Calculus

Problem 10001

Find the derivative $g^{\prime}(1)$ for the function $g(x)=\frac{4 x^{3}-5 x+1}{3 x-5}$ .

See Solution

Problem 10002

Find the derivative $f^{\prime}(1)$ for the function $f(x)=(2x^{3}-3x+7)(5x+\frac{2}{x})$ .

See Solution

Problem 10003

Find the limit of the population model $P(t)=\frac{6000}{40+60 e^{-0.03 t}}$ as $t$ approaches infinity.

See Solution

Problem 10004

Find the nature of the critical point $(20,20)$ for the function $f(x, y)=160 x-3 x^{2}-2 x y-2 y^{2}+120 y-18$ using the second derivative test:
$D = f_{xx}(x, y)f_{yy}(x, y) - [f_{xy}(x, y)]^2$
with $f_{xx} = -6$ , $f_{yy} = -4$ , and $f_{xy} = -2$ .

See Solution

Problem 10005

Given the function $f(x)=3 x^{4}-4 x^{3}-30 x^{2}-36 x$ , find local max/min points, inflection points, and intervals of increase/decrease and concavity.

See Solution

Problem 10006

Find critical points of $f(x) = x - x(\ln x)^{2}$ for $x>0$ by solving $(\ln x)^{2}+2 \ln x-1=0$ .

See Solution

Problem 10007

Gegeben ist die Funktion $f_{a}(x)$ . Bestimmen Sie die Ableitung und die Steigung bei $x=0$ . Wann ist die Steigung 1?

See Solution

Problem 10008

Find the derivative of $f(x)=e^{6 x}(x^{2}+8^{x})$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 10009

A spring stores energy as $E=\frac{1}{2} k x^{2}$ . If $k=0.2 \mathrm{Joules} / \mathrm{cm}^{2}$ and $x$ changes at $1.5 \mathrm{cm/sec}$ , find the energy rate at $x=10 \mathrm{cm}$ .

See Solution

Problem 10010

Given $F(3)=4$ , $F'(3)=1$ , $F(4)=2$ , $F'(4)=7$ , $G(3)=4$ , $G'(3)=1$ , $G(4)=1$ , $G'(4)=2$ , find:
A. $H(3)$ for $H(x)=F(G(x))$ B. $H'(3)$ for $H(x)=F(G(x))$ C. $H(3)$ for $H(x)=G(F(x))$ D. $H'(3)$ for $H(x)=G(F(x))$ E. $H'(3)$ for $H(x)=F(x) / G(x)$

See Solution

Problem 10011

Differentiate the function $g(x)=\frac{1}{\sqrt{x}}+\sqrt[5]{x}$ . Find $g^{\prime}(x)$ .

See Solution

Problem 10012

Find the derivative of $y=\left(\frac{x^{2}+1}{2}\right)^{4}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 10013

Differentiate the function $g(x)=\frac{1}{\sqrt{x}}+\sqrt[5]{x}$ and find $g^{\prime}(x)$ .

See Solution

Problem 10014

Given $F(3)=4$ , $F'(3)=4$ , $F(4)=7$ , $F'(4)=5$ , $G(3)=4$ , $G'(3)=7$ , $G(4)=1$ , $G'(4)=6$ , find: A. $H(3)$ for $H(x)=F(G(x))$ B. $H'(3)$ for $H(x)=F(G(x))$ C. $H(3)$ for $H(x)=G(F(x))$ D. $H'(3)$ for $H(x)=G(F(x))$ E. $H'(3)$ for $H(x)=F(x)/G(x)$ Use dne for any derivative that can't be computed.

See Solution

Problem 10015

Find the derivative of the function $f(x)=3 \cos (2 \ln (x))$ , i.e., compute $f^{\prime}(x)$ .

See Solution

Problem 10016

A cube of ice melts at $1.5 \mathrm{~cm/min}$ . Find the volume change rate when edge length is $65 \mathrm{~cm}$ .

See Solution

Problem 10017

Find the derivative $f'(x)$ of the function $f(x) = \ln[x^{4}(x+6)^{4}(x^{2}+3)^{7}]$ .

See Solution

Problem 10018

Find the derivative $f'(x)$ of the function $f(x)=(x+4)^{13} e^{13 x}$ and evaluate it at $x=-3$ .

See Solution

Problem 10019

Find the derivative of $f(x)=\frac{e^{x}}{(e^{x}+2)(x+1)}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 10020

Find the derivative $f'(x)$ of the function $f(x)=2 x \cos^{-1}(x+1)-\sqrt{13-3 x^{2}}$ .

See Solution

Problem 10021

Find the derivative of the function $f(x)=5 x \sin ^{-1}(x)$ , i.e., compute $f^{\prime}(x)$ .

See Solution

Problem 10022

Find $x$ where the slope of the tangent line to $y=x^{3}+6x^{2}$ equals 36.

See Solution

Problem 10023

A cone-shaped tank (height $16 \mathrm{~m}$ , radius $12 \mathrm{~m}$ ) fills at $2.1 \mathrm{~m}^3/\mathrm{min}$ . Find the height change rate when water diameter is $9 \mathrm{~m}$ (radius $=4.5$ ).

See Solution

Problem 10024

Find the horizontal asymptote of the function $f(x)=\frac{3 x^{20}}{4 e^{x}+8 x^{20}}$ for $x>0$ .

See Solution

Problem 10025

Find the integral of the function: $\int \frac{2 x+1}{3 x+4} \mathrm{~d} x$

See Solution

Problem 10026

A penny is dropped; its height after $t$ seconds is given by $h(t)=-16 t^{2}+123 t+3470$ . Find $h(10)$ .

See Solution

Problem 10027

Find the horizontal asymptote of the function $f(x)=\frac{3 x^{20}}{4 e^{x}+8 x^{20}}$ for $x>0$ .

See Solution

Problem 10028

Find the percentage error of the linear approximation for $y=\sin(3x)$ at $x=0$ with $\Delta x=0.2$ and $\Delta y \approx 0.6$ .

See Solution

Problem 10029

A block of mass $m=0.54 \mathrm{~kg}$ compresses a spring by $L=13 \mathrm{~cm}$ . Given $\mu=0.26$ and $k=10 \times 10^1 \mathrm{~N/m}$ , find: (a) work by spring, (b) work by friction, (c) speed when leaving spring, (d) distance until stop.

See Solution

Problem 10030

What number does $\left(1+\frac{1}{n}\right)^{n}$ approach as $n$ increases? Provide the value rounded to two decimal places.

See Solution

Problem 10031

Find the slope of the tangent line to the curve $5 x y^{7}+2 x y=7$ at the point $(1,1)$ using implicit differentiation.

See Solution

Problem 10032

Find the average rate of change for $f(x)=x^{2}$ over these intervals: (a) $0 \leq x \leq 2$ , (b) $2 \leq x \leq 4$ , (c) $4 \leq x \leq 6$ .

See Solution

Problem 10033

Find the slope of the tangent line to the curve $5 \sin (x)+2 \cos (y)-6 \sin (x) \cos (y)+x=6 \pi$ at $(6 \pi, 5 \pi / 2)$ .

See Solution

Problem 10034

Determine the vertical and horizontal asymptotes of the function $F(x)=\frac{8-x^{3}}{x^{2}+4 x+4}$ .

See Solution

Problem 10035

Find the limit of $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x^{2}-5$ .

See Solution

Problem 10036

Find $\frac{d y}{d t}$ when $x=1$ , $y=1$ given $x^{2}+y^{2}=2$ and $\frac{d x}{d t}=2$ .

See Solution

Problem 10037

Invest \ $10,000. Which is greater in 3 years:$ 12\% $quarterly or$ 11.90\%$ continuously?

See Solution

Problem 10038

Find $\frac{d x}{d t}$ when $y^{2}+x y-3 x=51$ , $\frac{d y}{d t}=-1$ , $x=-5$ , and $y=-4$ .

See Solution

Problem 10039

Find the rate of change of the distance from the particle at $(3,12)$ to the origin as $x$ increases at 5 units/sec on $y=3\sqrt{3x+7}$ .

See Solution

Problem 10040

A bottle of milk cools from $77^{\circ} \mathrm{F}$ to $55^{\circ} \mathrm{F}$ in 10 min. Find $T$ after $t$ min and when $T=51^{\circ} \mathrm{F}$ .

See Solution

Problem 10041

Find the limit as $t$ approaches 0 for the expression $\frac{1}{t}-\frac{1}{t^{2}-t}$ .

See Solution

Problem 10042

Find the derivative of $f(s) = \frac{\sqrt{3} s^{2}}{4}$ with respect to $s$ .

See Solution

Problem 10043

Approximate $\frac{1}{0.504}$ using linear approximation of $f(x)=\frac{1}{x}$ at a nearby "nice" point.

See Solution

Problem 10044

Complete the table and approximate $\lim _{h \rightarrow 0^{-}} \frac{2^{h}-1}{h}$ and $\lim _{h \rightarrow 0^{-}} \frac{3^{h}-1}{h}$ .

See Solution

Problem 10045

Approximate $\sqrt{25.2}$ using linear approximation with $f(x)=\sqrt{x}$ at $x=25$ . Find $m$ and $b$ for $y=mx+b$ . Provide 6 significant figures.

See Solution

Problem 10046

Calculate the integral: $\int \frac{48}{x^{2} \sqrt{x^{2}+16}} d x$

See Solution

Problem 10047

Find the average speed of an object from $t=2$ to $t=4$ using $g(t)=2 t^{2}+3 t$ . Provide the answer as a decimal.

See Solution

Problem 10048

Find the second derivative $y^{\prime \prime}(x)$ at the point (5,1) for the equation $x^{3}+y^{3}=126$ .

See Solution

Problem 10049

Find the derivative of $h(z)=(6-z^{2})(z^{3}-3z+1)$ using the product rule.

See Solution

Problem 10050

Find the derivative of $h(w)=\frac{w^{4}-w}{w}$ using the quotient rule.

See Solution

Problem 10051

Differentiate $f(x)=\frac{5}{x^{6}}$ using two methods.

See Solution

Problem 10052

Find the derivative of the function $f(x)=\frac{x}{x+15}$ and simplify your answer.

See Solution

Problem 10053

Find the second derivative $y^{\prime \prime}(x)$ for the equation $x^{3}+y^{3}=126$ at the point (5,1).

See Solution

Problem 10054

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{5 x^{2}-9 x+8}{7 x+1}$ .

See Solution

Problem 10055

Is it true or false that $\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=1$ ?

See Solution

Problem 10056

Is it true or false that $\lim _{x \rightarrow 0^{+}} \ln (x)=-\infty$ ?

See Solution

Problem 10057

Find max and min of $f(x)=x-\frac{27 x}{x+3}$ on $[0,7]$ . Min value = , Max value = .

See Solution

Problem 10058

A 5-m ladder leans against a wall. If the bottom slides away at $0.9 \mathrm{~m/s}$ , find the top's velocity at $t=2 \mathrm{~s}$ when $x=1.5 \mathrm{~m}$ . $\left.\frac{d h}{d t}\right|_{t=2}$ $\mathrm{m/s}$

See Solution

Problem 10059

Find the derivative of the function $f(x)=6x-11xe^{x}$ .

See Solution

Problem 10060

Find the derivative of $g(x)=e^{x}(5 x^{2}-10 x+10)$ and simplify it.

See Solution

Problem 10061

Find the derivative of $y=5 x^{2} \ln (3 x)$ . Choose the correct option from the list.

See Solution

Problem 10062

Find $g^{\prime}(x)$ for $g(x) = e^{x}(5x^{2}-10x+10)$ .

See Solution

Problem 10063

Find the value of $c$ for which the function $f(x)=\frac{\sqrt{x}-2}{x-4}$ (if $x \neq 4$ ) and $c$ (if $x=4$ ) is continuous on $(0, \infty)$ .

See Solution

Problem 10064

Find $f^{\prime}(x)$ for $y=f(x)=4 x e^{x}$ when it crosses the $y$ -axis. Options: 2, 4, -1, 0, does not cross.

See Solution

Problem 10065

Find the derivative and simplify for the function $f(x)=\frac{9 e^{x}}{2 e^{x}+5}$ .

See Solution

Problem 10066

An exponential function starts at 500 and decays at $15\%$ . Compare average rates of change for $0<x<4$ and $4<x<8$ . What about $x>8$ ? Explain.

See Solution

Problem 10067

Find the second derivative of $f(x)$ if $f'(x)=\frac{(2e^{x}+5)(9e^{x})'-(9e^{x})(2e^{x}+5)'}{(2e^{x}+5)^{2}}$ .

See Solution

Problem 10068

Simplify the derivative of the function $f(x)=\frac{9e^x}{2e^x + 5}$ using the quotient rule.

See Solution

Problem 10069

Differentiate the function $f(t)=\frac{\sqrt[3]{t}}{t-6}$ . Find $f^{\prime}(t)=$ .

See Solution

Problem 10070

Find the derivative of $y=(5t-1)(2t-4)^{-1}$ .

See Solution

Problem 10071

Find the limit: $\lim _{x \rightarrow+\infty}\left(x^{5}-x+2\right) e^{-x}$ .

See Solution

Problem 10072

Find the value(s) of $x$ where the tangent line to $f(x)=4 x^{3}-12 x^{2}-98 x+12$ is parallel to $y=-2 x+0.7$ .

See Solution

Problem 10073

Find where the tangent to $f(x)=2 x^{3}+3 x^{2}-120 x+14$ is horizontal. List the $x$ -values. $x \text{ value(s)} =$

See Solution

Problem 10074

Find the derivative of the function $f(x) = x^{6} e^{x}$ .

See Solution

Problem 10075

Find the stationary points of $y=x^{4}+x^{2}$ and determine their nature.

See Solution

Problem 10076

Find velocity and acceleration functions for $s=(1/3)t^3-3t^2+9t+2$ . Then, find acceleration at $t=1$ and when velocity=0.

See Solution

Problem 10077

Find the rate of volume change for a cone with $r=5$ cm and $h=20$ cm, both increasing at $7$ cm/s.

See Solution

Problem 10078

Find $F^{\prime}(\sqrt{\pi})$ where $F(x)=\int_{1+\cos(x^{2})}^{3} e^{3(t-1)^{2}} dt$ .

See Solution

Problem 10079

Find the derivative of $\frac{x^{6} e^{x}}{x^{6}+e^{x}}$ using the Quotient Rule with $g(x) = x^6 e^x$ and $h(x) = x^6 + e^x$ .

See Solution

Problem 10080

A particle's position is given by $s(t)=t^{4}-12 t+21$ . Find: (A) $v(t)$ , (B) $v(3)$ , (C) $t$ when at rest, (D) intervals moving positively, (E) total distance in 8 seconds.

See Solution

Problem 10081

Find $(f g)^{\prime}(5)$ given $f(5)=1$ , $f^{\prime}(5)=4$ , $g(5)=-6$ , $g^{\prime}(5)=9$ .

See Solution

Problem 10082

A ball is thrown from a 32 ft roof at 48 ft/sec. Height is $s(t)=32+48t-16t^{2}$ . Find max height and velocity at ground.

See Solution

Problem 10083

Find slant and vertical asymptotes for $f(x)=\frac{0.5 x^{2}-4 x+1}{x+2}$ . Choices for both types are given.

See Solution

Problem 10084

Given $f(5)=1$ , $f^{\prime}(5)=4$ , $g(5)=-6$ , $g^{\prime}(5)=9$ . Find $(f g)^{\prime}(5)$ and $\left(\frac{f}{g}\right)^{\prime}(5)$ .

See Solution

Problem 10085

Find the rate of increase of the surface area $S=4 \pi r^{2}$ of a balloon at $r=2, 3, 6$ inches.

See Solution

Problem 10086

Differentiate the function: $f(x)=\frac{x^{6} e^{x}}{x^{6}+e^{x}}$ . Find $f^{\prime}(x)$ .

See Solution

Problem 10087

A body moves along the $s$ -axis with $s=t^{3}-18 t^{2}+81 t$ .
(a) Find acceleration when velocity is zero.
Acceleration $=$
(b) Find velocity when acceleration is zero.
Velocity $=$
(c) Calculate total distance from $t=0$ to $t=6$ .
Total distance $=$

See Solution

Problem 10088

An elastic band with mass vibrates. Given $s(t)=2 \cos t+3 \sin t$ , find:
(a) Velocity function. (b) Time to first pass equilibrium. (c) Distance from equilibrium (round answers).

See Solution

Problem 10089

An elastic band vibrates with motion $s(t)=2 \cos t+3 \sin t$ . Find: (a) velocity, (b) first equilibrium time, (c) max distance from equilibrium. Round (b) and (c) to the nearest hundredth.

See Solution

Problem 10090

An object with weight $W=65 \mathrm{lb}$ is pulled by a force $F=\frac{0.4 W}{0.4 \sin t+\cos t}$ .
(a) Find $F^{\prime}(t)$ . (b) When is $F^{\prime}(t)=0$ ? Round to the nearest hundredth: $t=$ \# rad.

See Solution

Problem 10091

Find the $26^{\text{th}}$ derivative of $y=\cos(2x)$ by calculating the first few derivatives and identifying the pattern.

See Solution

Problem 10092

Find the rate of change of the amount $A$ for $r=3\%$ and $r=7.5\%$ in the formula $A=1200(1+\frac{1}{12} r)^{48}$ .

See Solution

Problem 10093

A car moves at $v=81 \mathrm{~km/h}$ , $3.7 \mathrm{~km}$ past a highway. How fast is its distance from a farmhouse ( $a=1.5 \mathrm{~km}$ ) increasing?

See Solution

Problem 10094

Find the unit tangent vector, unit normal vector, normal acceleration, and tangential acceleration for $\vec{r}(t)=\langle e^{-t}, 5t, e^{t}\rangle$ at $t=2$ .

See Solution

Problem 10095

Find the position at $t=4 \mathrm{~s}$ given initial position $x=10 \mathrm{~m}$ and displacement of $20 \mathrm{~m}$ .

See Solution

Problem 10096

A Cepheid variable star's brightness changes over time. For Cephei Joe, with a brightness model $B(t)=3.9+0.5 \sin (2 \pi t / 5.2)$ :
(a) Find the rate of change of brightness after $t$ days.
Rate of change =
(b) Find the rate of increase after one day.
Rate of increase =

See Solution

Problem 10097

Determine if the integral $\int_{1}^{\infty} \frac{1}{x^{2}+\sqrt{x}} d x$ converges or diverges using big $O$ analysis.

See Solution

Problem 10098

The value that $\left(1+\frac{1}{n}\right)^{n}$ approaches as $n$ increases is the base, approximately equal to (round to two decimal places).

See Solution

Problem 10099

Find the derivative of the function $g(x) = 8 - 3x$ .

See Solution

Problem 10100

Find the first and second derivatives of $f(x)=3 x^{2} \cos (4 x)$ and evaluate them at $x=3$ .

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 10001

Find the derivative g′(1)g^{\prime}(1)g′(1) for the function g(x)=4x3−5x+13x−5g(x)=\frac{4 x^{3}-5 x+1}{3 x-5}g(x)=3x−54x3−5x+1​.

Problem 10002

Find the derivative f′(1)f^{\prime}(1)f′(1) for the function f(x)=(2x3−3x+7)(5x+2x)f(x)=(2x^{3}-3x+7)(5x+\frac{2}{x})f(x)=(2x3−3x+7)(5x+x2​).

Problem 10003

Find the limit of the population model P(t)=600040+60e−0.03tP(t)=\frac{6000}{40+60 e^{-0.03 t}}P(t)=40+60e−0.03t6000​ as ttt approaches infinity.

Problem 10004

Problem 10005

Given the function f(x)=3x4−4x3−30x2−36xf(x)=3 x^{4}-4 x^{3}-30 x^{2}-36 xf(x)=3x4−4x3−30x2−36x, find local max/min points, inflection points, and intervals of increase/decrease and concavity.

Problem 10006

Find critical points of f(x)=x−x(ln⁡x)2f(x) = x - x(\ln x)^{2}f(x)=x−x(lnx)2 for x>0x>0x>0 by solving (ln⁡x)2+2ln⁡x−1=0(\ln x)^{2}+2 \ln x-1=0(lnx)2+2lnx−1=0.

Problem 10007

Gegeben ist die Funktion fa(x)f_{a}(x)fa​(x). Bestimmen Sie die Ableitung und die Steigung bei x=0x=0x=0. Wann ist die Steigung 1?

Problem 10008

Find the derivative of f(x)=e6x(x2+8x)f(x)=e^{6 x}(x^{2}+8^{x})f(x)=e6x(x2+8x). What is f′(x)f^{\prime}(x)f′(x)?

Problem 10009

A spring stores energy as E=12kx2E=\frac{1}{2} k x^{2}E=21​kx2. If k=0.2Joules/cm2k=0.2 \mathrm{Joules} / \mathrm{cm}^{2}k=0.2Joules/cm2 and xxx changes at 1.5cm/sec1.5 \mathrm{cm/sec}1.5cm/sec, find the energy rate at x=10cmx=10 \mathrm{cm}x=10cm.

Problem 10010

Problem 10011

Differentiate the function g(x)=1x+x5g(x)=\frac{1}{\sqrt{x}}+\sqrt[5]{x}g(x)=x​1​+5x​. Find g′(x)g^{\prime}(x)g′(x).

Problem 10012

Find the derivative of y=(x2+12)4y=\left(\frac{x^{2}+1}{2}\right)^{4}y=(2x2+1​)4. What is dydx\frac{d y}{d x}dxdy​?

Problem 10013

Differentiate the function g(x)=1x+x5g(x)=\frac{1}{\sqrt{x}}+\sqrt[5]{x}g(x)=x​1​+5x​ and find g′(x)g^{\prime}(x)g′(x).

Problem 10014

Problem 10015

Find the derivative of the function f(x)=3cos⁡(2ln⁡(x))f(x)=3 \cos (2 \ln (x))f(x)=3cos(2ln(x)), i.e., compute f′(x)f^{\prime}(x)f′(x).

Problem 10016

A cube of ice melts at 1.5 cm/min1.5 \mathrm{~cm/min}1.5 cm/min. Find the volume change rate when edge length is 65 cm65 \mathrm{~cm}65 cm.

Problem 10017

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=ln⁡[x4(x+6)4(x2+3)7]f(x) = \ln[x^{4}(x+6)^{4}(x^{2}+3)^{7}]f(x)=ln[x4(x+6)4(x2+3)7].

Problem 10018

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=(x+4)13e13xf(x)=(x+4)^{13} e^{13 x}f(x)=(x+4)13e13x and evaluate it at x=−3x=-3x=−3.

Problem 10019

Find the derivative of f(x)=ex(ex+2)(x+1)f(x)=\frac{e^{x}}{(e^{x}+2)(x+1)}f(x)=(ex+2)(x+1)ex​. What is f′(x)f^{\prime}(x)f′(x)?

Problem 10020

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=2xcos⁡−1(x+1)−13−3x2f(x)=2 x \cos^{-1}(x+1)-\sqrt{13-3 x^{2}}f(x)=2xcos−1(x+1)−13−3x2​.

Problem 10021

Find the derivative of the function f(x)=5xsin⁡−1(x)f(x)=5 x \sin ^{-1}(x)f(x)=5xsin−1(x), i.e., compute f′(x)f^{\prime}(x)f′(x).

Problem 10022

Find xxx where the slope of the tangent line to y=x3+6x2y=x^{3}+6x^{2}y=x3+6x2 equals 36.

Problem 10023

A cone-shaped tank (height 16 m16 \mathrm{~m}16 m, radius 12 m12 \mathrm{~m}12 m) fills at 2.1 m3/min2.1 \mathrm{~m}^3/\mathrm{min}2.1 m3/min. Find the height change rate when water diameter is 9 m9 \mathrm{~m}9 m (radius =4.5=4.5=4.5).

Problem 10024

Find the horizontal asymptote of the function f(x)=3x204ex+8x20f(x)=\frac{3 x^{20}}{4 e^{x}+8 x^{20}}f(x)=4ex+8x203x20​ for x>0x>0x>0.

Problem 10025

Find the integral of the function: ∫2x+13x+4 dx\int \frac{2 x+1}{3 x+4} \mathrm{~d} x∫3x+42x+1​ dx

Problem 10026

A penny is dropped; its height after ttt seconds is given by h(t)=−16t2+123t+3470h(t)=-16 t^{2}+123 t+3470h(t)=−16t2+123t+3470. Find h(10)h(10)h(10).

Problem 10027

Find the horizontal asymptote of the function f(x)=3x204ex+8x20f(x)=\frac{3 x^{20}}{4 e^{x}+8 x^{20}}f(x)=4ex+8x203x20​ for x>0x>0x>0.

Problem 10028

Find the percentage error of the linear approximation for y=sin⁡(3x)y=\sin(3x)y=sin(3x) at x=0x=0x=0 with Δx=0.2\Delta x=0.2Δx=0.2 and Δy≈0.6\Delta y \approx 0.6Δy≈0.6.

Problem 10029

Problem 10030

What number does (1+1n)n\left(1+\frac{1}{n}\right)^{n}(1+n1​)n approach as nnn increases? Provide the value rounded to two decimal places.

Problem 10031

Find the slope of the tangent line to the curve 5xy7+2xy=75 x y^{7}+2 x y=75xy7+2xy=7 at the point (1,1)(1,1)(1,1) using implicit differentiation.

Problem 10032

Find the average rate of change for f(x)=x2f(x)=x^{2}f(x)=x2 over these intervals: (a) 0≤x≤20 \leq x \leq 20≤x≤2, (b) 2≤x≤42 \leq x \leq 42≤x≤4, (c) 4≤x≤64 \leq x \leq 64≤x≤6.

Problem 10033

Find the slope of the tangent line to the curve 5sin⁡(x)+2cos⁡(y)−6sin⁡(x)cos⁡(y)+x=6π5 \sin (x)+2 \cos (y)-6 \sin (x) \cos (y)+x=6 \pi5sin(x)+2cos(y)−6sin(x)cos(y)+x=6π at (6π,5π/2)(6 \pi, 5 \pi / 2)(6π,5π/2).

Problem 10034

Determine the vertical and horizontal asymptotes of the function F(x)=8−x3x2+4x+4F(x)=\frac{8-x^{3}}{x^{2}+4 x+4}F(x)=x2+4x+48−x3​.

Problem 10035

Find the limit of f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=2x2−5f(x)=2x^{2}-5f(x)=2x2−5.

Problem 10036

Find dydt\frac{d y}{d t}dtdy​ when x=1x=1x=1, y=1y=1y=1 given x2+y2=2x^{2}+y^{2}=2x2+y2=2 and dxdt=2\frac{d x}{d t}=2dtdx​=2.

Problem 10037

Invest \10,000.Whichisgreaterin3years:10,000. Which is greater in 3 years: 10,000.Whichisgreaterin3years:12\%quarterlyor quarterly or quarterlyor11.90\%$ continuously?

Problem 10038

Find dxdt\frac{d x}{d t}dtdx​ when y2+xy−3x=51y^{2}+x y-3 x=51y2+xy−3x=51, dydt=−1\frac{d y}{d t}=-1dtdy​=−1, x=−5x=-5x=−5, and y=−4y=-4y=−4.

Problem 10039

Find the rate of change of the distance from the particle at (3,12)(3,12)(3,12) to the origin as xxx increases at 5 units/sec on y=33x+7y=3\sqrt{3x+7}y=33x+7​.

Problem 10040

A bottle of milk cools from 77∘F77^{\circ} \mathrm{F}77∘F to 55∘F55^{\circ} \mathrm{F}55∘F in 10 min. Find TTT after ttt min and when T=51∘FT=51^{\circ} \mathrm{F}T=51∘F.

Problem 10041

Find the limit as ttt approaches 0 for the expression 1t−1t2−t\frac{1}{t}-\frac{1}{t^{2}-t}t1​−t2−t1​.

Problem 10042

Find the derivative $g^{\prime}(1)$ for the function $g(x)=\frac{4 x^{3}-5 x+1}{3 x-5}$ .

Find the derivative $f^{\prime}(1)$ for the function $f(x)=(2x^{3}-3x+7)(5x+\frac{2}{x})$ .

Find the limit of the population model $P(t)=\frac{6000}{40+60 e^{-0.03 t}}$ as $t$ approaches infinity.

Given the function $f(x)=3 x^{4}-4 x^{3}-30 x^{2}-36 x$ , find local max/min points, inflection points, and intervals of increase/decrease and concavity.

Find critical points of $f(x) = x - x(\ln x)^{2}$ for $x>0$ by solving $(\ln x)^{2}+2 \ln x-1=0$ .

Gegeben ist die Funktion $f_{a}(x)$ . Bestimmen Sie die Ableitung und die Steigung bei $x=0$ . Wann ist die Steigung 1?

Find the derivative of $f(x)=e^{6 x}(x^{2}+8^{x})$ . What is $f^{\prime}(x)$ ?

A spring stores energy as $E=\frac{1}{2} k x^{2}$ . If $k=0.2 \mathrm{Joules} / \mathrm{cm}^{2}$ and $x$ changes at $1.5 \mathrm{cm/sec}$ , find the energy rate at $x=10 \mathrm{cm}$ .

Differentiate the function $g(x)=\frac{1}{\sqrt{x}}+\sqrt[5]{x}$ . Find $g^{\prime}(x)$ .

Find the derivative of $y=\left(\frac{x^{2}+1}{2}\right)^{4}$ . What is $\frac{d y}{d x}$ ?

Differentiate the function $g(x)=\frac{1}{\sqrt{x}}+\sqrt[5]{x}$ and find $g^{\prime}(x)$ .

Find the derivative of the function $f(x)=3 \cos (2 \ln (x))$ , i.e., compute $f^{\prime}(x)$ .

A cube of ice melts at $1.5 \mathrm{~cm/min}$ . Find the volume change rate when edge length is $65 \mathrm{~cm}$ .

Find the derivative $f'(x)$ of the function $f(x) = \ln[x^{4}(x+6)^{4}(x^{2}+3)^{7}]$ .

Find the derivative $f'(x)$ of the function $f(x)=(x+4)^{13} e^{13 x}$ and evaluate it at $x=-3$ .

Find the derivative of $f(x)=\frac{e^{x}}{(e^{x}+2)(x+1)}$ . What is $f^{\prime}(x)$ ?

Find the derivative $f'(x)$ of the function $f(x)=2 x \cos^{-1}(x+1)-\sqrt{13-3 x^{2}}$ .

Find the derivative of the function $f(x)=5 x \sin ^{-1}(x)$ , i.e., compute $f^{\prime}(x)$ .

Find $x$ where the slope of the tangent line to $y=x^{3}+6x^{2}$ equals 36.

A cone-shaped tank (height $16 \mathrm{~m}$ , radius $12 \mathrm{~m}$ ) fills at $2.1 \mathrm{~m}^3/\mathrm{min}$ . Find the height change rate when water diameter is $9 \mathrm{~m}$ (radius $=4.5$ ).

Find the horizontal asymptote of the function $f(x)=\frac{3 x^{20}}{4 e^{x}+8 x^{20}}$ for $x>0$ .

Find the integral of the function: $\int \frac{2 x+1}{3 x+4} \mathrm{~d} x$

A penny is dropped; its height after $t$ seconds is given by $h(t)=-16 t^{2}+123 t+3470$ . Find $h(10)$ .

Find the horizontal asymptote of the function $f(x)=\frac{3 x^{20}}{4 e^{x}+8 x^{20}}$ for $x>0$ .

Find the percentage error of the linear approximation for $y=\sin(3x)$ at $x=0$ with $\Delta x=0.2$ and $\Delta y \approx 0.6$ .

What number does $\left(1+\frac{1}{n}\right)^{n}$ approach as $n$ increases? Provide the value rounded to two decimal places.

Find the slope of the tangent line to the curve $5 x y^{7}+2 x y=7$ at the point $(1,1)$ using implicit differentiation.

Find the average rate of change for $f(x)=x^{2}$ over these intervals: (a) $0 \leq x \leq 2$ , (b) $2 \leq x \leq 4$ , (c) $4 \leq x \leq 6$ .

Find the slope of the tangent line to the curve $5 \sin (x)+2 \cos (y)-6 \sin (x) \cos (y)+x=6 \pi$ at $(6 \pi, 5 \pi / 2)$ .

Determine the vertical and horizontal asymptotes of the function $F(x)=\frac{8-x^{3}}{x^{2}+4 x+4}$ .

Find the limit of $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x^{2}-5$ .

Find $\frac{d y}{d t}$ when $x=1$ , $y=1$ given $x^{2}+y^{2}=2$ and $\frac{d x}{d t}=2$ .

Invest \ $10,000. Which is greater in 3 years:$ 12\% $quarterly or$ 11.90\%$ continuously?

Find $\frac{d x}{d t}$ when $y^{2}+x y-3 x=51$ , $\frac{d y}{d t}=-1$ , $x=-5$ , and $y=-4$ .

Find the rate of change of the distance from the particle at $(3,12)$ to the origin as $x$ increases at 5 units/sec on $y=3\sqrt{3x+7}$ .

A bottle of milk cools from $77^{\circ} \mathrm{F}$ to $55^{\circ} \mathrm{F}$ in 10 min. Find $T$ after $t$ min and when $T=51^{\circ} \mathrm{F}$ .

Find the limit as $t$ approaches 0 for the expression $\frac{1}{t}-\frac{1}{t^{2}-t}$ .

Find the derivative of $f(s) = \frac{\sqrt{3} s^{2}}{4}$ with respect to $s$ .

Approximate $\frac{1}{0.504}$ using linear approximation of $f(x)=\frac{1}{x}$ at a nearby "nice" point.

Complete the table and approximate $\lim _{h \rightarrow 0^{-}} \frac{2^{h}-1}{h}$ and $\lim _{h \rightarrow 0^{-}} \frac{3^{h}-1}{h}$ .

Approximate $\sqrt{25.2}$ using linear approximation with $f(x)=\sqrt{x}$ at $x=25$ . Find $m$ and $b$ for $y=mx+b$ . Provide 6 significant figures.

Calculate the integral: $\int \frac{48}{x^{2} \sqrt{x^{2}+16}} d x$

Find the average speed of an object from $t=2$ to $t=4$ using $g(t)=2 t^{2}+3 t$ . Provide the answer as a decimal.

Find the second derivative $y^{\prime \prime}(x)$ at the point (5,1) for the equation $x^{3}+y^{3}=126$ .

Find the derivative of $h(z)=(6-z^{2})(z^{3}-3z+1)$ using the product rule.

Find the derivative of $h(w)=\frac{w^{4}-w}{w}$ using the quotient rule.

Differentiate $f(x)=\frac{5}{x^{6}}$ using two methods.

Find the derivative of the function $f(x)=\frac{x}{x+15}$ and simplify your answer.

Find the second derivative $y^{\prime \prime}(x)$ for the equation $x^{3}+y^{3}=126$ at the point (5,1).

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{5 x^{2}-9 x+8}{7 x+1}$ .

Is it true or false that $\lim _{x \rightarrow 0} \frac{\sin (x)}{x}=1$ ?

Is it true or false that $\lim _{x \rightarrow 0^{+}} \ln (x)=-\infty$ ?

Find max and min of $f(x)=x-\frac{27 x}{x+3}$ on $[0,7]$ . Min value = , Max value = .

A 5-m ladder leans against a wall. If the bottom slides away at $0.9 \mathrm{~m/s}$ , find the top's velocity at $t=2 \mathrm{~s}$ when $x=1.5 \mathrm{~m}$ . $\left.\frac{d h}{d t}\right|_{t=2}$ $\mathrm{m/s}$

Find the derivative of the function $f(x)=6x-11xe^{x}$ .

Find the derivative of $g(x)=e^{x}(5 x^{2}-10 x+10)$ and simplify it.

Find the derivative of $y=5 x^{2} \ln (3 x)$ . Choose the correct option from the list.

Find $g^{\prime}(x)$ for $g(x) = e^{x}(5x^{2}-10x+10)$ .

Find the value of $c$ for which the function $f(x)=\frac{\sqrt{x}-2}{x-4}$ (if $x \neq 4$ ) and $c$ (if $x=4$ ) is continuous on $(0, \infty)$ .

Find $f^{\prime}(x)$ for $y=f(x)=4 x e^{x}$ when it crosses the $y$ -axis. Options: 2, 4, -1, 0, does not cross.

Find the derivative and simplify for the function $f(x)=\frac{9 e^{x}}{2 e^{x}+5}$ .

An exponential function starts at 500 and decays at $15\%$ . Compare average rates of change for $0<x<4$ and $4<x<8$ . What about $x>8$ ? Explain.

Find the second derivative of $f(x)$ if $f'(x)=\frac{(2e^{x}+5)(9e^{x})'-(9e^{x})(2e^{x}+5)'}{(2e^{x}+5)^{2}}$ .

Simplify the derivative of the function $f(x)=\frac{9e^x}{2e^x + 5}$ using the quotient rule.

Differentiate the function $f(t)=\frac{\sqrt[3]{t}}{t-6}$ . Find $f^{\prime}(t)=$ .

Find the derivative of $y=(5t-1)(2t-4)^{-1}$ .

Find the limit: $\lim _{x \rightarrow+\infty}\left(x^{5}-x+2\right) e^{-x}$ .

Find the value(s) of $x$ where the tangent line to $f(x)=4 x^{3}-12 x^{2}-98 x+12$ is parallel to $y=-2 x+0.7$ .

Find where the tangent to $f(x)=2 x^{3}+3 x^{2}-120 x+14$ is horizontal. List the $x$ -values. $x \text{ value(s)} =$

Find the derivative of the function $f(x) = x^{6} e^{x}$ .