Calculus

Problem 14001

Find intervals where the drug concentration function $K(t)=\frac{8 t}{t^{2}+4}$ is increasing or decreasing for $0<t<\infty$ .

See Solution

Problem 14002

Find the points on the graph of $f(x)=x^{3}-2 x^{2}+x$ where the gradient equals 1.

See Solution

Problem 14003

Show that the polynomial $f(x)=6x^{3}+6x^{2}-4x+6$ has a zero in $[-6,-1]$ using the intermediate value theorem. Find $f(-6)$ and $f(-1)$ .

See Solution

Problem 14004

Show that the polynomial $f(x)=8 x^{4}-3 x^{2}+4 x-1$ has a zero in the interval [0,1]. Find $f(0)$ .

See Solution

Problem 14005

Find the derivative $f^{\prime}(x)$ , partition numbers, and critical numbers for $f(x)=x^{3}-48x-4$ .

See Solution

Problem 14006

Find $\lim _{x \rightarrow 1} g(f(x))$ for $f(x)=5-2x$ and $g(x)=x^{3}$ . Options: (A) 64 (B) 9 (C) 27 (D) none.

See Solution

Problem 14007

Find the area between the function $f(x)=2x+6$ and the $x$ -axis from $x=-7$ to $x=0$ .

See Solution

Problem 14008

Find the derivative of the function $f(x)=2 \sin (x) \sin ^{-1}(x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 14009

Find the area between the line $f(x)=-x-4$ and the $x$ -axis from $x=-8$ to $x=6$ .

See Solution

Problem 14010

Calculate the area between $f(x)=3x+8$ and the $x$ -axis from $x=-4$ to $x=-1$ . Provide the answer as a fraction.

See Solution

Problem 14011

Find the area between the line $f(x)=2x$ and the $x$ -axis from $x=-5$ to $x=5$ .

See Solution

Problem 14012

Find $\frac{d y}{d x}$ for the equation $y^{7} \ln(x^{2}) - 10x^{2} - y^{9} = -4$ .

See Solution

Problem 14013

Find the global max and min of $g(\theta)=3 \theta-5 \sin (\theta)$ on $[0, \pi]$ . Round to the nearest thousandth.

See Solution

Problem 14014

Find the limit using L'Hospital's rule: $\lim _{x \rightarrow 0} \frac{1-\cos 8 x}{1-\cos 7 x}.$

See Solution

Problem 14015

Find the absolute max and min of $f(x)=3x^{2}-10x+7$ on $[0,6]$ . Give $x$ values and their corresponding function values.

See Solution

Problem 14016

The bacteria count is $n(t)=990 e^{0.25 t}$ . Find the growth rate, initial population, and count at $t=5$ .

See Solution

Problem 14017

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

See Solution

Problem 14018

Find the absolute maximum and minimum of $f(x)=\frac{x}{x^{2}-x+25}$ for $0 \leq x \leq 27$ .

See Solution

Problem 14019

Find the absolute extreme values of the function $f(x)=3x+\frac{243}{x}$ for $0<x<\infty$ .

See Solution

Problem 14020

Find the derivative of the function $f(x)=2 x^{3}+24 x^{2}-120 x+9$ on $[-10,3]$ and identify its min and max values.

See Solution

Problem 14021

Find the derivative $\frac{d y}{d x}$ for the equation $y=\ln(6 x^{2}+y^{2})$ .

See Solution

Problem 14022

Find the first derivative of $g(x)=8 x^{3}+60 x^{2}+96 x$ and check concavity at $x=-4$ using $g^{\prime \prime}(x)$ .

See Solution

Problem 14023

1. Find the line integral $\oint_{c} \bar{z} d z$ for the curve $c:|z|=2$ .
2. Compute the line integral $\oint_{c}(1+z+z^{2}+z^{3}) d z$ .
3. Evaluate $\oint \frac{1-i z}{z^{2}+n^{2}-z} d z$ for complex $z$ and real $n$ .

See Solution

Problem 14024

Find the first and second derivatives of $f(x)=x^{3}+6 x^{2}-96 x+19$ . Determine intervals where $f$ is increasing, decreasing, concave up, and concave down.

See Solution

Problem 14025

Find $\frac{d y}{d x}$ for the equation $y^{7} \ln(x^{2}) - 10x^{2} - y^{9} = -4$ .

See Solution

Problem 14026

Find the min and max of $f(x)=x^{4}-32x^{2}+3$ on $[-3,9]$ using derivatives and endpoint values.

See Solution

Problem 14027

Find the limit of the sequence $a_{n}=\sqrt{n^{4}+3 n^{2}+5}-\sqrt{n^{4}-n^{2}+n}$ , or explain why it doesn't exist.

See Solution

Problem 14028

An electron starts far from a stationary proton. What is its speed at $0.07 \mathrm{~m}$ from the proton?

See Solution

Problem 14029

Analyze and sketch the graph of $y=\frac{x}{x^{2}+49}$ . Find intercepts, extrema, inflection points, and asymptotes.

See Solution

Problem 14030

Analyze and sketch the graph of $y=\frac{x}{x^{2}+49}$ , finding intercepts, extrema, inflection points, and asymptotes.

See Solution

Problem 14031

Find relative extrema for $f(x)=x^{3}-9x^{2}+9$ using the Second Derivative Test. Enter DNE if none exist.

See Solution

Problem 14032

Find the integral $\int_{1}^{2} f(x) d x$ given $\int_{0}^{1} f(x) d x=-1$ , $\int_{0}^{2} f(x) d x=2$ , and $\int_{1}^{4} f(x) d x=11$ .

See Solution

Problem 14033

Find $\int_{4}^{-2}(f(x)-9 g(x)) d x$ given $\int_{-2}^{1} f(x) d x=4$ , $\int_{1}^{4} f(x) d x=-6$ , and $\int_{-2}^{4} g(x) d x=4$ .

See Solution

Problem 14034

Estimate the number of people hearing a rumor on day 14 using the model $f(x)=\frac{1,000}{1+499 e^{-0.6030 x}}$ . Round to the nearest person.

See Solution

Problem 14035

Find the integral $\int_{0}^{4} f(x) d x$ given $\int_{0}^{1} f(x) d x=-2$ , $\int_{0}^{2} f(x) d x=5$ , and $\int_{1}^{4} f(x) d x=7$ .

See Solution

Problem 14036

Calculate the Riemann sum $R(f, P, C)$ for $f(x)=31x+28$ , $P=\{-4,-1,1,4,8\}$ , and $C=\{-2,0,3,7\}$ .

See Solution

Problem 14037

Find the integrals $\int_{0}^{4} g(t) dt$ and $\int_{1}^{5} g(t) dt$ for the triangle $g(t)$ with vertices (1,0), (4,0), (3,2).

See Solution

Problem 14038

Find the Riemann sum $R(f, P, C)$ for $f(x)=x^{2}+7x$ , with partition $P=\{4,7,9,12\}$ and points $C=\{4,7.5,11.5\}$ . Give your answer to two decimal places. $R(f, P, C)=$

See Solution

Problem 14039

Find the inflection points of $f(x)=x+7 \cos x$ on $[0,2\pi]$ . Describe concavity in interval notation.

See Solution

Problem 14040

Berechne die Fallzeit und Endgeschwindigkeit eines Steins vom Freiburger Münster ( $h=116 \mathrm{~m}$ ) und Eiffelturm ( $h=300 \mathrm{~m}$ ).

See Solution

Problem 14041

Evaluate the integral $\int_{1}^{\infty} 21 x^{-3} dx$ using limits.

See Solution

Problem 14042

Untersuche die Stetigkeit der Funktion $f(x)=\left\{\begin{array}{cl} \sin (x) & \text { für } x \leq 0 \\ \frac{1}{x} & \text { für } x>0 \end{array}\right.$ . Gibt es eine Unstetigkeit?

See Solution

Problem 14043

Max springt aus 10 m Höhe. Berechne die Sprungdauer und Eintauchgeschwindigkeit ( $t=1,43 \mathrm{~s}$ ). Paula springt 3 m hoch. Wie lange nach Max kommt sie ins Wasser? ( $\Delta t=0,54 \mathrm{~s}$ )

See Solution

Problem 14044

A coffee at $130^{\circ} \mathrm{F}$ cools in a freezer at $0^{\circ} \mathrm{F}$ . After 15 min, it's $39^{\circ} \mathrm{F}$ . Find its temp after 20 min. Answer: $\square^{\circ} \mathrm{F}$ .

See Solution

Problem 14045

Find the function $f(x)$ if its derivative is $f'(x)=e^{2} x+e x^{2}+2 e x$ .

See Solution

Problem 14046

Bestimmen Sie die Ableitung von $x^{2}+2 e+1$ .

See Solution

Problem 14047

Find the function $f(x)$ if its derivative is $f'(x)=e^x+x+2$ .

See Solution

Problem 14048

Bestimmen Sie die Ableitung von $f(x)=e x^{3}+e^{3}+e^{x}$ .

See Solution

Problem 14049

A 17 ft ladder leans against a wall. If the top slips down at $2 \mathrm{ft/s}$ , how fast is the foot moving when the top is 16 ft up?

See Solution

Problem 14050

A triangle's height increases at $2 \mathrm{~cm/min}$ and area at $3 \mathrm{~cm}^2/\mathrm{min}$ . Find base's rate of change when height is 8 cm and area is 91 cm².

See Solution

Problem 14051

Invest \$6000 at 2% interest compounded continuously. Find account value after 6 years and time to reach \$75000.

See Solution

Problem 14052

Invest \$10000 at 7% interest compounded continuously. (a) Find the value after 6 years. (b) When will it reach \$24000?

See Solution

Problem 14053

Analyze the curve $y=3 x^{4}-24 x^{3}$ for concavity, inflection points, and local maxima.

See Solution

Problem 14054

Analyze the curve $y=3 x^{4}-24 x^{3}$ for concavity, inflection points, and local extrema. Sketch the curve.

See Solution

Problem 14055

Find critical numbers of $f(x)=3x^{4}-24x^{3}$ by solving $f^{\prime}(x)=0$ . Use the Second Derivative Test for concavity.

See Solution

Problem 14056

Find the limit: $\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}$ .

See Solution

Problem 14057

Given the function $f(x)=x^{3}-3 x^{2}-9 x+2$ , determine:
(a) Intervals where $f$ is increasing and decreasing. (b) Local min and max values. (c) Inflection point $(x, y)=(\square)$ and intervals for concavity.

See Solution

Problem 14058

Find the inflection point of the function $f(x)=x^{3}-3x^{2}-9x+2$ .

See Solution

Problem 14059

Find the interval where the function $f$ is increasing given its derivative $f^{\prime}(x)=(x+1)^{2}(x-4)^{7}(x-6)^{4}$ .

See Solution

Problem 14060

Find the interval where the function $f$ is increasing, given its derivative $f^{\prime}(x)=(x+1)^{2}(x-4)^{7}(x-6)^{4}$ .

See Solution

Problem 14061

Find the interval where the function $f$ is increasing given $f^{\prime}(x)=(x+1)^{2}(x-4)^{7}(x-6)^{4}$ . Answer: $(-1,4) \cup(6, \infty)$ .

See Solution

Problem 14062

Find the average value $g_{\text{ave}}$ of $g(t)=\frac{9}{1+t^{2}}$ over the interval $[0,5]$ .

See Solution

Problem 14063

Analyze the function $f(x)=-x^{3}$ as $x \rightarrow -\infty$ . What does $f(x)$ approach? $\infty$ , $-\infty$ , or 0?

See Solution

Problem 14064

Find the average value of the function $f(x)=3x^2+8x$ over the interval $[-1,3]$ .

See Solution

Problem 14065

Given the function $f(x)=e^{4 x}+e^{-x}$ , find where $f$ is increasing and decreasing, and its local min and max values.

See Solution

Problem 14066

Analyze the limits of $f(x)=-x^{3}$ as $x \to -\infty$ and $x \to \infty$ . What do they approach?

See Solution

Problem 14067

Evaluate the limits of the polynomial function $f(r)=-3 r^{9}+r^{8}+3 r^{6}+2$ as $r \rightarrow -\infty$ and $r \rightarrow \infty$ .

See Solution

Problem 14068

Given the function $f(x)=e^{4 x}+e^{-x}$ , find:
(a) Intervals where $f$ is increasing and decreasing.
(b) Local minimum and maximum values.
(c) Inflection point $(x, y)=(\square)$ and intervals of concavity.

See Solution

Problem 14069

Evaluate the integral from 0 to 4 of $\frac{x^{2}+x+1}{(x+1)^{2}(x+2)}$ dx.

See Solution

Problem 14070

Calculate the integral: $\int \frac{8 t-3}{t+1} d t$

See Solution

Problem 14071

Calculate the integral: $\int_{0}^{9} \frac{d t}{\sqrt{81+t^{2}}}$ .

See Solution

Problem 14072

Find the interval where the function $f(x)=(x^{2}-12)e^{-x}$ is concave up for $x>0$ . Answer in the form $(, )$ .

See Solution

Problem 14073

Find the initial number of flu cases using the function $N(t)=\frac{10,000}{1+999 e^{-t}}$ at $t=0$ . Round to the nearest person.

See Solution

Problem 14074

Find the increase in car sales over 6 years with advertising: from $5 + 0.5 t^{3/2}$ to $5 e^{0.3 t}$ .

See Solution

Problem 14075

Find how many times faster the growth rate of U.S. citizens aged 65+ from 2000-2050 is than from 1910-1960 using $R(t)=0.063 t^{2}-0.48 t+3.87$ .

See Solution

Problem 14076

A rare isotope decays at $12\%$ per second. How much remains after 8 seconds from an initial 6 grams? Round to the nearest gram.

See Solution

Problem 14077

Find $F'(3)$ given $F(x)=f(g(x))$ , $g(3)=6$ , $g'(3)=4$ , $f'(3)=-7$ , and $f'(6)=7$ .

See Solution

Problem 14078

Find how many times faster the growth rate of U.S. citizens aged 65+ from 2000-2050 is compared to 1910-1960. Round to two decimals.

See Solution

Problem 14079

Esquissez le graphique de la fonction $f(x)$ avec les informations suivantes : $f'(x)$ change de signe à -1 et 1, $f''(x)$ à 0.

See Solution

Problem 14080

Find how many times faster the growth rate of U.S. citizens aged 65+ from 2000 to 2050 is compared to 1910 to 1960 using $R(t)=0.063 t^{2}-0.48 t+3.87$ . Round to two decimal places.

See Solution

Problem 14081

Find $f'(1)$ for $f(x)=h(g(x) k(x))$ given $g(1)=7$ , $k(1)=0$ , $h(1)=-2$ , $g'(1)=5$ , $k'(1)=-1$ , $h'(1)=1$ , $h'(0)=3$ .

See Solution

Problem 14082

Find the expectation $\mathbb{E}$ and variance $\mathbb{V}$ of the PDF $\rho(x)=\frac{2}{\pi} \frac{1}{x^{2}+1}$ on $[0,+\infty)$ . Check for convergence.

See Solution

Problem 14083

Evaluate the integrals: (i) $\int \sec ^{5}\left(\frac{\theta}{2}\right) \tan \left(\frac{\theta}{2}\right) d \theta$ (ii) $\int \frac{\cos t}{1+\cos t} d t$

See Solution

Problem 14084

Given $F(x)=f(f(x))$ and $G(x)=(F(x))^{2}$ with $f(8)=5$ , $f(5)=2$ , $f^{\prime}(5)=10$ , $f^{\prime}(8)=2$ , find $F^{\prime}(8)$ and $G^{\prime}(8)$ .

See Solution

Problem 14085

Find $\left(f^{-1}\right)^{\prime}(-8)$ for the function $f(x)=x^{3}+6x-1$ . If no answer, enter DNE.

See Solution

Problem 14086

Find the linear approximation for $f(-0.5)$ using the values of $f(x)$ and $f^{\prime}(x)$ from the table.

See Solution

Problem 14087

Find the derivative of $y=\ln(t^2)-\arctan\left(\frac{t}{2}\right)$ .

See Solution

Problem 14088

Find the derivative of $f(x)=\operatorname{arccsc}(7x)$ . What is $f'(x)=\square$ ?

See Solution

Problem 14089

Find the derivative of $f(x)=\sqrt{\sin(e^{x^{5} \sin(x)})}$ .

See Solution

Problem 14090

Given $F(x)=f(f(x))$ and $G(x)=(F(x))^{2}$ with $f(7)=9$ , $f(9)=3$ , $f^{\prime}(9)=15$ , $f^{\prime}(7)=10$ , find $F^{\prime}(7)$ and $G^{\prime}(7)$ .

See Solution

Problem 14091

Find the derivative of $f(x)=\sqrt{\cos(e^{x^{4} \sin(x)})}$ .

See Solution

Problem 14092

Find the derivative of $y=\frac{1}{2}\left[x \sqrt{25-x^{2}}+25 \arcsin \left(\frac{x}{5}\right)\right]$ .

See Solution

Problem 14093

Find $\left(f^{-1}\right)^{\prime}(-8)$ for the function $f(x)=x^{3}+6x-1$ . Enter DNE if it doesn't exist.

See Solution

Problem 14094

Evaluate the integral: $\int_{-3}^{-2} x^{2} e^{x^{3}+27} d x$ using a change of variables.

See Solution

Problem 14095

Find the limit as $x$ approaches 3 for the expression $3 x^{3}-7 x-2$ .

See Solution

Problem 14096

Evaluate $\lim _{x \rightarrow \infty} \frac{8 x^{3}-5 x^{2}}{2 x^{3}+7 x^{2}}$ . Options: A. $\infty$ , B. 8, C. 4, D. 0.

See Solution

Problem 14097

Find the limit as $x$ approaches 3 for the expression $\frac{2 x^{2}-x-15}{x-3}$ .

See Solution

Problem 14098

What is the best choice for $dv$ in $\int x^{n} e^{a x} dx$ for integration by parts?
A. $x^{n} dx$ B. $x^{n}$ C. $e^{ax}$ D. $e^{2x} dx$

See Solution

Problem 14099

Given curve $C$ : $2 x^{2}+a x y+y^{2}=56$ .
(a) Prove $\frac{d y}{d x}=-\frac{4 x+a y}{a x+2 y}$ . (b) For normal line $L: x+y+4=0$ at $P(2,-6)$ , find gradient and value of $a$ . (c) Show points $Q$ and $R$ on $C$ with horizontal tangents are $(2,8)$ and $(-2,-8)$ .

See Solution

Problem 14100

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{6 x^{4}-3 x^{3}+2 x^{2}+4}{6 x^{3}+1}$ .

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 14001

Find intervals where the drug concentration function K(t)=8tt2+4K(t)=\frac{8 t}{t^{2}+4}K(t)=t2+48t​ is increasing or decreasing for 0<t<∞0<t<\infty0<t<∞.

Problem 14002

Find the points on the graph of f(x)=x3−2x2+xf(x)=x^{3}-2 x^{2}+xf(x)=x3−2x2+x where the gradient equals 1.

Problem 14003

Show that the polynomial f(x)=6x3+6x2−4x+6f(x)=6x^{3}+6x^{2}-4x+6f(x)=6x3+6x2−4x+6 has a zero in [−6,−1][-6,-1][−6,−1] using the intermediate value theorem. Find f(−6)f(-6)f(−6) and f(−1)f(-1)f(−1).

Problem 14004

Show that the polynomial f(x)=8x4−3x2+4x−1f(x)=8 x^{4}-3 x^{2}+4 x-1f(x)=8x4−3x2+4x−1 has a zero in the interval [0,1]. Find f(0)f(0)f(0).

Problem 14005

Find the derivative f′(x)f^{\prime}(x)f′(x), partition numbers, and critical numbers for f(x)=x3−48x−4f(x)=x^{3}-48x-4f(x)=x3−48x−4.

Problem 14006

Find lim⁡x→1g(f(x))\lim _{x \rightarrow 1} g(f(x))limx→1​g(f(x)) for f(x)=5−2xf(x)=5-2xf(x)=5−2x and g(x)=x3g(x)=x^{3}g(x)=x3. Options: (A) 64 (B) 9 (C) 27 (D) none.

Problem 14007

Find the area between the function f(x)=2x+6f(x)=2x+6f(x)=2x+6 and the xxx-axis from x=−7x=-7x=−7 to x=0x=0x=0.

Problem 14008

Find the derivative of the function f(x)=2sin⁡(x)sin⁡−1(x)f(x)=2 \sin (x) \sin ^{-1}(x)f(x)=2sin(x)sin−1(x). What is f′(x)f^{\prime}(x)f′(x)?

Problem 14009

Find the area between the line f(x)=−x−4f(x)=-x-4f(x)=−x−4 and the xxx-axis from x=−8x=-8x=−8 to x=6x=6x=6.

Problem 14010

Calculate the area between f(x)=3x+8f(x)=3x+8f(x)=3x+8 and the xxx-axis from x=−4x=-4x=−4 to x=−1x=-1x=−1. Provide the answer as a fraction.

Problem 14011

Find the area between the line f(x)=2xf(x)=2xf(x)=2x and the xxx-axis from x=−5x=-5x=−5 to x=5x=5x=5.

Problem 14012

Find dydx\frac{d y}{d x}dxdy​ for the equation y7ln⁡(x2)−10x2−y9=−4y^{7} \ln(x^{2}) - 10x^{2} - y^{9} = -4y7ln(x2)−10x2−y9=−4.

Problem 14013

Find the global max and min of g(θ)=3θ−5sin⁡(θ)g(\theta)=3 \theta-5 \sin (\theta)g(θ)=3θ−5sin(θ) on [0,π][0, \pi][0,π]. Round to the nearest thousandth.

Problem 14014

Find the limit using L'Hospital's rule: lim⁡x→01−cos⁡8x1−cos⁡7x.\lim _{x \rightarrow 0} \frac{1-\cos 8 x}{1-\cos 7 x}.x→0lim​1−cos7x1−cos8x​.

Problem 14015

Find the absolute max and min of f(x)=3x2−10x+7f(x)=3x^{2}-10x+7f(x)=3x2−10x+7 on [0,6][0,6][0,6]. Give xxx values and their corresponding function values.

Problem 14016

The bacteria count is n(t)=990e0.25tn(t)=990 e^{0.25 t}n(t)=990e0.25t. Find the growth rate, initial population, and count at t=5t=5t=5.

Problem 14017

Evaluate the integrals: a. ∮C1−izz2+2iz+3dz\oint_{C} \frac{1-i z}{z^{2}+2 i z+3} d z∮C​z2+2iz+31−iz​dz and b. ∮ccos⁡(z)z5dz\oint_{c} \frac{\cos (z)}{z^{5}} d z∮c​z5cos(z)​dz.

Problem 14018

Find the absolute maximum and minimum of f(x)=xx2−x+25f(x)=\frac{x}{x^{2}-x+25}f(x)=x2−x+25x​ for 0≤x≤270 \leq x \leq 270≤x≤27.

Problem 14019

Find the absolute extreme values of the function f(x)=3x+243xf(x)=3x+\frac{243}{x}f(x)=3x+x243​ for 0<x<∞0<x<\infty0<x<∞.

Problem 14020

Find the derivative of the function f(x)=2x3+24x2−120x+9f(x)=2 x^{3}+24 x^{2}-120 x+9f(x)=2x3+24x2−120x+9 on [−10,3][-10,3][−10,3] and identify its min and max values.

Problem 14021

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation y=ln⁡(6x2+y2)y=\ln(6 x^{2}+y^{2})y=ln(6x2+y2).

Problem 14022

Find the first derivative of g(x)=8x3+60x2+96xg(x)=8 x^{3}+60 x^{2}+96 xg(x)=8x3+60x2+96x and check concavity at x=−4x=-4x=−4 using g′′(x)g^{\prime \prime}(x)g′′(x).

Problem 14023

Problem 14024

Find the first and second derivatives of f(x)=x3+6x2−96x+19f(x)=x^{3}+6 x^{2}-96 x+19f(x)=x3+6x2−96x+19. Determine intervals where fff is increasing, decreasing, concave up, and concave down.

Problem 14025

Find dydx\frac{d y}{d x}dxdy​ for the equation y7ln⁡(x2)−10x2−y9=−4y^{7} \ln(x^{2}) - 10x^{2} - y^{9} = -4y7ln(x2)−10x2−y9=−4.

Problem 14026

Find the min and max of f(x)=x4−32x2+3f(x)=x^{4}-32x^{2}+3f(x)=x4−32x2+3 on [−3,9][-3,9][−3,9] using derivatives and endpoint values.

Problem 14027

Find the limit of the sequence an=n4+3n2+5−n4−n2+na_{n}=\sqrt{n^{4}+3 n^{2}+5}-\sqrt{n^{4}-n^{2}+n}an​=n4+3n2+5​−n4−n2+n​, or explain why it doesn't exist.

Problem 14028

An electron starts far from a stationary proton. What is its speed at 0.07 m0.07 \mathrm{~m}0.07 m from the proton?

Problem 14029

Analyze and sketch the graph of y=xx2+49y=\frac{x}{x^{2}+49}y=x2+49x​. Find intercepts, extrema, inflection points, and asymptotes.

Problem 14030

Analyze and sketch the graph of y=xx2+49y=\frac{x}{x^{2}+49}y=x2+49x​, finding intercepts, extrema, inflection points, and asymptotes.

Problem 14031

Find relative extrema for f(x)=x3−9x2+9f(x)=x^{3}-9x^{2}+9f(x)=x3−9x2+9 using the Second Derivative Test. Enter DNE if none exist.

Problem 14032

Find the integral ∫12f(x)dx\int_{1}^{2} f(x) d x∫12​f(x)dx given ∫01f(x)dx=−1\int_{0}^{1} f(x) d x=-1∫01​f(x)dx=−1, ∫02f(x)dx=2\int_{0}^{2} f(x) d x=2∫02​f(x)dx=2, and ∫14f(x)dx=11\int_{1}^{4} f(x) d x=11∫14​f(x)dx=11.

Problem 14033

Find ∫4−2(f(x)−9g(x))dx\int_{4}^{-2}(f(x)-9 g(x)) d x∫4−2​(f(x)−9g(x))dx given ∫−21f(x)dx=4\int_{-2}^{1} f(x) d x=4∫−21​f(x)dx=4, ∫14f(x)dx=−6\int_{1}^{4} f(x) d x=-6∫14​f(x)dx=−6, and ∫−24g(x)dx=4\int_{-2}^{4} g(x) d x=4∫−24​g(x)dx=4.

Problem 14034

Estimate the number of people hearing a rumor on day 14 using the model f(x)=1,0001+499e−0.6030xf(x)=\frac{1,000}{1+499 e^{-0.6030 x}}f(x)=1+499e−0.6030x1,000​. Round to the nearest person.

Problem 14035

Find the integral ∫04f(x)dx\int_{0}^{4} f(x) d x∫04​f(x)dx given ∫01f(x)dx=−2\int_{0}^{1} f(x) d x=-2∫01​f(x)dx=−2, ∫02f(x)dx=5\int_{0}^{2} f(x) d x=5∫02​f(x)dx=5, and ∫14f(x)dx=7\int_{1}^{4} f(x) d x=7∫14​f(x)dx=7.

Problem 14036

Calculate the Riemann sum R(f,P,C)R(f, P, C)R(f,P,C) for f(x)=31x+28f(x)=31x+28f(x)=31x+28, P={−4,−1,1,4,8}P=\{-4,-1,1,4,8\}P={−4,−1,1,4,8}, and C={−2,0,3,7}C=\{-2,0,3,7\}C={−2,0,3,7}.

Problem 14037

Find the integrals ∫04g(t)dt\int_{0}^{4} g(t) dt∫04​g(t)dt and ∫15g(t)dt\int_{1}^{5} g(t) dt∫15​g(t)dt for the triangle g(t)g(t)g(t) with vertices (1,0), (4,0), (3,2).

Problem 14038

Find the Riemann sum R(f,P,C)R(f, P, C)R(f,P,C) for f(x)=x2+7xf(x)=x^{2}+7xf(x)=x2+7x, with partition P={4,7,9,12}P=\{4,7,9,12\}P={4,7,9,12} and points C={4,7.5,11.5}C=\{4,7.5,11.5\}C={4,7.5,11.5}. Give your answer to two decimal places. R(f,P,C)= R(f, P, C)= R(f,P,C)=

Problem 14039

Find the inflection points of f(x)=x+7cos⁡xf(x)=x+7 \cos xf(x)=x+7cosx on [0,2π][0,2\pi][0,2π]. Describe concavity in interval notation.

Problem 14040

Berechne die Fallzeit und Endgeschwindigkeit eines Steins vom Freiburger Münster (h=116 mh=116 \mathrm{~m}h=116 m) und Eiffelturm (h=300 mh=300 \mathrm{~m}h=300 m).

Find intervals where the drug concentration function $K(t)=\frac{8 t}{t^{2}+4}$ is increasing or decreasing for $0<t<\infty$ .

Find the points on the graph of $f(x)=x^{3}-2 x^{2}+x$ where the gradient equals 1.

Show that the polynomial $f(x)=6x^{3}+6x^{2}-4x+6$ has a zero in $[-6,-1]$ using the intermediate value theorem. Find $f(-6)$ and $f(-1)$ .

Show that the polynomial $f(x)=8 x^{4}-3 x^{2}+4 x-1$ has a zero in the interval [0,1]. Find $f(0)$ .

Find the derivative $f^{\prime}(x)$ , partition numbers, and critical numbers for $f(x)=x^{3}-48x-4$ .

Find $\lim _{x \rightarrow 1} g(f(x))$ for $f(x)=5-2x$ and $g(x)=x^{3}$ . Options: (A) 64 (B) 9 (C) 27 (D) none.

Find the area between the function $f(x)=2x+6$ and the $x$ -axis from $x=-7$ to $x=0$ .

Find the derivative of the function $f(x)=2 \sin (x) \sin ^{-1}(x)$ . What is $f^{\prime}(x)$ ?

Find the area between the line $f(x)=-x-4$ and the $x$ -axis from $x=-8$ to $x=6$ .

Calculate the area between $f(x)=3x+8$ and the $x$ -axis from $x=-4$ to $x=-1$ . Provide the answer as a fraction.

Find the area between the line $f(x)=2x$ and the $x$ -axis from $x=-5$ to $x=5$ .

Find $\frac{d y}{d x}$ for the equation $y^{7} \ln(x^{2}) - 10x^{2} - y^{9} = -4$ .

Find the global max and min of $g(\theta)=3 \theta-5 \sin (\theta)$ on $[0, \pi]$ . Round to the nearest thousandth.

Find the limit using L'Hospital's rule: $\lim _{x \rightarrow 0} \frac{1-\cos 8 x}{1-\cos 7 x}.$

Find the absolute max and min of $f(x)=3x^{2}-10x+7$ on $[0,6]$ . Give $x$ values and their corresponding function values.

The bacteria count is $n(t)=990 e^{0.25 t}$ . Find the growth rate, initial population, and count at $t=5$ .

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

Find the absolute maximum and minimum of $f(x)=\frac{x}{x^{2}-x+25}$ for $0 \leq x \leq 27$ .

Find the absolute extreme values of the function $f(x)=3x+\frac{243}{x}$ for $0<x<\infty$ .

Find the derivative of the function $f(x)=2 x^{3}+24 x^{2}-120 x+9$ on $[-10,3]$ and identify its min and max values.

Find the derivative $\frac{d y}{d x}$ for the equation $y=\ln(6 x^{2}+y^{2})$ .

Find the first derivative of $g(x)=8 x^{3}+60 x^{2}+96 x$ and check concavity at $x=-4$ using $g^{\prime \prime}(x)$ .

Find the first and second derivatives of $f(x)=x^{3}+6 x^{2}-96 x+19$ . Determine intervals where $f$ is increasing, decreasing, concave up, and concave down.

Find $\frac{d y}{d x}$ for the equation $y^{7} \ln(x^{2}) - 10x^{2} - y^{9} = -4$ .

Find the min and max of $f(x)=x^{4}-32x^{2}+3$ on $[-3,9]$ using derivatives and endpoint values.

Find the limit of the sequence $a_{n}=\sqrt{n^{4}+3 n^{2}+5}-\sqrt{n^{4}-n^{2}+n}$ , or explain why it doesn't exist.

An electron starts far from a stationary proton. What is its speed at $0.07 \mathrm{~m}$ from the proton?

Analyze and sketch the graph of $y=\frac{x}{x^{2}+49}$ . Find intercepts, extrema, inflection points, and asymptotes.

Analyze and sketch the graph of $y=\frac{x}{x^{2}+49}$ , finding intercepts, extrema, inflection points, and asymptotes.

Find relative extrema for $f(x)=x^{3}-9x^{2}+9$ using the Second Derivative Test. Enter DNE if none exist.

Find the integral $\int_{1}^{2} f(x) d x$ given $\int_{0}^{1} f(x) d x=-1$ , $\int_{0}^{2} f(x) d x=2$ , and $\int_{1}^{4} f(x) d x=11$ .

Find $\int_{4}^{-2}(f(x)-9 g(x)) d x$ given $\int_{-2}^{1} f(x) d x=4$ , $\int_{1}^{4} f(x) d x=-6$ , and $\int_{-2}^{4} g(x) d x=4$ .

Estimate the number of people hearing a rumor on day 14 using the model $f(x)=\frac{1,000}{1+499 e^{-0.6030 x}}$ . Round to the nearest person.

Find the integral $\int_{0}^{4} f(x) d x$ given $\int_{0}^{1} f(x) d x=-2$ , $\int_{0}^{2} f(x) d x=5$ , and $\int_{1}^{4} f(x) d x=7$ .

Calculate the Riemann sum $R(f, P, C)$ for $f(x)=31x+28$ , $P=\{-4,-1,1,4,8\}$ , and $C=\{-2,0,3,7\}$ .

Find the integrals $\int_{0}^{4} g(t) dt$ and $\int_{1}^{5} g(t) dt$ for the triangle $g(t)$ with vertices (1,0), (4,0), (3,2).

Find the Riemann sum $R(f, P, C)$ for $f(x)=x^{2}+7x$ , with partition $P=\{4,7,9,12\}$ and points $C=\{4,7.5,11.5\}$ . Give your answer to two decimal places. $R(f, P, C)=$

Find the inflection points of $f(x)=x+7 \cos x$ on $[0,2\pi]$ . Describe concavity in interval notation.

Berechne die Fallzeit und Endgeschwindigkeit eines Steins vom Freiburger Münster ( $h=116 \mathrm{~m}$ ) und Eiffelturm ( $h=300 \mathrm{~m}$ ).

Evaluate the integral $\int_{1}^{\infty} 21 x^{-3} dx$ using limits.

Untersuche die Stetigkeit der Funktion $f(x)=\left\{\begin{array}{cl} \sin (x) & \text { für } x \leq 0 \\ \frac{1}{x} & \text { für } x>0 \end{array}\right.$ . Gibt es eine Unstetigkeit?

Max springt aus 10 m Höhe. Berechne die Sprungdauer und Eintauchgeschwindigkeit ( $t=1,43 \mathrm{~s}$ ). Paula springt 3 m hoch. Wie lange nach Max kommt sie ins Wasser? ( $\Delta t=0,54 \mathrm{~s}$ )

A coffee at $130^{\circ} \mathrm{F}$ cools in a freezer at $0^{\circ} \mathrm{F}$ . After 15 min, it's $39^{\circ} \mathrm{F}$ . Find its temp after 20 min. Answer: $\square^{\circ} \mathrm{F}$ .

Find the function $f(x)$ if its derivative is $f'(x)=e^{2} x+e x^{2}+2 e x$ .

Bestimmen Sie die Ableitung von $x^{2}+2 e+1$ .

Find the function $f(x)$ if its derivative is $f'(x)=e^x+x+2$ .

Bestimmen Sie die Ableitung von $f(x)=e x^{3}+e^{3}+e^{x}$ .

A 17 ft ladder leans against a wall. If the top slips down at $2 \mathrm{ft/s}$ , how fast is the foot moving when the top is 16 ft up?

A triangle's height increases at $2 \mathrm{~cm/min}$ and area at $3 \mathrm{~cm}^2/\mathrm{min}$ . Find base's rate of change when height is 8 cm and area is 91 cm².

Analyze the curve $y=3 x^{4}-24 x^{3}$ for concavity, inflection points, and local maxima.

Analyze the curve $y=3 x^{4}-24 x^{3}$ for concavity, inflection points, and local extrema. Sketch the curve.

Find critical numbers of $f(x)=3x^{4}-24x^{3}$ by solving $f^{\prime}(x)=0$ . Use the Second Derivative Test for concavity.

Find the limit: $\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}$ .

Given the function $f(x)=x^{3}-3 x^{2}-9 x+2$ , determine:
(a) Intervals where $f$ is increasing and decreasing. (b) Local min and max values. (c) Inflection point $(x, y)=(\square)$ and intervals for concavity.

Find the inflection point of the function $f(x)=x^{3}-3x^{2}-9x+2$ .

Find the interval where the function $f$ is increasing given its derivative $f^{\prime}(x)=(x+1)^{2}(x-4)^{7}(x-6)^{4}$ .

Find the interval where the function $f$ is increasing, given its derivative $f^{\prime}(x)=(x+1)^{2}(x-4)^{7}(x-6)^{4}$ .

Find the interval where the function $f$ is increasing given $f^{\prime}(x)=(x+1)^{2}(x-4)^{7}(x-6)^{4}$ . Answer: $(-1,4) \cup(6, \infty)$ .

Find the average value $g_{\text{ave}}$ of $g(t)=\frac{9}{1+t^{2}}$ over the interval $[0,5]$ .

Analyze the function $f(x)=-x^{3}$ as $x \rightarrow -\infty$ . What does $f(x)$ approach? $\infty$ , $-\infty$ , or 0?

Find the average value of the function $f(x)=3x^2+8x$ over the interval $[-1,3]$ .

Given the function $f(x)=e^{4 x}+e^{-x}$ , find where $f$ is increasing and decreasing, and its local min and max values.

Analyze the limits of $f(x)=-x^{3}$ as $x \to -\infty$ and $x \to \infty$ . What do they approach?

Evaluate the limits of the polynomial function $f(r)=-3 r^{9}+r^{8}+3 r^{6}+2$ as $r \rightarrow -\infty$ and $r \rightarrow \infty$ .

Given the function $f(x)=e^{4 x}+e^{-x}$ , find:
(a) Intervals where $f$ is increasing and decreasing.
(b) Local minimum and maximum values.
(c) Inflection point $(x, y)=(\square)$ and intervals of concavity.

Evaluate the integral from 0 to 4 of $\frac{x^{2}+x+1}{(x+1)^{2}(x+2)}$ dx.

Calculate the integral: $\int \frac{8 t-3}{t+1} d t$

Calculate the integral: $\int_{0}^{9} \frac{d t}{\sqrt{81+t^{2}}}$ .

Find the interval where the function $f(x)=(x^{2}-12)e^{-x}$ is concave up for $x>0$ . Answer in the form $(, )$ .

Find the initial number of flu cases using the function $N(t)=\frac{10,000}{1+999 e^{-t}}$ at $t=0$ . Round to the nearest person.

Find the increase in car sales over 6 years with advertising: from $5 + 0.5 t^{3/2}$ to $5 e^{0.3 t}$ .