Find $\frac{d y}{d x}$ for $\frac{3 x^{2}}{4 y}=x$ using three methods. Do results match? Explain differences.

See Solution

Problem 7579

Zeichnen Sie den Graphen von $f(x)=\frac{1}{4} x^{3}-x$ und schätzen Sie die Steigungen. Berechnen Sie $f^{\prime}(0)$ .

See Solution

Problem 7580

Find the relative extrema of $g(x) = 12 + 8x + x^2$ and determine where it is increasing and decreasing.

See Solution

Problem 7581

Gegeben sind die Funktionen $f_{a}(x)=\frac{1}{a} x^{3}+3 x^{2}+5 x+2 a$ und $h(x)=-\frac{1}{2} x^{-3}$ .
a) Bestimmen Sie die Symmetrie von $K$ und das Verhalten von $h$ für $x \rightarrow+\infty$ . Warum gilt $\lim _{x \rightarrow+\infty} h(x) \neq \lim _{-x \rightarrow+\infty} f_{a}(x)$ ?
b) Berechnen Sie den Flächeninhalt des Dreiecks, das von der Tangente an $K$ bei $P(-1, h(-1))$ und den Koordinatenachsen begrenzt wird.
c) Zeigen Sie, dass $K$ keine lokalen Extrempunkte hat.
d) Finden Sie die zwei Punkte auf $G_{2}$ mit Tangentensteigung $m=1,5$ .
e) Bestimmen Sie den Parameterwert $a$ , für den $G_{a}$ eine waagerechte Tangente hat, und erläutern Sie den Sattelpunkt.
f) Zeigen Sie, dass $P_{1}(-4, 0)$ und $P_{2}(-0,64, 1,9)$ auf den Graphen liegen. Berechnen Sie die minimalen Seitenlängen einer rechteckigen Plane, die den Teich abdeckt.

See Solution

Problem 7582

Find the limit: $\lim _{x \rightarrow-\infty} 5+\frac{8}{x}$ .

See Solution

Problem 7583

Find $c$ for the function $f(x)=c x^{2}-2 x^{-2}$ to have an inflection point at $(1, f(1))$ . Choices: $c=6$ , $c=12$ , $c=-6$ , $c=0$ , $c=-12$ .

See Solution

Problem 7584

Find the one-sided limits of $f(x)$ at $x=-5$ and check if $f$ is continuous at -5 (YES/NO).

See Solution

Problem 7585

Check if the Mean Value Theorem applies to $f(x)=x^{3}-3x+9$ on $[-2,2]$ and find values of $c$ . If not, write DNE.

See Solution

Problem 7586

Given the function $f(x)=x^{3}-3x+9$ on $[-2,2]$ , is $f$ continuous there? Find $f'(-2)$ , $f'(-2)$ , and $\frac{f(2)-f(-2)}{4}$ .

See Solution

Problem 7587

Find $f(1)$ for the polynomial $f(x)=x^{3}+A x^{2}+B x$ given critical point $x=3$ and inflection point $x=-2$ .

See Solution

Problem 7588

Determine the concavity and points of inflection for $f(x)=\frac{1}{12} x^{4}-2 x^{2}$ .

See Solution

Problem 7589

Solve the differential equation: $\frac{d y}{d x}=\frac{-y}{\sqrt{1-y^{2}}}$ .

See Solution

Problem 7590

Determine the concavity and inflection points of $f(x)=3(x-2)^{5/3}$ .

See Solution

Problem 7591

Find the minimum value of $f(7)$ given $f(3)=9$ and $f^{\prime}(x) \geq 2$ for $3 \leq x \leq 7$ .

See Solution

Problem 7592

Find the derivative of $y=\ln \left((7 x)^{3}\right)$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 7593

Berechnen Sie die Ableitungen von $f$ für: a) $f(x)=2 x^{2}-1$ , b) $f(x)=-\frac{3}{2} x+2$ , c) $f(x)=x^{3}+x$ , d) $f(x)=-1+x+\frac{x^{2}}{4}$ .

See Solution

Problem 7594

Find the $x$ -coordinates of points of inflection for $f(x)$ given $f^{\prime \prime}(x)=\frac{x^{3}-3 x^{2}}{(x-2)^{3}(x-4)^{3}}$ .

See Solution

Problem 7595

Find the limits for the piecewise function $f(x)$ defined as:
$f(x)=\begin{cases} \sqrt{-4-x}+4, & x<-5 \\ 4, & x=-5 \\ 2x+15, & x>-5 \end{cases}$
Calculate: $\lim_{x \to -5^-} f(x)$ , $\lim_{x \to -5^+} f(x)$ , $\lim_{x \to -5} f(x)$ .

See Solution

Problem 7596

Analyze the concavity of $f(x)=\frac{x}{x-6}$ and identify any points of inflection.

See Solution

Problem 7597

Analyze the concavity of $f(x)=x^{3}-x-2$ and locate any points of inflection.

See Solution

Problem 7598

Given the function $f(x)=x^{3}-4 x^{2}-16 x+2$ on $[-4,4]$ , is $f$ continuous there? If differentiable on $(-4,4)$ , find $f^{\prime}(x)$ . Also, find $f(-4)$ and $f(4)$ .

See Solution

Problem 7599

Find the first and second derivatives of $k(x)=b^{\log_b(x)}$ , where $b > 1$ .

See Solution

Problem 7600

Find $c$ for which the graph of $f(x)=c x^{2}-x^{-2}$ has an inflection point at $(2, f(2))$ . Options: $c=\frac{3}{8}$ , $c=0$ , $c=-\frac{3}{8}$ , $c=\frac{3}{16}$ , $c=-\frac{3}{16}$ .

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337

Calculus

Problem 7501

Untersuche die Funktion f(x)=(x−4)⋅xf(x)=(x-4) \cdot \sqrt{x}f(x)=(x−4)⋅x​ für x≥0x \geq 0x≥0 auf Monotonieverhalten und finde die innere Extremstelle.

Problem 7502

Find the rate of change of the revenue function R(Q)=100Q2+500R(Q)=100 Q^{2}+500R(Q)=100Q2+500 at production level Q0Q_{0}Q0​: ΔR(Q0)/ΔQ\Delta R(Q_{0}) / \Delta QΔR(Q0​)/ΔQ.

Problem 7503

Problem 7504

Evaluate the integral ∫(3x2+6x+5)tan⁡−1x dx\int(3 x^{2}+6 x+5) \tan^{-1} x \, dx∫(3x2+6x+5)tan−1xdx.

Problem 7505

Evaluate the integral ∫1edxx(ln⁡x)1/3\int_{1}^{e} \frac{d x}{x(\ln x)^{1 / 3}}∫1e​x(lnx)1/3dx​.

Problem 7506

Prove that ∫1edxx(ln⁡x)1/3=32\int_{1}^{e} \frac{d x}{x(\ln x)^{1 / 3}}=\frac{3}{2}∫1e​x(lnx)1/3dx​=23​.

Problem 7507

Evaluate the integral from 0 to 9: ∫09x12dx\int_{0}^{9} x^{\frac{1}{2}} d x∫09​x21​dx.

Problem 7508

Evaluate the integral: ∫−∞∞1x2+2x+5 dx\int_{-\infty}^{\infty} \frac{1}{x^{2}+2 x+5} \, dx∫−∞∞​x2+2x+51​dx

Problem 7509

Calculate the integral ∫01x dx\int_{0}^{1} \sqrt{x} \, dx∫01​x​dx.

Problem 7510

Calculate the integral from -1 to 0 of exe^{x}ex, i.e., ∫−10exdx\int_{-1}^{0} e^{x} d x∫−10​exdx.

Problem 7511

Berechne ∫−23(4x2−3x+5)dx+∫−23(3x−5)dx\int_{-2}^{3}\left(4 x^{2}-3 x+5\right) d x + \int_{-2}^{3}(3 x-5) d x∫−23​(4x2−3x+5)dx+∫−23​(3x−5)dx.

Problem 7512

Choose negative values for aaa and ddd (a<d<0a < d < 0a<d<0) and observe which axis solution trajectories approach as they stabilize at the origin. Prove your observation.

Problem 7513

Problem 7514

Problem 7515

Bestimmen Sie den Wert von k∈R+k \in \mathbb{R}^{+}k∈R+ für a) ∫−kk3x2dx=16\int_{-k}^{k} 3 x^{2} d x=16∫−kk​3x2dx=16 und b) ∫−24kxdx=18\int_{-2}^{4} k x d x=18∫−24​kxdx=18.

Problem 7516

Problem 7517

Calculate the integral ∫sin⁡4xcos⁡6x dx\int \sin^{4} x \cos^{6} x \, dx∫sin4xcos6xdx.

Problem 7518

Calculate the integrals: 1) ∫12x3+x2+2x2dx\int_{1}^{2} \frac{x^{3}+x^{2}+2}{x^{2}} dx∫12​x2x3+x2+2​dx and 2) ∫−π4π4cos⁡xdx\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos x dx∫−4π​4π​​cosxdx.

Problem 7519

Finde den Wert von k∈R+k \in \mathbb{R}^{+}k∈R+ für: a) ∫k3x2dx=16\int^{k} 3 x^{2} d x=16∫k3x2dx=16 und b) ∫−24kxdx=18\int_{-2}^{4} k x d x=18∫−24​kxdx=18.

Problem 7520

Berechnen Sie die 1. und 2. Ableitung der Funktionen und die Definitionsmengen Dt,DfD_{t}, D_{f}Dt​,Df​. a) f(x)=x4−2xf(x)=\sqrt[4]{x}-2xf(x)=4x​−2x b) f(x)=x2+1x2f(x)=\frac{x^{2}+1}{x^{2}}f(x)=x2x2+1​

Problem 7521

Find the limit: lim⁡Δx→04x+Δx−4xΔx\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{\sqrt{x+\Delta x}}-\frac{4}{\sqrt{x}}}{\Delta x}limΔx→0​Δxx+Δx​4​−x​4​​.

Problem 7522

Problem 7523

Calculate the integral: ∫x+x2/3+x1/6x(1+x1/3)dx\int \frac{x+x^{2 / 3}+x^{1 / 6}}{x\left(1+x^{1 / 3}\right)} d x∫x(1+x1/3)x+x2/3+x1/6​dx

Problem 7524

Find g′(0)g^{\prime}(0)g′(0) if g(x)=f(x)3f(x)g(x)=f(x)^{3 f(x)}g(x)=f(x)3f(x) and the tangent line at x=0x=0x=0 is y=3x+2y=3x+2y=3x+2.

Problem 7525

Problem 7526

Find the tangent line equation using implicit differentiation for y2(y2−4)=x2(x2−5)y^{2}(y^{2}-4)=x^{2}(x^{2}-5)y2(y2−4)=x2(x2−5) at (0,−2)(0,-2)(0,−2).

Problem 7527

Find the derivative of the inverse function for f(x)=ex−e−x2f(x)=\frac{e^{x}-e^{-x}}{2}f(x)=2ex−e−x​ using (f−1)′(x)=1f′(f−1(x))\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}(f−1)′(x)=f′(f−1(x))1​.

Problem 7528

Trova le equazioni degli asintoti per la curva: f(x)=x+x2−2x+5f(x)=x+\sqrt{x^{2}-2 x+5}f(x)=x+x2−2x+5​.

Problem 7529

Find the limit: lim⁡x→1(11−x−21−x2)\lim _{x \rightarrow 1}\left(\frac{1}{1-x}-\frac{2}{1-x^{2}}\right)limx→1​(1−x1​−1−x22​).

Problem 7530

Find the average value fave f_{\text {ave }}fave ​ of f(x)=2bax2−a2f(x)=2 \frac{b}{a} \sqrt{x^{2}-a^{2}}f(x)=2ab​x2−a2​ over [a,2a][a, 2a][a,2a].

Problem 7531

Calculate the integral of sin⁡4xcos⁡6x\sin^{4} x \cos^{6} xsin4xcos6x with respect to xxx: ∫sin⁡4xcos⁡6x dx\int \sin^{4} x \cos^{6} x \, dx∫sin4xcos6xdx.

Problem 7532

Find the tangent line equation for y=1−3xy=\sqrt{1-3x}y=1−3x​ at the point (−1,2)(-1,2)(−1,2).

Problem 7533

Berechnen Sie die Punkte, wo der Graph von f(x)=2x2+...f(x)=2 x^{2}+...f(x)=2x2+... a) eine Tangente mit Steigung 4 hat, b) dieselbe Steigung wie g(x)=x3−4x−1g(x)=x^{3}-4 x-1g(x)=x3−4x−1.

Problem 7534

Find the derivative of f(t)=1t2+1f(t)=\frac{1}{t^{2}+1}f(t)=t2+11​ using the limit definition: f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​.

Problem 7535

Find points ccc where the Mean Value Theorem applies for f(x)=x3f(x)=x^{3}f(x)=x3 on [−19,19][-19,19][−19,19]. Answer: c=\mathrm{c}=c= (exact values).

Problem 7536

Calculate the average rate of change of f(x)=5x3−3x2+7f(x)=5 x^{3}-3 x^{2}+7f(x)=5x3−3x2+7 from x=−3x=-3x=−3 to x=2x=2x=2.

Problem 7537

Find the derivative of y=(x3+4)53x4−2y=\frac{(x^{3}+4)^{5}}{3 x^{4}-2}y=3x4−2(x3+4)5​ using the chain rule.

Problem 7538

Find the slope and equation of the tangent line for f(x)=x2−5f(x)=x^{2}-5f(x)=x2−5 at x=3x=3x=3.

Problem 7539

Find the tangent line equation for g(x)=xg(x)=\sqrt{x}g(x)=x​ at x=16x=16x=16 using g′(x)=12xg^{\prime}(x)=\frac{1}{2 \sqrt{x}}g′(x)=2x​1​.

Problem 7540

Find the tangent line equation for f(x)=4xf(x)=\frac{4}{x}f(x)=x4​ at x=−7x=-7x=−7. Use f′(x)=−4x2f^{\prime}(x)=\frac{-4}{x^{2}}f′(x)=x2−4​.

Problem 7541

Find the derivative of y=((x+5)5−1)4y=((x+5)^{5}-1)^{4}y=((x+5)5−1)4 using the chain rule.

Problem 7542

Check if Rolle's theorem applies to f(x)=−sin⁡8xf(x)=-\sin 8 xf(x)=−sin8x on [π8,π4]\left[\frac{\pi}{8}, \frac{\pi}{4}\right][8π​,4π​] and find guaranteed point(s).

Problem 7543

Untersuche die Funktion $f(x)=(x-4) \cdot \sqrt{x}$ für $x \geq 0$ auf Monotonieverhalten und finde die innere Extremstelle.

Find the rate of change of the revenue function $R(Q)=100 Q^{2}+500$ at production level $Q_{0}$ : $\Delta R(Q_{0}) / \Delta Q$ .

Evaluate the integral $\int(3 x^{2}+6 x+5) \tan^{-1} x \, dx$ .

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

Prove that $\int_{1}^{e} \frac{d x}{x(\ln x)^{1 / 3}}=\frac{3}{2}$ .

Evaluate the integral from 0 to 9: $\int_{0}^{9} x^{\frac{1}{2}} d x$ .

Evaluate the integral: $\int_{-\infty}^{\infty} \frac{1}{x^{2}+2 x+5} \, dx$

Calculate the integral $\int_{0}^{1} \sqrt{x} \, dx$ .

Calculate the integral from -1 to 0 of $e^{x}$ , i.e., $\int_{-1}^{0} e^{x} d x$ .

Berechne $\int_{-2}^{3}\left(4 x^{2}-3 x+5\right) d x + \int_{-2}^{3}(3 x-5) d x$ .

Choose negative values for $a$ and $d$ ( $a < d < 0$ ) and observe which axis solution trajectories approach as they stabilize at the origin. Prove your observation.

Bestimmen Sie den Wert von $k \in \mathbb{R}^{+}$ für a) $\int_{-k}^{k} 3 x^{2} d x=16$ und b) $\int_{-2}^{4} k x d x=18$ .

Calculate the integral $\int \sin^{4} x \cos^{6} x \, dx$ .

Calculate the integrals: 1) $\int_{1}^{2} \frac{x^{3}+x^{2}+2}{x^{2}} dx$ and 2) $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos x dx$ .

Finde den Wert von $k \in \mathbb{R}^{+}$ für: a) $\int^{k} 3 x^{2} d x=16$ und b) $\int_{-2}^{4} k x d x=18$ .

Berechnen Sie die 1. und 2. Ableitung der Funktionen und die Definitionsmengen $D_{t}, D_{f}$ . a) $f(x)=\sqrt[4]{x}-2x$ b) $f(x)=\frac{x^{2}+1}{x^{2}}$

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{\frac{4}{\sqrt{x+\Delta x}}-\frac{4}{\sqrt{x}}}{\Delta x}$ .

Calculate the integral: $\int \frac{x+x^{2 / 3}+x^{1 / 6}}{x\left(1+x^{1 / 3}\right)} d x$

Find $g^{\prime}(0)$ if $g(x)=f(x)^{3 f(x)}$ and the tangent line at $x=0$ is $y=3x+2$ .

Find the tangent line equation using implicit differentiation for $y^{2}(y^{2}-4)=x^{2}(x^{2}-5)$ at $(0,-2)$ .

Find the derivative of the inverse function for $f(x)=\frac{e^{x}-e^{-x}}{2}$ using $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$ .

Trova le equazioni degli asintoti per la curva: $f(x)=x+\sqrt{x^{2}-2 x+5}$ .

Find the limit: $\lim _{x \rightarrow 1}\left(\frac{1}{1-x}-\frac{2}{1-x^{2}}\right)$ .

Find the average value $f_{\text {ave }}$ of $f(x)=2 \frac{b}{a} \sqrt{x^{2}-a^{2}}$ over $[a, 2a]$ .

Calculate the integral of $\sin^{4} x \cos^{6} x$ with respect to $x$ : $\int \sin^{4} x \cos^{6} x \, dx$ .

Find the tangent line equation for $y=\sqrt{1-3x}$ at the point $(-1,2)$ .

Berechnen Sie die Punkte, wo der Graph von $f(x)=2 x^{2}+...$ a) eine Tangente mit Steigung 4 hat, b) dieselbe Steigung wie $g(x)=x^{3}-4 x-1$ .

Find the derivative of $f(t)=\frac{1}{t^{2}+1}$ using the limit definition: $f'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}$ .

Find points $c$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on $[-19,19]$ . Answer: $\mathrm{c}=$ (exact values).

Calculate the average rate of change of $f(x)=5 x^{3}-3 x^{2}+7$ from $x=-3$ to $x=2$ .

Find the derivative of $y=\frac{(x^{3}+4)^{5}}{3 x^{4}-2}$ using the chain rule.

Find the slope and equation of the tangent line for $f(x)=x^{2}-5$ at $x=3$ .

Find the tangent line equation for $g(x)=\sqrt{x}$ at $x=16$ using $g^{\prime}(x)=\frac{1}{2 \sqrt{x}}$ .

Find the tangent line equation for $f(x)=\frac{4}{x}$ at $x=-7$ . Use $f^{\prime}(x)=\frac{-4}{x^{2}}$ .

Find the derivative of $y=((x+5)^{5}-1)^{4}$ using the chain rule.

Check if Rolle's theorem applies to $f(x)=-\sin 8 x$ on $\left[\frac{\pi}{8}, \frac{\pi}{4}\right]$ and find guaranteed point(s).

Find the derivative of $y=\left(5 x^{3}-3\right)^{5} \sqrt[4]{-4 x^{5}-3}$ using the chain rule.

Check if Rolle's theorem applies to $f(x)=-\sin 7 x$ on $\left[\frac{\pi}{7}, \frac{2 \pi}{7}\right]$ and find guaranteed point(s).

Find all functions $y$ such that $y' = 9x^8 - 9$ , including the constant $C$ .

Find points $c$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on $[-18,18]$ . Answer: $\mathrm{c}=$

Find the critical points of $f$ if $f^{\prime}(x)=x+13$ and determine where $f$ is increasing or decreasing.

Find critical points of $f$ where $f^{\prime}(x)=x-11$ . Determine intervals where $f$ is increasing or decreasing.

Determine where the function $f(x)=-4 \sin ^{2} x$ is increasing or decreasing on the interval $[-\pi, \pi]$ . Find $f^{\prime}(x)$ .

Find critical points of $f$ if $f^{\prime}(x)=x+11$ and determine where $f$ is increasing or decreasing.

Ein ICE bremst ab. Gegeben ist die Funktion $s(t)=-\frac{1}{5} t^{2}+80 t$ . Berechne: a) Bremszeit, b) Bremsweg, c) Anfangsgeschwindigkeit, d) Zeitpunkt bei $100 \mathrm{~km/h}$ , e) Geschwindigkeit nach $2000 \mathrm{~m}$ .

Given $g^{\prime \prime}(x)=5-x$ , find intervals where $g$ is concave up/down and identify inflection points.

Calculate the average value $f_{avg}$ of $f(x)=2 \frac{b}{a}-\sqrt{x^{2}-a^{2}}$ over $[a, 2a]$ for positive $a$ and $b$ .

Bestimme die Ableitung $f_{a}^{\prime}$ und berechne $f_{2}^{\prime}(1)$ für die Funktionen a) bis f).

Find functions whose derivative is $y' = 9x^8 - 4$ . Express $y$ using the constant $\mathrm{C}$ .

Determine where the function $f(t)=t^{\frac{1}{5}}(t^{2}-99)$ is increasing and decreasing.

Determine where the function $f(x)=-\cos ^{2} x$ is increasing and decreasing on the interval $[-\pi, \pi]$ .

Warum erscheint der Term $\cos(a x) \cdot a$ in der Ableitung der Funktion $f a(x)=(\sin (a x))^{2}$ ?

Gegeben sind die Funktionen $f(x)=-x^{3}+3 x^{2}-3 x+4$ und $g(x)=\frac{1}{x^{2}}+2 x$ . Bestimme den Winkel des Graphen von $f$ bei $P(1 \mid f(1))$ und prüfe, ob $f$ und $g$ sich dort berühren.

Evaluate the Riemann sum for $I=\int_{0}^{2} x \, dx$ and determine the optimal error $E(n)$ as $n \to \infty$ .

Determine where the function $f(t)=t^{\frac{1}{5}}(t^{2}-99)$ is increasing or decreasing. Provide intervals in notation.

Find the average value of $f(x)=2 \frac{b}{a}-\sqrt{x^{2}-a^{2}}$ over the interval $[a, 2a]$ .

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=3 x^{2}-7$ , with $h \neq 0$ .

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=4 x^{2}-1$ , where $h \neq 0$ .

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=5 x^{2}-3 x+8$ , where $h \neq 0$ .

Evaluate the integral $\int \sqrt{1-7 w^{2}} d w$ using the substitution $w=\frac{1}{\sqrt{7}} \sin (\theta)$ .

Find the relative extrema of the function $f(x)=5x \ln\left(\frac{x}{2}\right)$ .

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=9x-2$ , where $h \neq 0$ .

Calculate the integral $\int_{3}^{6}\left(2 x^{\frac{2}{3}}-\frac{4}{3} x\right) dx$ .

Evaluate the integral from 3 to 6 of the function $2 x^{\frac{2}{3}} - \frac{4}{3 x}$ .

Find the horizontal asymptote of the drug concentration $C(t)=\frac{t}{6 t^{2}+6}$ . $C=\square$ (Simplify your answer.)

Find the horizontal asymptote of $C(t)=\frac{t}{6t^{2}+6}$ and what value $C(t)$ approaches as $t$ increases.

Find the volume of revolution when the area under the curve $y=f(x)$ from $x=0$ to $x=k$ is rotated around the $x$ -axis, given $\frac{d y}{d x}=\frac{-y}{\sqrt{1-y^{2}}}$ , $f(0)=1$ , and $f(k)=a$ .