Calculus

Problem 13601

Find the derivative of $f(x)=a x^{3}+c$ and $f(x)=\frac{1}{x^{2}}$ .

See Solution

Problem 13602

Estimate the total light intensity $\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta$ using the Midpoint Rule with $n=10$ .

See Solution

Problem 13603

See Solution

Problem 13604

Estimate the total light intensity $\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta$ using the Midpoint Rule with $n=10$ .

See Solution

Problem 13605

Convert $D(t)$ from hours to seconds. Use Simpson's Rule to estimate $A(43200) \approx \frac{3600}{3}[\text{sum of } D(t)]$ . Round to the nearest thousand megabits.

See Solution

Problem 13606

Find the tangent equation to the curve at $t=0$ for $x=\ln \frac{a t+b}{c t+d}$ and $y=\frac{a t+b}{c t+d}$ .

See Solution

Problem 13607

Find the value of $\int_{2}^{14}[-3 f(x)+2 g(x)-h(x)] d x$ given $\int_{2}^{14} f(x) d x=28$ , $\int_{2}^{14} g(x) d x=39$ , $\int_{2}^{14} h(x) d x=8$ .

See Solution

Problem 13608

Find $\int_{2}^{7} f(x) d x$ given $\int_{1}^{7} f(x) d x=-6$ and $\int_{1}^{2} f(x) d x=10$ .

See Solution

Problem 13609

Compare the integrals $F=\int_{1}^{5}(\sqrt{x+5}+4) dx$ and $G=\int_{1}^{5}(\sqrt{x+1}+1) dx$ using $\leq$ or $\geq$ .
Provide your answer below: $F \square G$

See Solution

Problem 13610

Find lower and upper bounds for $F=\int_{-4}^{3}(\sqrt{x+6}+2) d x$ using the comparison theorem.

See Solution

Problem 13611

A passenger on the London Eye rises at $0.032 \mathrm{~m/s}$ after 1 minute. Find their horizontal speed at that moment.

See Solution

Problem 13612

Find the sum: $\sum_{k=0}^{\infty} \frac{(-2)^{k} x^{k+1}}{k+1}$ .

See Solution

Problem 13613

Find $\int_{1}^{6} 8 f(x) d x$ given $\int_{9}^{1} f(x) d x=-9$ and $\int_{6}^{9} f(x) d x=-8$ .

See Solution

Problem 13614

Find the value of $\int_{1}^{5} 2 f(x) d x$ given $\int_{6}^{1} f(x) d x=13$ and $\int_{5}^{6} f(x) d x=-2$ .

See Solution

Problem 13615

Find lower and upper bounds for $F$ using the comparison theorem, where $F=\int_{2}^{3}(6 x+3) d x.$

See Solution

Problem 13616

Find the best lower and upper bounds for $F$ using the comparison theorem for the integral $F=\int_{0}^{6}(\sqrt{x+2}+3) d x$ .

See Solution

Problem 13617

Find lower and upper bounds for $F = \int_{0}^{6}(\sqrt{x+2}+3) dx$ using the comparison theorem. $\square \leq F \leq \square$

See Solution

Problem 13618

Find the tangent line(s) to the curve $x=t^{5}-4 t^{3}$ and $y=t^{5}$ at $(0,32)$ . Write as a Cartesian equation.

See Solution

Problem 13619

Given the function $f(x)=x^{3}-4x$ , find local max/min values and corresponding $x$ values, rounded to two decimal places.

See Solution

Problem 13620

Evaluate the integrals $F = \int_{1}^{5}(2x^2 + 5)dx$ and $G = \int_{1}^{5}(4x^2 + 6)dx$ . Which statements are true?

See Solution

Problem 13621

Find the best lower and upper bounds for $F = \int_{1}^{5}(6x+5) \, dx$ using the comparison theorem.

See Solution

Problem 13622

Given the parametric equations $x=\sin(t), y=\sin(2t)$ :
a) Find where the tangent line is horizontal in the first quadrant.
b) Determine the Cartesian equations of the two tangents at the origin.

See Solution

Problem 13623

Find the value of the integral $\int_{-6}^{0} f(x) d x$ given that $\int_{0}^{6} f(x) d x = -4$ and $f(x)$ is odd.

See Solution

Problem 13624

Evaluate the integral $\int_{-a}^{a}\left(-4 x^{3}+5 x\right) d x$ and select the true statement from the options provided.

See Solution

Problem 13625

Find the limit of the sequence: $a_1 = \sqrt{3}, a_2 = \sqrt{3 a_1}, a_3 = \sqrt{3 a_2}, \ldots$

See Solution

Problem 13626

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\int_{2 x}^{x^{3}}\left(t^{3}-18\right)^{5} d t$ .

See Solution

Problem 13627

Evaluate the integral $\int_{-a}^{a}(2 x^{7}+2 x^{5}) dx$ and identify the correct equality from the options.

See Solution

Problem 13628

Approximate $\tan^{-1} 0.1$ to three decimal places using the Maclaurin series for $\tan^{-1} x$ .

See Solution

Problem 13629

Find the antiderivative of $\int \sqrt{\frac{1}{(x-2)^{2}}-\frac{1}{(x-4)^{2}}} d x$ .

See Solution

Problem 13630

A right triangle has legs of 36 in and 48 in. Short leg decreases by $5 \mathrm{in/sec}$ , long leg increases by $7 \mathrm{in/sec}$ . Find the hypotenuse's rate of change.

See Solution

Problem 13631

Find the radius of convergence for the series $\sum_{k=0}^{\infty} \frac{x^{k+2}}{k !}$ .

See Solution

Problem 13632

Find the values of $t \geq 0$ for which the speed of the particle, given by $v(t)=-t^{3}+2 t^{2}+2^{-t}$ , is increasing.

See Solution

Problem 13633

Find when the speed of a particle with velocity $v(t)=-t^{3}+2 t^{2}+2^{-t}$ is increasing for $t \geq 0$ . Options: (A) $(0,0.177)$ and $(1.256, \infty)$ (B) $(0,1.256)$ (C) $(0,2.057)$ (D) $(0.177,1.256)$ (E) $(0.177,1.256)$ and $(2.057, \infty)$ .

See Solution

Problem 13634

Find the limit: $\lim _{x \rightarrow \frac{\pi}{6}} \cot (x)$ .

See Solution

Problem 13635

Evaluate the integral: $\int 4 x^{2} e^{\frac{-x^{3}}{6}} d x$

See Solution

Problem 13636

Find the value of the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \sec (x)$ .

See Solution

Problem 13637

Differentiate the function $f(x) = x(x - 9 \sqrt{x})$ .

See Solution

Problem 13638

Find the limit: $\lim _{x \rightarrow 0} \tan (x)$ .

See Solution

Problem 13639

Find the limit as $x$ approaches $\pi$ for $\cos(x)$ . What is it?

See Solution

Problem 13640

Berechnen Sie die Durchschnittsgeschwindigkeit des Schlittens für $s(t)=\frac{1}{2} \cdot t^{2}$ in den ersten 5 Sekunden.

See Solution

Problem 13641

Find the derivative of the function $f(x)=\frac{x^{3}+7 x^{2}}{x}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 13642

Find the rate of water level rise in a 20-ft equilateral triangle trough when water is pumped in at $4 \mathrm{ft}^{3} / \mathrm{min}$ and is 1 ft deep.

See Solution

Problem 13643

Determine where the function $f(x)=\frac{x-3}{x^{2}}$ is increasing or decreasing.

See Solution

Problem 13644

Find the derivative of the function $f(x)=\sqrt{x}(8x^{2}+6x+4)$ . What is $f'(x)$ ?

See Solution

Problem 13645

Find the limit as $x$ approaches 3 for the expression $\frac{x^{3}-9 x}{x^{2}-3 x}$ .

See Solution

Problem 13646

Analyze the vertical motion of a softball with initial velocity $29 \mathrm{~m/s}$ and gravity $g=-9.8 \mathrm{~m/s}^2$ . Find: a. Velocity over time. b. Position over time. c. Time and height at the highest point. d. Time when it hits the ground.

See Solution

Problem 13647

Determine which inequality shows that the number of people in a building is increasing: $f(t)$ in, $g(t)$ out. Choices: (A) $f(t)>0$ , (B) $f^{\prime}(t)>0$ , (C) $f(t)-g(t)>0$ , (D) $f(t)-g^{\prime}(t)>0$ .

See Solution

Problem 13648

Find intervals where $y=f(x)$ is concave up/down and identify points of inflection for $f(x)=(x^{2}-9)e^{x}$ .

See Solution

Problem 13649

Calculate the left and right Riemann sums for $f(x)=\frac{1}{x}+2$ on $[1,5]$ with $n=4$ . Round to two decimal places.

See Solution

Problem 13650

Given the velocities (in mi/hr) of a car over 2 hours, sketch a curve through the points. Then, find the midpoint Riemann sum for $n=2$ and $n=4$ .

See Solution

Problem 13651

Find the function F such that F''(x) = cos x, F'(0) = 4, and F(π) = 4.

See Solution

Problem 13652

Determine which inequality shows that the rate of change of people in a building is increasing at time $t$ : (A) $f(t)>0$ (B) $f'(t)>0$ (C) $f(t)-g(t)>0$ (D) $f'(t)-g'(t)>0$

See Solution

Problem 13653

A cube's volume $V(x)=x^{3}$ is increasing at 40 cm³/min. Find the rate of change of $x$ when $x=2$ .

See Solution

Problem 13654

How fast is the area of a circular paint spill increasing at a radius of 18 inches, given the radius grows at 2.5 in/min?

See Solution

Problem 13655

Find the limit using l'Hôpital's Rule:
$\lim _{x \rightarrow-3} \frac{x^{2}+11 x+24}{-15-2 x+x^{2}}$
Rewrite it as:
$\lim _{x \rightarrow-3}(\square)$
Then, evaluate:
\lim _{x \rightarrow-3} \frac{x^{2}+11 x+24}{-15-2 x+x^{2}}=\square($ Simplify your answer. $)

See Solution

Problem 13656

A paint spill grows circularly with a radius increase of 2.5 in/min. Find the area increase rate when the radius is $r$ in. Options: (A) $5 \pi \mathrm{in}^{2} / \mathrm{min}$ (B) $36 \pi \mathrm{in}^{2} / \mathrm{min}$ (C) $45 \pi \mathrm{in}^{2} / \mathrm{min}$ (D) $90 \pi \mathrm{in}^{2} / \mathrm{min}$

See Solution

Problem 13657

What is the marginal cost of producing the $50,001^{\text{th}}$ pizza if total cost is \ $10,000.10? Average total cost is$ \square$.

See Solution

Problem 13658

Find the function $F$ such that $F''(x)=\cos x$ , $F'(0)=7$ , and $F(\pi)=7$ . What is $F(x)=\square$ ?

See Solution

Problem 13659

Calculate the integral from -8 to 5 of the function $-|x|$ .

See Solution

Problem 13660

A circle's radius increases at 0.5 m/s. Find the circumference's increase rate when the radius is 3 m. Choices: (A) 0.5 m/s, (B) 1.5 m/s, (C) $\pi$ m/s, (D) $3\pi$ m/s.

See Solution

Problem 13661

Evaluate the integrals: a. $\oint_{C} \frac{1-i z}{z^{2}+2 i z+3} d z = -4 \pi$ ; b. $\oint_{c} \frac{\cos (z)}{z^{5}} d z = 2 \pi i$ .

See Solution

Problem 13662

Evaluate the limit:
$\lim _{x \rightarrow-1} \frac{6 x^{4}+4 x^{3}+4 x+2}{x+1}$
using l'Hôpital's Rule. What is the result?

See Solution

Problem 13663

Find the dimensions and maximum area of a rectangle under the parabola $y=441-x^{2}$ on the $x$ -axis.
Area function: $A=\square$ .
Interval: $\square$ .
Shorter dimension: $\square$ , longer dimension: $\square$ .
Maximum area: $\square$ .

See Solution

Problem 13664

Find the rate of change of wave velocity $v(h) = 3 \sqrt{h}$ at depth $h = 2$ m. Choices: (A) $-\frac{3}{4 \sqrt{2}}$ , (B) $-\frac{3}{8 \sqrt{2}}$ , (C) $\frac{3}{2 \sqrt{2}}$ , (D) $\frac{3}{\sqrt{2}}$ , (E) $4 \sqrt{2}$ .

See Solution

Problem 13665

Find the critical points of $y(x)=39 x^{27/13}-81 \sqrt{x}$ . Provide answers as a comma-separated list or DNE if none exist.

See Solution

Problem 13666

Find the critical point of $f(x)=3 e^{-x}-15 e^{-2 x}$ . Is it a local minimum? True or False?

See Solution

Problem 13667

Find the critical points of $y(x)=39 x^{27/13}-81 \sqrt{x}$ . List them as $x=$ or DNE if none exist.

See Solution

Problem 13668

Deposit \$7000 at 5.6% annual interest compounded continuously. How long to double the investment? Round to nearest hundredth.

See Solution

Problem 13669

What indeterminate form does $\lim _{x \rightarrow \infty} e^{-x} x^{2}$ represent? a. $\infty-\infty$ b. $\infty+\infty$ c. $\frac{\infty}{\infty}$ d. $0 \cdot \infty$ e. $\frac{0}{0}$

See Solution

Problem 13670

Find the derivative of $f(x)=x^{3}$ and evaluate $f^{\prime}(-6)$ , $f^{\prime}(-0.5)$ , $f^{\prime}\left(\frac{2}{3}\right)$ , and $f^{\prime}(2)$ .

See Solution

Problem 13671

Find the integral that computes the moment of inertia for a unit-density spherical shell of radius $R$ about the $z$ -axis.

See Solution

Problem 13672

Find the correct integral for the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc $x^{2}+y^{2} \leq 1$ .

See Solution

Problem 13673

Find the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc using cylindrical coordinates.

See Solution

Problem 13674

Find the horizontal asymptote of the function $h(x)=\frac{x^{2}-8}{7 x^{4}+3}$ .

See Solution

Problem 13675

Find the curve passing through $(-1,2)$ with slope $\frac{d y}{d x}=6 x^{2}-8 x$ . What is the equation for $y$ ?

See Solution

Problem 13676

Find the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc $x^{2}+y^{2} \leq 1$ using cylindrical coordinates.

See Solution

Problem 13677

Find the largest rectangle's maximum area inscribed in $y=\frac{9-x}{5+x}$ and the axes. Round to three decimal places.

See Solution

Problem 13678

Find the derivative of the function $f(x)=2 x(\sin x+\cos x)$ , i.e., $f^{\prime}(x)=$ .

See Solution

Problem 13679

Use the intermediate value theorem to check if $f(x)=x^{3}+2 x^{2}-8 x-3$ has a real zero between $a=-8$ and $b=-3$ .

See Solution

Problem 13680

Find critical points of $f(x)=x^{3}-18 x^{2}+96 x$ and use the Second Derivative Test to classify them.

See Solution

Problem 13681

Differentiate the function $y=x^{5} \sin x \tan x$ . What is $y^{\prime}=?$

See Solution

Problem 13682

Find the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc using cylindrical coordinates.

See Solution

Problem 13683

Find the local maximum and minimum of the function $f(x)=x^{3}-48x$ . What are the $x$ values?

See Solution

Problem 13684

Explain with examples what "The derivative does not exist" means in calculus.

See Solution

Problem 13685

Find the absolute max and min of $f(x)=x-\frac{64 x}{x+4}$ on $[0,13]$ . What are the $x$ values?

See Solution

Problem 13686

Find critical points of $f(x)=-4 x^{4}+4 x^{2}-5$ and use the Second Derivative Test for local min/max.

See Solution

Problem 13687

Find the critical points for $y(x)=39 x^{27/13}-81 \sqrt{x}$ . Provide your answer in the form of a comma-separated list.

See Solution

Problem 13688

Find the critical point of $z = f(x, y) = x^{4} + y^{4} - x^{2} - xy - y^{2}$ where $f_x = 4x^{3} - 2x - y$ .

See Solution

Problem 13689

Bestimme die Tangentensteigung von $f(x)=\frac{1}{2}x^{3}$ bei $x_{0}=2$ .

See Solution

Problem 13690

Evaluate $\frac{f(a+h)-f(a)}{h}$ for $f(x)=2x^{2}-5x+1$ and $h \neq 0$ .

See Solution

Problem 13691

How far did a canteen fall in 4 seconds? Use the formula $d = \frac{1}{2}gt^2$ where $g \approx 9.8 \, m/s^2$ .

See Solution

Problem 13692

Find the limit: $\lim _{x \rightarrow+4} \frac{x^{2}-4 x}{x^{2}-2 x-8}$ .

See Solution

Problem 13693

Bestimmen Sie die Ableitungen der Funktionen: a) $f(x)=x^{3}+x^{2}$ b) $f(x)=x^{4}-x^{3}+5$ c) $f(x)=x+\sqrt{x}$ d) $f(x)=x^{2}-\frac{1}{x}-5$ e) $f(x)=5+x^{4}-\sqrt{x}$ f) $f(x)=x^{-6}+x^{-2}$

See Solution

Problem 13694

Beweisen Sie die Intervalladditivität für Integrale: $\int_{a}^{b} f(x) d x+\int_{b}^{c} f(x) d x=\int_{a}^{c} f(x) d x$

See Solution

Problem 13695

If $y=\left(x+\sqrt{x^{2}+a^{2}}\right)^{m}$ , prove that $(x^{2}+a^{2}) y_{n+2}+(2n+1) x y_{n+1}+(n^{2}-m^{2}) y_{n}=0$ .

See Solution

Problem 13696

Find the derivative of $f(x)=x^{2}-5x$ using first principles.

See Solution

Problem 13697

Gegeben ist $V(r)=\frac{4 \pi}{3} \cdot r^{3}$ für $r$ in $\mathrm{dm}$ .
Aufgabe: a) Berechne die Volumsänderung von $1 \mathrm{dm}$ auf $3 \mathrm{dm}$ . b) Bestimme die mittlere Volumsänderungsrate von $1 \mathrm{dm}$ auf $3 \mathrm{dm}$ . c) Finde die Volumsänderungsrate bei $r=2 \mathrm{dm}$ .

See Solution

Problem 13698

Bestimme die durchschnittliche Änderungsrate von $V(r)=\frac{4}{3} \pi r^3$ im Intervall $[r; z]$ und die Änderungsrate bei $r$ .

See Solution

Problem 13699

Find the derivative of $f(x)=3 x^{-4}-2 x^{7}$ .

See Solution

Problem 13700

Gegeben ist $V(t)=(10-t)^{2}$ . Berechne und interpretiere: a) $V(6)-V(0)$ , b) $\frac{V(6)-V(0)}{V(0)}$ , c) $\frac{V(6)-V(0)}{6}$ , d) $\frac{V(3,01)-V(3)}{3,01-3}$ . Erkläre $\lim _{z \rightarrow 3} \frac{V(z)-V(3)}{z-3}$ . Gib Ausdrücke für mittlere und momentane Volumsänderungsgeschwindigkeit an.

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 13601

Find the derivative of f(x)=ax3+cf(x)=a x^{3}+cf(x)=ax3+c and f(x)=1x2f(x)=\frac{1}{x^{2}}f(x)=x21​.

Problem 13602

Estimate the total light intensity ∫−10−610−6I(θ)dθ\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta∫−10−610−6​I(θ)dθ using the Midpoint Rule with n=10n=10n=10.

Problem 13603

Problem 13604

Estimate the total light intensity ∫−10−610−6I(θ)dθ\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta∫−10−610−6​I(θ)dθ using the Midpoint Rule with n=10n=10n=10.

Problem 13605

Convert D(t)D(t)D(t) from hours to seconds. Use Simpson's Rule to estimate A(43200)≈36003[sum of D(t)]A(43200) \approx \frac{3600}{3}[\text{sum of } D(t)]A(43200)≈33600​[sum of D(t)]. Round to the nearest thousand megabits.

Problem 13606

Find the tangent equation to the curve at t=0t=0t=0 for x=ln⁡at+bct+dx=\ln \frac{a t+b}{c t+d}x=lnct+dat+b​ and y=at+bct+dy=\frac{a t+b}{c t+d}y=ct+dat+b​.

Problem 13607

Problem 13608

Find ∫27f(x)dx\int_{2}^{7} f(x) d x∫27​f(x)dx given ∫17f(x)dx=−6\int_{1}^{7} f(x) d x=-6∫17​f(x)dx=−6 and ∫12f(x)dx=10\int_{1}^{2} f(x) d x=10∫12​f(x)dx=10.

Problem 13609

Compare the integrals F=∫15(x+5+4)dxF=\int_{1}^{5}(\sqrt{x+5}+4) dxF=∫15​(x+5​+4)dx and G=∫15(x+1+1)dxG=\int_{1}^{5}(\sqrt{x+1}+1) dxG=∫15​(x+1​+1)dx using ≤\leq≤ or ≥\geq≥. Provide your answer below: F□G F \square G F□G

Problem 13610

Find lower and upper bounds for F=∫−43(x+6+2)dxF=\int_{-4}^{3}(\sqrt{x+6}+2) d xF=∫−43​(x+6​+2)dx using the comparison theorem.

Problem 13611

A passenger on the London Eye rises at 0.032 m/s0.032 \mathrm{~m/s}0.032 m/s after 1 minute. Find their horizontal speed at that moment.

Problem 13612

Find the sum: ∑k=0∞(−2)kxk+1k+1\sum_{k=0}^{\infty} \frac{(-2)^{k} x^{k+1}}{k+1}∑k=0∞​k+1(−2)kxk+1​.

Problem 13613

Find ∫168f(x)dx\int_{1}^{6} 8 f(x) d x∫16​8f(x)dx given ∫91f(x)dx=−9\int_{9}^{1} f(x) d x=-9∫91​f(x)dx=−9 and ∫69f(x)dx=−8\int_{6}^{9} f(x) d x=-8∫69​f(x)dx=−8.

Problem 13614

Find the value of ∫152f(x)dx\int_{1}^{5} 2 f(x) d x∫15​2f(x)dx given ∫61f(x)dx=13\int_{6}^{1} f(x) d x=13∫61​f(x)dx=13 and ∫56f(x)dx=−2\int_{5}^{6} f(x) d x=-2∫56​f(x)dx=−2.

Problem 13615

Find lower and upper bounds for FFF using the comparison theorem, where F=∫23(6x+3)dx.F=\int_{2}^{3}(6 x+3) d x.F=∫23​(6x+3)dx.

Problem 13616

Find the best lower and upper bounds for FFF using the comparison theorem for the integral F=∫06(x+2+3)dxF=\int_{0}^{6}(\sqrt{x+2}+3) d xF=∫06​(x+2​+3)dx.

Problem 13617

Find lower and upper bounds for F=∫06(x+2+3)dxF = \int_{0}^{6}(\sqrt{x+2}+3) dxF=∫06​(x+2​+3)dx using the comparison theorem. □≤F≤□\square \leq F \leq \square□≤F≤□

Problem 13618

Find the tangent line(s) to the curve x=t5−4t3x=t^{5}-4 t^{3}x=t5−4t3 and y=t5y=t^{5}y=t5 at (0,32)(0,32)(0,32). Write as a Cartesian equation.

Problem 13619

Given the function f(x)=x3−4xf(x)=x^{3}-4xf(x)=x3−4x, find local max/min values and corresponding xxx values, rounded to two decimal places.

Problem 13620

Evaluate the integrals F=∫15(2x2+5)dxF = \int_{1}^{5}(2x^2 + 5)dxF=∫15​(2x2+5)dx and G=∫15(4x2+6)dxG = \int_{1}^{5}(4x^2 + 6)dxG=∫15​(4x2+6)dx. Which statements are true?

Problem 13621

Find the best lower and upper bounds for F=∫15(6x+5) dxF = \int_{1}^{5}(6x+5) \, dxF=∫15​(6x+5)dx using the comparison theorem.

Problem 13622

Given the parametric equations x=sin⁡(t),y=sin⁡(2t)x=\sin(t), y=\sin(2t)x=sin(t),y=sin(2t):a) Find where the tangent line is horizontal in the first quadrant. b) Determine the Cartesian equations of the two tangents at the origin.

Problem 13623

Find the value of the integral ∫−60f(x)dx\int_{-6}^{0} f(x) d x∫−60​f(x)dx given that ∫06f(x)dx=−4\int_{0}^{6} f(x) d x = -4∫06​f(x)dx=−4 and f(x)f(x)f(x) is odd.

Problem 13624

Evaluate the integral ∫−aa(−4x3+5x)dx\int_{-a}^{a}\left(-4 x^{3}+5 x\right) d x∫−aa​(−4x3+5x)dx and select the true statement from the options provided.

Problem 13625

Find the limit of the sequence: a1=3,a2=3a1,a3=3a2,…a_1 = \sqrt{3}, a_2 = \sqrt{3 a_1}, a_3 = \sqrt{3 a_2}, \ldotsa1​=3​,a2​=3a1​​,a3​=3a2​​,…

Problem 13626

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=∫2xx3(t3−18)5dtf(x)=\int_{2 x}^{x^{3}}\left(t^{3}-18\right)^{5} d tf(x)=∫2xx3​(t3−18)5dt.

Problem 13627

Evaluate the integral ∫−aa(2x7+2x5)dx\int_{-a}^{a}(2 x^{7}+2 x^{5}) dx∫−aa​(2x7+2x5)dx and identify the correct equality from the options.

Problem 13628

Approximate tan⁡−10.1\tan^{-1} 0.1tan−10.1 to three decimal places using the Maclaurin series for tan⁡−1x\tan^{-1} xtan−1x.

Problem 13629

Find the antiderivative of ∫1(x−2)2−1(x−4)2dx\int \sqrt{\frac{1}{(x-2)^{2}}-\frac{1}{(x-4)^{2}}} d x∫(x−2)21​−(x−4)21​​dx.

Problem 13630

A right triangle has legs of 36 in and 48 in. Short leg decreases by 5in/sec5 \mathrm{in/sec}5in/sec, long leg increases by 7in/sec7 \mathrm{in/sec}7in/sec. Find the hypotenuse's rate of change.

Problem 13631

Find the radius of convergence for the series ∑k=0∞xk+2k!\sum_{k=0}^{\infty} \frac{x^{k+2}}{k !}∑k=0∞​k!xk+2​.

Problem 13632

Find the values of t≥0t \geq 0t≥0 for which the speed of the particle, given by v(t)=−t3+2t2+2−tv(t)=-t^{3}+2 t^{2}+2^{-t}v(t)=−t3+2t2+2−t, is increasing.

Problem 13633

Problem 13634

Find the limit: lim⁡x→π6cot⁡(x)\lim _{x \rightarrow \frac{\pi}{6}} \cot (x)limx→6π​​cot(x).

Problem 13635

Evaluate the integral: ∫4x2e−x36dx\int 4 x^{2} e^{\frac{-x^{3}}{6}} d x∫4x2e6−x3​dx

Problem 13636

Find the value of the limit: lim⁡x→π2sec⁡(x)\lim _{x \rightarrow \frac{\pi}{2}} \sec (x)limx→2π​​sec(x).

Problem 13637

Differentiate the function f(x)=x(x−9x)f(x) = x(x - 9 \sqrt{x})f(x)=x(x−9x​).

Problem 13638

Find the limit: lim⁡x→0tan⁡(x)\lim _{x \rightarrow 0} \tan (x)limx→0​tan(x).

Problem 13639

Find the limit as xxx approaches π\piπ for cos⁡(x)\cos(x)cos(x). What is it?

Problem 13640

Berechnen Sie die Durchschnittsgeschwindigkeit des Schlittens für s(t)=12⋅t2s(t)=\frac{1}{2} \cdot t^{2}s(t)=21​⋅t2 in den ersten 5 Sekunden.

Problem 13641

Find the derivative of the function f(x)=x3+7x2xf(x)=\frac{x^{3}+7 x^{2}}{x}f(x)=xx3+7x2​. What is f′(x)f^{\prime}(x)f′(x)?

Find the derivative of $f(x)=a x^{3}+c$ and $f(x)=\frac{1}{x^{2}}$ .

Estimate the total light intensity $\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta$ using the Midpoint Rule with $n=10$ .

Estimate the total light intensity $\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta$ using the Midpoint Rule with $n=10$ .

Convert $D(t)$ from hours to seconds. Use Simpson's Rule to estimate $A(43200) \approx \frac{3600}{3}[\text{sum of } D(t)]$ . Round to the nearest thousand megabits.

Find the tangent equation to the curve at $t=0$ for $x=\ln \frac{a t+b}{c t+d}$ and $y=\frac{a t+b}{c t+d}$ .

Find $\int_{2}^{7} f(x) d x$ given $\int_{1}^{7} f(x) d x=-6$ and $\int_{1}^{2} f(x) d x=10$ .

Compare the integrals $F=\int_{1}^{5}(\sqrt{x+5}+4) dx$ and $G=\int_{1}^{5}(\sqrt{x+1}+1) dx$ using $\leq$ or $\geq$ .
Provide your answer below: $F \square G$

Find lower and upper bounds for $F=\int_{-4}^{3}(\sqrt{x+6}+2) d x$ using the comparison theorem.

A passenger on the London Eye rises at $0.032 \mathrm{~m/s}$ after 1 minute. Find their horizontal speed at that moment.

Find the sum: $\sum_{k=0}^{\infty} \frac{(-2)^{k} x^{k+1}}{k+1}$ .

Find $\int_{1}^{6} 8 f(x) d x$ given $\int_{9}^{1} f(x) d x=-9$ and $\int_{6}^{9} f(x) d x=-8$ .

Find the value of $\int_{1}^{5} 2 f(x) d x$ given $\int_{6}^{1} f(x) d x=13$ and $\int_{5}^{6} f(x) d x=-2$ .

Find lower and upper bounds for $F$ using the comparison theorem, where $F=\int_{2}^{3}(6 x+3) d x.$

Find the best lower and upper bounds for $F$ using the comparison theorem for the integral $F=\int_{0}^{6}(\sqrt{x+2}+3) d x$ .

Find lower and upper bounds for $F = \int_{0}^{6}(\sqrt{x+2}+3) dx$ using the comparison theorem. $\square \leq F \leq \square$

Find the tangent line(s) to the curve $x=t^{5}-4 t^{3}$ and $y=t^{5}$ at $(0,32)$ . Write as a Cartesian equation.

Given the function $f(x)=x^{3}-4x$ , find local max/min values and corresponding $x$ values, rounded to two decimal places.

Evaluate the integrals $F = \int_{1}^{5}(2x^2 + 5)dx$ and $G = \int_{1}^{5}(4x^2 + 6)dx$ . Which statements are true?

Find the best lower and upper bounds for $F = \int_{1}^{5}(6x+5) \, dx$ using the comparison theorem.

Given the parametric equations $x=\sin(t), y=\sin(2t)$ :
a) Find where the tangent line is horizontal in the first quadrant.
b) Determine the Cartesian equations of the two tangents at the origin.

Find the value of the integral $\int_{-6}^{0} f(x) d x$ given that $\int_{0}^{6} f(x) d x = -4$ and $f(x)$ is odd.

Evaluate the integral $\int_{-a}^{a}\left(-4 x^{3}+5 x\right) d x$ and select the true statement from the options provided.

Find the limit of the sequence: $a_1 = \sqrt{3}, a_2 = \sqrt{3 a_1}, a_3 = \sqrt{3 a_2}, \ldots$

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\int_{2 x}^{x^{3}}\left(t^{3}-18\right)^{5} d t$ .

Evaluate the integral $\int_{-a}^{a}(2 x^{7}+2 x^{5}) dx$ and identify the correct equality from the options.

Approximate $\tan^{-1} 0.1$ to three decimal places using the Maclaurin series for $\tan^{-1} x$ .

Find the antiderivative of $\int \sqrt{\frac{1}{(x-2)^{2}}-\frac{1}{(x-4)^{2}}} d x$ .

A right triangle has legs of 36 in and 48 in. Short leg decreases by $5 \mathrm{in/sec}$ , long leg increases by $7 \mathrm{in/sec}$ . Find the hypotenuse's rate of change.

Find the radius of convergence for the series $\sum_{k=0}^{\infty} \frac{x^{k+2}}{k !}$ .

Find the values of $t \geq 0$ for which the speed of the particle, given by $v(t)=-t^{3}+2 t^{2}+2^{-t}$ , is increasing.

Find the limit: $\lim _{x \rightarrow \frac{\pi}{6}} \cot (x)$ .

Evaluate the integral: $\int 4 x^{2} e^{\frac{-x^{3}}{6}} d x$

Find the value of the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \sec (x)$ .

Differentiate the function $f(x) = x(x - 9 \sqrt{x})$ .

Find the limit: $\lim _{x \rightarrow 0} \tan (x)$ .

Find the limit as $x$ approaches $\pi$ for $\cos(x)$ . What is it?

Berechnen Sie die Durchschnittsgeschwindigkeit des Schlittens für $s(t)=\frac{1}{2} \cdot t^{2}$ in den ersten 5 Sekunden.

Find the derivative of the function $f(x)=\frac{x^{3}+7 x^{2}}{x}$ . What is $f^{\prime}(x)$ ?

Find the rate of water level rise in a 20-ft equilateral triangle trough when water is pumped in at $4 \mathrm{ft}^{3} / \mathrm{min}$ and is 1 ft deep.

Determine where the function $f(x)=\frac{x-3}{x^{2}}$ is increasing or decreasing.

Find the derivative of the function $f(x)=\sqrt{x}(8x^{2}+6x+4)$ . What is $f'(x)$ ?

Find the limit as $x$ approaches 3 for the expression $\frac{x^{3}-9 x}{x^{2}-3 x}$ .

Analyze the vertical motion of a softball with initial velocity $29 \mathrm{~m/s}$ and gravity $g=-9.8 \mathrm{~m/s}^2$ . Find: a. Velocity over time. b. Position over time. c. Time and height at the highest point. d. Time when it hits the ground.

Determine which inequality shows that the number of people in a building is increasing: $f(t)$ in, $g(t)$ out. Choices: (A) $f(t)>0$ , (B) $f^{\prime}(t)>0$ , (C) $f(t)-g(t)>0$ , (D) $f(t)-g^{\prime}(t)>0$ .

Find intervals where $y=f(x)$ is concave up/down and identify points of inflection for $f(x)=(x^{2}-9)e^{x}$ .

Calculate the left and right Riemann sums for $f(x)=\frac{1}{x}+2$ on $[1,5]$ with $n=4$ . Round to two decimal places.

Given the velocities (in mi/hr) of a car over 2 hours, sketch a curve through the points. Then, find the midpoint Riemann sum for $n=2$ and $n=4$ .

Determine which inequality shows that the rate of change of people in a building is increasing at time $t$ : (A) $f(t)>0$ (B) $f'(t)>0$ (C) $f(t)-g(t)>0$ (D) $f'(t)-g'(t)>0$

A cube's volume $V(x)=x^{3}$ is increasing at 40 cm³/min. Find the rate of change of $x$ when $x=2$ .

What is the marginal cost of producing the $50,001^{\text{th}}$ pizza if total cost is \ $10,000.10? Average total cost is$ \square$.

Find the function $F$ such that $F''(x)=\cos x$ , $F'(0)=7$ , and $F(\pi)=7$ . What is $F(x)=\square$ ?

Calculate the integral from -8 to 5 of the function $-|x|$ .

A circle's radius increases at 0.5 m/s. Find the circumference's increase rate when the radius is 3 m. Choices: (A) 0.5 m/s, (B) 1.5 m/s, (C) $\pi$ m/s, (D) $3\pi$ m/s.

Evaluate the integrals: a. $\oint_{C} \frac{1-i z}{z^{2}+2 i z+3} d z = -4 \pi$ ; b. $\oint_{c} \frac{\cos (z)}{z^{5}} d z = 2 \pi i$ .

Evaluate the limit:
$\lim _{x \rightarrow-1} \frac{6 x^{4}+4 x^{3}+4 x+2}{x+1}$
using l'Hôpital's Rule. What is the result?

Find the dimensions and maximum area of a rectangle under the parabola $y=441-x^{2}$ on the $x$ -axis.
Area function: $A=\square$ .
Interval: $\square$ .
Shorter dimension: $\square$ , longer dimension: $\square$ .
Maximum area: $\square$ .

Find the rate of change of wave velocity $v(h) = 3 \sqrt{h}$ at depth $h = 2$ m. Choices: (A) $-\frac{3}{4 \sqrt{2}}$ , (B) $-\frac{3}{8 \sqrt{2}}$ , (C) $\frac{3}{2 \sqrt{2}}$ , (D) $\frac{3}{\sqrt{2}}$ , (E) $4 \sqrt{2}$ .

Find the critical points of $y(x)=39 x^{27/13}-81 \sqrt{x}$ . Provide answers as a comma-separated list or DNE if none exist.

Find the critical point of $f(x)=3 e^{-x}-15 e^{-2 x}$ . Is it a local minimum? True or False?

Find the critical points of $y(x)=39 x^{27/13}-81 \sqrt{x}$ . List them as $x=$ or DNE if none exist.

What indeterminate form does $\lim _{x \rightarrow \infty} e^{-x} x^{2}$ represent? a. $\infty-\infty$ b. $\infty+\infty$ c. $\frac{\infty}{\infty}$ d. $0 \cdot \infty$ e. $\frac{0}{0}$

Find the derivative of $f(x)=x^{3}$ and evaluate $f^{\prime}(-6)$ , $f^{\prime}(-0.5)$ , $f^{\prime}\left(\frac{2}{3}\right)$ , and $f^{\prime}(2)$ .

Find the integral that computes the moment of inertia for a unit-density spherical shell of radius $R$ about the $z$ -axis.

Find the correct integral for the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc $x^{2}+y^{2} \leq 1$ .

Find the surface area of $z=e^{-(x^{2}+y^{2})}$ over the unit disc using cylindrical coordinates.

Find the horizontal asymptote of the function $h(x)=\frac{x^{2}-8}{7 x^{4}+3}$ .