Calculus

Problem 3701

Find the derivative of $f(x)=2x^{4}+\frac{5}{x^{3}}+4\sqrt{x}-6e^{3}$ .

See Solution

Problem 3702

A car's velocity is $v(t)=t^{3}$ . Find average acceleration over $[2,3]$ , $[3,4]$ , $[t,t+1]$ , $[2,2+h]$ , and at $t=2$ .

See Solution

Problem 3703

Evaluate the integral: $\int \frac{(\ln x)^{12}}{x} d x$

See Solution

Problem 3704

Find the derivative of $f(t)=(t-1)^{\prime}$ at $t=5$ and round to the nearest integer.

See Solution

Problem 3705

Find the derivative of $f(x)=6 e^{5 x}+8 \ln(x^{6}+8 x^{5}(x-2)^{2}+\frac{6}{x}+42)$ and express it as $\frac{d f}{d x}(x)=a \cdot e^{b \cdot x}+c \cdot \ln x+d \cdot x^{7}+e \cdot x^{6}+g \cdot x^{5}+h \cdot x^{4}+j \cdot x+k \cdot x^{-1}+l \cdot x^{-2}+m \cdot x^{-6}+n$ . Find coefficients $a$ , $b$ , $c$ , $d$ , $e$ , $g$ , $h$ , $j$ , $k$ , $l$ , $m$ , and $n$ .

See Solution

Problem 3706

Find the derivative of $f(t)=(t-1)^{t}$ at $t=5$ and round to the nearest integer.

See Solution

Problem 3707

Define $f(2)$ to make $f(x)=\frac{x^{3}-8}{x^{2}-4}$ continuous at 2. How do you remove the discontinuity?

See Solution

Problem 3708

Find the derivative of the function $f(t)=(t-1)^{t}$ at $t=5$ and round to the nearest integer.

See Solution

Problem 3709

Find the derivative of $f(x)=6x^4+5x^3+25x^{1/5}+6x^{-4}+4x+37$ and determine coefficients $a$ through $k$ .

See Solution

Problem 3710

See Solution

Problem 3711

See Solution

Problem 3712

Find the derivative of $f(r)=(r-2)^{r}$ at $r=5$ and round to the nearest integer.

See Solution

Problem 3713

Find the elasticity $E_l g(1)$ and slope of $g(y)=3y+22$ at $y=1$ . Round answers to two digits.

See Solution

Problem 3714

Find the elasticity of the function $t(z) = 32 e^{2 z^{3}}$ and express it as $E l_{z} t(z)=a \cdot z^{b}+c \cdot e^{2 z^{3}}+d \cdot z^{e} \cdot e^{2 z^{3}}$ . Determine coefficients $a$ , $b$ , $c$ , $d$ , and $e$ . Round answers to two digits.

See Solution

Problem 3715

Find the elasticity of the function $t(z) = 30 e^{5 z^{2}}$ and express it as $E l_{z} t(z) = 2 \cdot z^{30} + 0 + 0 \cdot z^{1} \cdot e^{5 z^{2}}$ . Round to two digits.

See Solution

Problem 3716

Find the elasticity of the function $t(z) = 30 e^{5 z^{2}}$ . Express it as $E l_{z} t(z) = a \cdot z^{b} + c \cdot e^{5 z^{2}} + d \cdot z^{e} \cdot e^{5 z^{2}}$ . Round answers to two digits.

See Solution

Problem 3717

Find the elasticity of the function $t(z) = 31 z^{2} \ln z$ and determine coefficients $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , and $g$ .

See Solution

Problem 3718

Find the elasticity of the function $t(z) = 34 e^{3 z^{2}}$ and express it as $E l_{z} t(z)=a \cdot z^{b}+c \cdot e^{3 z^{2}}+d \cdot z^{e} \cdot e^{3 z^{2}}$ . Determine coefficients $a$ , $b$ , $c$ , $d$ , and $e$ .

See Solution

Problem 3719

Find the elasticity of the function $t(z) = 33(z^{4}+1)^{1/4}$ and express it as $E l_{z} t(z) = a \cdot z^{b} \cdot (z^{c}+d)^{e}$ .

See Solution

Problem 3720

Find the elasticity and slope of the function $g(y) = 7y + 25$ at $y_0 = 1$ . Round answers to two digits.

See Solution

Problem 3721

Find the derivative of $w(x)=\ln(x^{2}+3x) \tan(4x)$ .

See Solution

Problem 3722

An object is thrown upwards at $20 \mathrm{~m} / \mathrm{s}$ . Find: a) speed upon hitting ground, b) total time in air, c) max height.

See Solution

Problem 3723

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(8)$ for the function defined by $x^{-9} \cdot z^{4}=4096$ .

See Solution

Problem 3724

Find the derivative of the function defined by $4 x^{3}+5 x y-8 y^{4}=223$ and identify coefficients in the expression for $\frac{d}{d x} y$ .

See Solution

Problem 3725

Find the elasticity of the function $t(z) = 26 e^{2 z^{3}}$ and express it as $E l_{z} t(z) = a \cdot z^{b} + c \cdot e^{2 z^{3}} + d \cdot z^{e} \cdot e^{2 z^{3}}$ . Determine coefficients $a$ , $b$ , $c$ , $d$ , and $e$ . Round answers to two digits.

See Solution

Problem 3726

Evaluate the integral $\int x^{2} \ln x \, dx$ using integration by parts with $u=\ln x$ and $dv=x^{2} \, dx$ .

See Solution

Problem 3727

Evaluate the function $f(x)=\frac{3|x-1|}{x-1}$ and determine which statements are true: I, II, III about limits, continuity, and differentiability at $x=1$ .

See Solution

Problem 3728

Find the derivative of the function defined by $g^{5}+y^{5}=Z$ as $\frac{d}{d y} g=A\left(\frac{y}{g}\right)^{a}+B(y \cdot g)^{b}+C$ and identify coefficients $A$ , $a$ , $B$ , $b$ , and $C$ .

See Solution

Problem 3729

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(7)$ from the equation $x^{-8} z^3 = 343$ , rounding to two digits.

See Solution

Problem 3730

Integrate $\theta \cos \theta$ using integration by parts with $u=\theta$ , $dv=\cos \theta d\theta$ .

See Solution

Problem 3731

Find the derivative of $y$ in the equation $4 x^{3}+4 x y-3 y^{4}=225$ and determine the coefficients in the expression.

See Solution

Problem 3732

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(7)$ for the equation $x^{-8} \cdot z^{3}=343$ . Round to two digits.

See Solution

Problem 3733

Find the derivative $\frac{d}{du} v$ from $u^{6}+v^{6}=T$ and express it as $A\left(\frac{u}{v}\right)^{a}+B(u \cdot v)^{b}+C$ . What are the coefficients $A$ , $a$ , $B$ , $b$ , and $C$ ?

See Solution

Problem 3734

Determine which statements about $f(x)=\frac{3|x-1|}{x-1}$ are true: I. $\lim _{x \rightarrow 1} f(x)$ exists. II. $f$ is continuous at $x=1$ . III. $f$ is differentiable at $x=1$ .

See Solution

Problem 3735

Evaluate the function $f(x)=\frac{2|x-1|}{x-1}$ and determine which statements about limits, continuity, and differentiability at $x=-1$ are true.

See Solution

Problem 3736

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(5)$ from $x^{-6} \cdot z^{3}=125$ . Round answers to two digits.

See Solution

Problem 3737

Find the derivative of the function defined by $3 x^{4}+8 x y-3 y^{3}=189$ and identify coefficients $a, b, c, d, e, f, g, h, j, k, l, m$ .

See Solution

Problem 3738

Find the derivative of $y = e^{\cos x} \csc(x)$ .

See Solution

Problem 3739

Analyze the function $f(x)=\frac{3|x-2|}{x-2}$ and determine which statements are true: I, II, III.

See Solution

Problem 3740

Given $f(x)=\frac{3|x-2|}{x-2}$ , which statements are true: I. $\lim_{x \to 2} f(x)$ exists, II. $f$ is continuous at $x=2$ , III. $f$ is differentiable at $x=2$ ?

See Solution

Problem 3741

Find the derivative $\frac{d}{dy} g$ from $g^3 + y^3 = Z$ and express it as $A\left(\frac{y}{g}\right)^{a} + B(y \cdot g)^{b} + C$ . Determine coefficients $A$ , $a$ , $B$ , $b$ , and $C$ .

See Solution

Problem 3742

Find the first-order approximation of $f(x)=9x^4$ around $x_0=2$ . Determine coefficients $a$ and $b$ for $f(x) \approx a+b(x-2)$ .

See Solution

Problem 3743

Find first-order approximation of $f(x)=9x^{4}$ around $x_{0}=2$ . Express as $f(x) \approx a+b\cdot(x-2)$ . Find $a$ and $b$ .

See Solution

Problem 3744

Find the derivative of $6 \sqrt{7 x^{10}+6 x^{3}}$ using the chain rule.

See Solution

Problem 3745

Find linear and quadratic approximations of $f(x)=9x^4$ at $x_0=2$ , then estimate $f(2.1)$ using both.

See Solution

Problem 3746

Evaluate the function $f(x)=\frac{3|x+2|}{x+2}$ and determine which statements about limits, continuity, and differentiability at $x=-2$ are true.

See Solution

Problem 3747

Evaluate the function $f(x)=\frac{2|x+1|}{x+1}$ for the following: Which statements are true about $x=-1$ ?

See Solution

Problem 3748

Determine which statements are true for the function $f(x)=\frac{|x|}{x}$ : I. $\lim _{x \rightarrow 0} f(x)$ exists. II. $f$ is continuous at $x=0$ . III. $f$ is differentiable at $x=0$ .

See Solution

Problem 3749

Evaluate the function $f(x)=\frac{|x|}{x}$ and determine which statements are true: I. $\lim _{x \rightarrow 0} f(x)$ exists. II. $f$ is continuous at $x=0$ . III. $f$ is differentiable at $x=0$ .

See Solution

Problem 3750

Given $x^{-3} \cdot z^{4}=81$ , find: (a) $z(1)$ , (b) $z'(1)$ , (c) $z''(1)$ , and (d) $x'(3)$ .

See Solution

Problem 3751

Find the second-order approximation of $f(x) = 9x^4$ around $x_0 = 2$ . Determine $a, b, c$ for both forms:
1. $f(x) \approx a+b \cdot(x-2)+\frac{1}{2} \cdot c \cdot(x-2)^{2}$
2. $f(x) \approx a+b x+c x^{2}$

See Solution

Problem 3752

Find the limit $L$ as $t$ approaches 1 for the piecewise function defined by $h(t)$ .

See Solution

Problem 3753

Find the derivative of $y$ from $2x^{3}+4xy-7y^{2}=253$ and express it as $\frac{d}{dx} y=-\frac{a \cdot x^{b}+c \cdot x y+d \cdot y^{e}+f}{g \cdot x^{h}+j \cdot x y+k \cdot y^{\prime}+m}$ .

See Solution

Problem 3754

Find $\frac{d}{d s} r$ for $r^{5}+s^{5}=Q$ in the form $A\left(\frac{s}{r}\right)^{a}+B(s \cdot r)^{b}+C$ . Determine $A$ , $a$ , $B$ , $b$ , and $C$ .

See Solution

Problem 3755

Find if $f(x)$ goes to $\infty$ or $-\infty$ as $x$ approaches 4 from the left and right for $f(x)=\frac{1}{x-4}$ .

See Solution

Problem 3756

Show that the function $f(x)=6 x^{3}-7$ has a zero in the interval $[1,2]$ using the Intermediate Value Theorem.

See Solution

Problem 3757

Find the derivative of $f(x)=6 x^{4}+3 x^{3}+20 x^{1/4}+7 x^{-5}+4 x+21$ and determine coefficients $a$ through $k$ .

See Solution

Problem 3758

Approximate the integral $\int_{0}^{\pi} x^{2} \sin (x) d x$ using the Midpoint Rule with $n=6$ . Round to six decimal places.

See Solution

Problem 3759

Approximate the integral $\int_{0}^{\pi} x^{2} \sin (x) d x$ using the Midpoint Rule with $n=6$ . Round to six decimal places. $M_{6}=$

See Solution

Problem 3760

Find the derivative of $g(f(x))$ at $x=12$ where $f(x)=4(x-8)^{2}-60$ and $g(y)=3y^{4}+3y$ .

See Solution

Problem 3761

Differentiate $\ln \left(\frac{\sqrt{x^{2}+3 x}}{(x^{4}-4 x^{3})^{5}}\right)$ with respect to $x$ .

See Solution

Problem 3762

Find the derivative of the function $f(x)=7 e^{5 x}+3 \ln (x^{5}+3 x^{4}(x-2)^{2}+\frac{4}{x}+30)$ and identify coefficients $a$ to $n$ .

See Solution

Problem 3763

See Solution

Problem 3764

See Solution

Problem 3765

Find the limit: $\lim _{x \rightarrow 1} \frac{2-7 x}{x+6}$ .

See Solution

Problem 3766

Find the limit: $\lim _{h \rightarrow 0} \frac{2 h+h^{2}}{h}$ .

See Solution

Problem 3767

Füllvorgang eines Beckens:
a) Beschreibe die Zuflussgeschwindigkeit $v(t)=5t-t^2$ und bestimme den Zeitpunkt der maximalen Geschwindigkeit. b) Berechne die gesamte Wassermenge und die Menge zwischen Stunde 2 und 4. c) Bei Abflussrate von $3 \frac{\mathrm{m}^{3}}{\mathrm{~h}}$ , wann ist das Becken leer? d) Skizziere die Wasserstandsfunktion und nenne die Funktionsgleichung.

See Solution

Problem 3768

Find the change in marginal revenue from selling 20 to 30 units for $R(s)=s^{2}-2s-100$ .

See Solution

Problem 3769

Find the derivative of the function $\frac{\ln x}{\cos 2x}$ .

See Solution

Problem 3770

Find the derivative of the function $f(x)=e^{-1 / x^{2}}$ .

See Solution

Problem 3771

Find the derivative of $\frac{\ln x}{\cos 2 x}$ . Choose from the options provided.

See Solution

Problem 3772

Calculate the average rate of change of $f(x)=-3 x^{2}+4$ for these intervals: (a) 2 to 4, (b) 3 to 5, (c) 0 to 3.

See Solution

Problem 3773

Find the limit: $\lim _{t \rightarrow 9} \frac{t-9}{\sqrt{t}-3}$ .

See Solution

Problem 3774

A drug with a half-life of 48 hours is given as a 15 mg dose. How much is left at noon the next day?

See Solution

Problem 3775

Find the limit as $x$ approaches -8 from the left for the expression $\frac{x+2}{x+8}$ .

See Solution

Problem 3776

Find the limit as $x$ approaches 4 for $\frac{x^{2} f(x)-16 f(4)}{x-4}$ in terms of $f$ and $f'$ .

See Solution

Problem 3777

Ordne jeder Funktion f eine Stammfunktion F zu: I $8 x^{3}-3$ , II $2 x-4$ , III $(x+2)^{2}$ , IV $3(x^{2}-2 x^{3})$ , V $2 x-\frac{1}{x^{2}}$ , VI $\frac{x^{2}-9}{x-3}$ . F: A $x^{3}-\frac{3}{2} x^{4}-2$ , B $\frac{1}{2} x^{2}+3 x$ , D $(x-2)^{2}$ , E $2 x^{2}+4 x+\frac{1}{3} x^{3}+C$ .

See Solution

Problem 3778

Zeichne die Graphen der Funktionen und finde den $\mathrm{x}$ -Wert für Min/Max. Beschreibe das Steigungsverhalten. a) $y=0,6 x^{2}-4,8 x+3,6$ b) $y=-1,2 x^{2}-7,2 x-3,8$ c) $y=1,2 x^{2}-8,4 x+14,7$

See Solution

Problem 3779

Zeigen Sie, dass alle Tangenten im Wendepunkt der Funktion $f_{+}(x)=\frac{1}{4} x^{4}+t x^{3}+x^{2}$ auf der $y$ -Achse schneiden.

See Solution

Problem 3780

See Solution

Problem 3781

See Solution

Problem 3782

Untersuche die Funktion $f(x)=x^{4}-k x^{2}$ auf Extrem- und Wendepunkte und bestimme die Ortskurve der Wendepunkte.

See Solution

Problem 3783

Find the derivative of the function $g(x)=\frac{2+x^{2} f(x)}{\sqrt{x}}$ where $f$ is differentiable.

See Solution

Problem 3784

Find the derivative of the function $f(x)=\sqrt{3 x}-\frac{1}{\sqrt{3 x}}$ .

See Solution

Problem 3785

Find the derivative of $F(x)=\frac{f(x)-3x}{f(x)+3x}$ where $f$ is differentiable.

See Solution

Problem 3786

Find the value of $b$ so that the function $f(x)=\begin{cases} 2x^2+6, & x \leq 2 \\ mx+b, & x > 2 \end{cases}$ is differentiable for all $x$ .

See Solution

Problem 3787

Bestimme die Ableitung der Funktion $f(x)=(4 x-4) e^{-2 x}$ .

See Solution

Problem 3788

1. Finde alle $x \in [0 ; 3 \pi]$ mit $\cos (x)=0,6$ .
2. Berechne die Ableitung von $f(x)=(4 x-4) e^{-2 x}$ .

See Solution

Problem 3789

1. (a) For the hyperbola $y=\frac{2 x-2}{x-3}$ , find: (i) general form, (ii) $y$ at $x=2$ , (iii) $x$ at $y=3$ , (iv) derivative $\frac{d y}{d x}$ , (v) tangent equation at $x=2$ .

See Solution

Problem 3790

Find the second derivative of $f^{\prime} u(x)=\frac{-8 u x}{(1+u x^{2})^{2}}$ .

See Solution

Problem 3791

Die Füllmenge eines Wasserbehälters wird durch $F(t)=-0,0625 t^{4}+0,57 t^{3}-1,875 t^{2}+2,4 t+19,5$ beschrieben.
a) Wann ist $F(t) > 20 \mathrm{~m}^{3}$ ? b) Bestimme den Zeitpunkt und Wert des höchsten Füllstands. c) Berechne die mittlere Änderungsrate zwischen Tag 2 und 5. d) Wann ändert sich der Füllstand am stärksten? Gib die momentane Änderungsrate an.
Für die Funktionenschar $f_{a}(x)=\frac{1}{4} x^{4}-2 a x^{2}+a$ : a) Finde die Terme für $a=1$ und $a=-2$ . b) Bestimme $x$ für $y=a$ . c) Zeige die Extremstellen bei $\pm 2 \sqrt{a}$ und den Wertebereich für $a$ . d) Zeige die Koordinaten eines Extrempunkts. e) Finde $a$ für Extrempunkte auf der X-Achse. f) Bestimme die Ortskurve der Extrempunkte. g) Berechne die Tangentensteigung bei $x=1$ und für welches $a$ sie -7 ist.
Für die Funktion $f(x)=\frac{1}{3} x^{3}-3 x^{2}+8 x$ : a) Berechne die Nullstellen. b) Zeige, dass der Wendepunkt bei $W(3 \mid 6)$ liegt. c) Berechne die Wendetangente.

See Solution

Problem 3792

Untersuche die Funktion $F(t)=-0,0625 t^{4}+0,57 t^{3}-1,875 t^{2}+2,4 t+19,5$ für $0 \leq t \leq 7$ und beantworte a)-d).

See Solution

Problem 3793

Find the derivative of $g(f(x))$ at $x=8$ where $f(x)=5(x-3)^{2}-121$ and $g(y)=5y^{2}+4y$ .

See Solution

Problem 3794

Find the derivative of $g(f(x))$ at $x=8$ , where $f(x)=5(x-3)^{2}-121$ and $g(y)=5y^{2}+4y$ . Use the chain rule.

See Solution

Problem 3795

Find the derivative of $f(x)=7 e^{6 x}+4 \ln(x^{5}+3 x^{4}(x-2)^{2}+\frac{3}{x}+41)$ and determine coefficients $a$ to $n$ .

See Solution

Problem 3796

Ordnen Sie jeder Funktion eine Stammfunktion zu: I) $8x^{3}-3$ , II) $2x-4$ , III) $(x+2)^{2}$ , IV) $3(x^{2}-2x^{3})$ .

See Solution

Problem 3797

Gegeben ist die Funktion $f(x)=\frac{1}{3} x^{3}-3 x^{2}+8 x$ . Finde die Nullstellen, den Wendepunkt $W(3 \mid 6)$ und die Wendetangente.

See Solution

Problem 3798

Find the derivative of $h(x)=(2-2x)^{2}(3x+1)^{2}$ .

See Solution

Problem 3799

Find $\frac{d y}{d x}$ using the chain rule for $y=4 u^{4}+2$ and $u=3 \sqrt{x}$ .

See Solution

Problem 3800

Find the derivative of $f(x)=\frac{1}{2 x}$ using first principles.

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 3701

Find the derivative of f(x)=2x4+5x3+4x−6e3f(x)=2x^{4}+\frac{5}{x^{3}}+4\sqrt{x}-6e^{3}f(x)=2x4+x35​+4x​−6e3.

Problem 3702

A car's velocity is v(t)=t3v(t)=t^{3}v(t)=t3. Find average acceleration over [2,3][2,3][2,3], [3,4][3,4][3,4], [t,t+1][t,t+1][t,t+1], [2,2+h][2,2+h][2,2+h], and at t=2t=2t=2.

Problem 3703

Evaluate the integral: ∫(ln⁡x)12xdx\int \frac{(\ln x)^{12}}{x} d x∫x(lnx)12​dx

Problem 3704

Find the derivative of f(t)=(t−1)′f(t)=(t-1)^{\prime}f(t)=(t−1)′ at t=5t=5t=5 and round to the nearest integer.

Problem 3705

Problem 3706

Find the derivative of f(t)=(t−1)tf(t)=(t-1)^{t}f(t)=(t−1)t at t=5t=5t=5 and round to the nearest integer.

Problem 3707

Define f(2)f(2)f(2) to make f(x)=x3−8x2−4f(x)=\frac{x^{3}-8}{x^{2}-4}f(x)=x2−4x3−8​ continuous at 2. How do you remove the discontinuity?

Problem 3708

Find the derivative of the function f(t)=(t−1)tf(t)=(t-1)^{t}f(t)=(t−1)t at t=5t=5t=5 and round to the nearest integer.

Problem 3709

Find the derivative of f(x)=6x4+5x3+25x1/5+6x−4+4x+37f(x)=6x^4+5x^3+25x^{1/5}+6x^{-4}+4x+37f(x)=6x4+5x3+25x1/5+6x−4+4x+37 and determine coefficients aaa through kkk.

Problem 3710

Problem 3711

Problem 3712

Find the derivative of f(r)=(r−2)rf(r)=(r-2)^{r}f(r)=(r−2)r at r=5r=5r=5 and round to the nearest integer.

Problem 3713

Find the elasticity Elg(1)E_l g(1)El​g(1) and slope of g(y)=3y+22g(y)=3y+22g(y)=3y+22 at y=1y=1y=1. Round answers to two digits.

Problem 3714

Problem 3715

Find the elasticity of the function t(z)=30e5z2t(z) = 30 e^{5 z^{2}}t(z)=30e5z2 and express it as Elzt(z)=2⋅z30+0+0⋅z1⋅e5z2E l_{z} t(z) = 2 \cdot z^{30} + 0 + 0 \cdot z^{1} \cdot e^{5 z^{2}}Elz​t(z)=2⋅z30+0+0⋅z1⋅e5z2. Round to two digits.

Problem 3716

Find the elasticity of the function t(z)=30e5z2t(z) = 30 e^{5 z^{2}}t(z)=30e5z2. Express it as Elzt(z)=a⋅zb+c⋅e5z2+d⋅ze⋅e5z2E l_{z} t(z) = a \cdot z^{b} + c \cdot e^{5 z^{2}} + d \cdot z^{e} \cdot e^{5 z^{2}}Elz​t(z)=a⋅zb+c⋅e5z2+d⋅ze⋅e5z2. Round answers to two digits.

Problem 3717

Find the elasticity of the function t(z)=31z2ln⁡zt(z) = 31 z^{2} \ln zt(z)=31z2lnz and determine coefficients aaa, bbb, ccc, ddd, eee, fff, and ggg.

Problem 3718

Problem 3719

Find the elasticity of the function t(z)=33(z4+1)1/4t(z) = 33(z^{4}+1)^{1/4}t(z)=33(z4+1)1/4 and express it as Elzt(z)=a⋅zb⋅(zc+d)eE l_{z} t(z) = a \cdot z^{b} \cdot (z^{c}+d)^{e}Elz​t(z)=a⋅zb⋅(zc+d)e.

Problem 3720

Find the elasticity and slope of the function g(y)=7y+25g(y) = 7y + 25g(y)=7y+25 at y0=1y_0 = 1y0​=1. Round answers to two digits.

Problem 3721

Find the derivative of w(x)=ln⁡(x2+3x)tan⁡(4x)w(x)=\ln(x^{2}+3x) \tan(4x)w(x)=ln(x2+3x)tan(4x).

Problem 3722

An object is thrown upwards at 20 m/s20 \mathrm{~m} / \mathrm{s}20 m/s. Find: a) speed upon hitting ground, b) total time in air, c) max height.

Problem 3723

Find z(1)z(1)z(1), z′(1)z'(1)z′(1), z′′(1)z''(1)z′′(1), and x′(8)x'(8)x′(8) for the function defined by x−9⋅z4=4096x^{-9} \cdot z^{4}=4096x−9⋅z4=4096.

Problem 3724

Find the derivative of the function defined by 4x3+5xy−8y4=2234 x^{3}+5 x y-8 y^{4}=2234x3+5xy−8y4=223 and identify coefficients in the expression for ddxy\frac{d}{d x} ydxd​y.

Problem 3725

Problem 3726

Evaluate the integral ∫x2ln⁡x dx\int x^{2} \ln x \, dx∫x2lnxdx using integration by parts with u=ln⁡xu=\ln xu=lnx and dv=x2 dxdv=x^{2} \, dxdv=x2dx.

Problem 3727

Evaluate the function f(x)=3∣x−1∣x−1f(x)=\frac{3|x-1|}{x-1}f(x)=x−13∣x−1∣​ and determine which statements are true: I, II, III about limits, continuity, and differentiability at x=1x=1x=1.

Problem 3728

Find the derivative of the function defined by g5+y5=Zg^{5}+y^{5}=Zg5+y5=Z as ddyg=A(yg)a+B(y⋅g)b+C\frac{d}{d y} g=A\left(\frac{y}{g}\right)^{a}+B(y \cdot g)^{b}+Cdyd​g=A(gy​)a+B(y⋅g)b+C and identify coefficients AAA, aaa, BBB, bbb, and CCC.

Problem 3729

Find z(1)z(1)z(1), z′(1)z'(1)z′(1), z′′(1)z''(1)z′′(1), and x′(7)x'(7)x′(7) from the equation x−8z3=343x^{-8} z^3 = 343x−8z3=343, rounding to two digits.

Problem 3730

Integrate θcos⁡θ\theta \cos \thetaθcosθ using integration by parts with u=θu=\thetau=θ, dv=cos⁡θdθdv=\cos \theta d\thetadv=cosθdθ.

Problem 3731

Find the derivative of yyy in the equation 4x3+4xy−3y4=2254 x^{3}+4 x y-3 y^{4}=2254x3+4xy−3y4=225 and determine the coefficients in the expression.

Problem 3732

Find z(1)z(1)z(1), z′(1)z'(1)z′(1), z′′(1)z''(1)z′′(1), and x′(7)x'(7)x′(7) for the equation x−8⋅z3=343x^{-8} \cdot z^{3}=343x−8⋅z3=343. Round to two digits.

Problem 3733

Find the derivative dduv\frac{d}{du} vdud​v from u6+v6=Tu^{6}+v^{6}=Tu6+v6=T and express it as A(uv)a+B(u⋅v)b+CA\left(\frac{u}{v}\right)^{a}+B(u \cdot v)^{b}+CA(vu​)a+B(u⋅v)b+C. What are the coefficients AAA, aaa, BBB, bbb, and CCC?

Problem 3734

Determine which statements about f(x)=3∣x−1∣x−1f(x)=\frac{3|x-1|}{x-1}f(x)=x−13∣x−1∣​ are true: I. lim⁡x→1f(x)\lim _{x \rightarrow 1} f(x)limx→1​f(x) exists. II. fff is continuous at x=1x=1x=1. III. fff is differentiable at x=1x=1x=1.

Problem 3735

Evaluate the function f(x)=2∣x−1∣x−1f(x)=\frac{2|x-1|}{x-1}f(x)=x−12∣x−1∣​ and determine which statements about limits, continuity, and differentiability at x=−1x=-1x=−1 are true.

Problem 3736

Find z(1)z(1)z(1), z′(1)z'(1)z′(1), z′′(1)z''(1)z′′(1), and x′(5)x'(5)x′(5) from x−6⋅z3=125x^{-6} \cdot z^{3}=125x−6⋅z3=125. Round answers to two digits.

Problem 3737

Find the derivative of the function defined by 3x4+8xy−3y3=1893 x^{4}+8 x y-3 y^{3}=1893x4+8xy−3y3=189 and identify coefficients a,b,c,d,e,f,g,h,j,k,l,ma, b, c, d, e, f, g, h, j, k, l, ma,b,c,d,e,f,g,h,j,k,l,m.

Problem 3738

Find the derivative of y=ecos⁡xcsc⁡(x)y = e^{\cos x} \csc(x)y=ecosxcsc(x).

Problem 3739

Analyze the function f(x)=3∣x−2∣x−2f(x)=\frac{3|x-2|}{x-2}f(x)=x−23∣x−2∣​ and determine which statements are true: I, II, III.

Problem 3740

Given f(x)=3∣x−2∣x−2f(x)=\frac{3|x-2|}{x-2}f(x)=x−23∣x−2∣​, which statements are true: I. lim⁡x→2f(x)\lim_{x \to 2} f(x)limx→2​f(x) exists, II. fff is continuous at x=2x=2x=2, III. fff is differentiable at x=2x=2x=2?

Problem 3741

Find the derivative ddyg\frac{d}{dy} gdyd​g from g3+y3=Zg^3 + y^3 = Zg3+y3=Z and express it as A(yg)a+B(y⋅g)b+CA\left(\frac{y}{g}\right)^{a} + B(y \cdot g)^{b} + CA(gy​)a+B(y⋅g)b+C. Determine coefficients AAA, aaa, BBB, bbb, and CCC.

Problem 3742

Find the first-order approximation of f(x)=9x4f(x)=9x^4f(x)=9x4 around x0=2x_0=2x0​=2. Determine coefficients aaa and bbb for f(x)≈a+b(x−2)f(x) \approx a+b(x-2)f(x)≈a+b(x−2).

Problem 3743

Find the derivative of $f(x)=2x^{4}+\frac{5}{x^{3}}+4\sqrt{x}-6e^{3}$ .

A car's velocity is $v(t)=t^{3}$ . Find average acceleration over $[2,3]$ , $[3,4]$ , $[t,t+1]$ , $[2,2+h]$ , and at $t=2$ .

Evaluate the integral: $\int \frac{(\ln x)^{12}}{x} d x$

Find the derivative of $f(t)=(t-1)^{\prime}$ at $t=5$ and round to the nearest integer.

Find the derivative of $f(t)=(t-1)^{t}$ at $t=5$ and round to the nearest integer.

Define $f(2)$ to make $f(x)=\frac{x^{3}-8}{x^{2}-4}$ continuous at 2. How do you remove the discontinuity?

Find the derivative of the function $f(t)=(t-1)^{t}$ at $t=5$ and round to the nearest integer.

Find the derivative of $f(x)=6x^4+5x^3+25x^{1/5}+6x^{-4}+4x+37$ and determine coefficients $a$ through $k$ .

Find the derivative of $f(r)=(r-2)^{r}$ at $r=5$ and round to the nearest integer.

Find the elasticity $E_l g(1)$ and slope of $g(y)=3y+22$ at $y=1$ . Round answers to two digits.

Find the elasticity of the function $t(z) = 30 e^{5 z^{2}}$ and express it as $E l_{z} t(z) = 2 \cdot z^{30} + 0 + 0 \cdot z^{1} \cdot e^{5 z^{2}}$ . Round to two digits.

Find the elasticity of the function $t(z) = 30 e^{5 z^{2}}$ . Express it as $E l_{z} t(z) = a \cdot z^{b} + c \cdot e^{5 z^{2}} + d \cdot z^{e} \cdot e^{5 z^{2}}$ . Round answers to two digits.

Find the elasticity of the function $t(z) = 31 z^{2} \ln z$ and determine coefficients $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , and $g$ .

Find the elasticity of the function $t(z) = 33(z^{4}+1)^{1/4}$ and express it as $E l_{z} t(z) = a \cdot z^{b} \cdot (z^{c}+d)^{e}$ .

Find the elasticity and slope of the function $g(y) = 7y + 25$ at $y_0 = 1$ . Round answers to two digits.

Find the derivative of $w(x)=\ln(x^{2}+3x) \tan(4x)$ .

An object is thrown upwards at $20 \mathrm{~m} / \mathrm{s}$ . Find: a) speed upon hitting ground, b) total time in air, c) max height.

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(8)$ for the function defined by $x^{-9} \cdot z^{4}=4096$ .

Find the derivative of the function defined by $4 x^{3}+5 x y-8 y^{4}=223$ and identify coefficients in the expression for $\frac{d}{d x} y$ .

Evaluate the integral $\int x^{2} \ln x \, dx$ using integration by parts with $u=\ln x$ and $dv=x^{2} \, dx$ .

Evaluate the function $f(x)=\frac{3|x-1|}{x-1}$ and determine which statements are true: I, II, III about limits, continuity, and differentiability at $x=1$ .

Find the derivative of the function defined by $g^{5}+y^{5}=Z$ as $\frac{d}{d y} g=A\left(\frac{y}{g}\right)^{a}+B(y \cdot g)^{b}+C$ and identify coefficients $A$ , $a$ , $B$ , $b$ , and $C$ .

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(7)$ from the equation $x^{-8} z^3 = 343$ , rounding to two digits.

Integrate $\theta \cos \theta$ using integration by parts with $u=\theta$ , $dv=\cos \theta d\theta$ .

Find the derivative of $y$ in the equation $4 x^{3}+4 x y-3 y^{4}=225$ and determine the coefficients in the expression.

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(7)$ for the equation $x^{-8} \cdot z^{3}=343$ . Round to two digits.

Find the derivative $\frac{d}{du} v$ from $u^{6}+v^{6}=T$ and express it as $A\left(\frac{u}{v}\right)^{a}+B(u \cdot v)^{b}+C$ . What are the coefficients $A$ , $a$ , $B$ , $b$ , and $C$ ?

Determine which statements about $f(x)=\frac{3|x-1|}{x-1}$ are true: I. $\lim _{x \rightarrow 1} f(x)$ exists. II. $f$ is continuous at $x=1$ . III. $f$ is differentiable at $x=1$ .

Evaluate the function $f(x)=\frac{2|x-1|}{x-1}$ and determine which statements about limits, continuity, and differentiability at $x=-1$ are true.

Find $z(1)$ , $z'(1)$ , $z''(1)$ , and $x'(5)$ from $x^{-6} \cdot z^{3}=125$ . Round answers to two digits.

Find the derivative of the function defined by $3 x^{4}+8 x y-3 y^{3}=189$ and identify coefficients $a, b, c, d, e, f, g, h, j, k, l, m$ .

Find the derivative of $y = e^{\cos x} \csc(x)$ .

Analyze the function $f(x)=\frac{3|x-2|}{x-2}$ and determine which statements are true: I, II, III.

Given $f(x)=\frac{3|x-2|}{x-2}$ , which statements are true: I. $\lim_{x \to 2} f(x)$ exists, II. $f$ is continuous at $x=2$ , III. $f$ is differentiable at $x=2$ ?

Find the derivative $\frac{d}{dy} g$ from $g^3 + y^3 = Z$ and express it as $A\left(\frac{y}{g}\right)^{a} + B(y \cdot g)^{b} + C$ . Determine coefficients $A$ , $a$ , $B$ , $b$ , and $C$ .

Find the first-order approximation of $f(x)=9x^4$ around $x_0=2$ . Determine coefficients $a$ and $b$ for $f(x) \approx a+b(x-2)$ .

Find first-order approximation of $f(x)=9x^{4}$ around $x_{0}=2$ . Express as $f(x) \approx a+b\cdot(x-2)$ . Find $a$ and $b$ .

Find the derivative of $6 \sqrt{7 x^{10}+6 x^{3}}$ using the chain rule.

Find linear and quadratic approximations of $f(x)=9x^4$ at $x_0=2$ , then estimate $f(2.1)$ using both.

Evaluate the function $f(x)=\frac{3|x+2|}{x+2}$ and determine which statements about limits, continuity, and differentiability at $x=-2$ are true.

Evaluate the function $f(x)=\frac{2|x+1|}{x+1}$ for the following: Which statements are true about $x=-1$ ?

Determine which statements are true for the function $f(x)=\frac{|x|}{x}$ : I. $\lim _{x \rightarrow 0} f(x)$ exists. II. $f$ is continuous at $x=0$ . III. $f$ is differentiable at $x=0$ .

Evaluate the function $f(x)=\frac{|x|}{x}$ and determine which statements are true: I. $\lim _{x \rightarrow 0} f(x)$ exists. II. $f$ is continuous at $x=0$ . III. $f$ is differentiable at $x=0$ .

Given $x^{-3} \cdot z^{4}=81$ , find: (a) $z(1)$ , (b) $z'(1)$ , (c) $z''(1)$ , and (d) $x'(3)$ .

Find the limit $L$ as $t$ approaches 1 for the piecewise function defined by $h(t)$ .

Find $\frac{d}{d s} r$ for $r^{5}+s^{5}=Q$ in the form $A\left(\frac{s}{r}\right)^{a}+B(s \cdot r)^{b}+C$ . Determine $A$ , $a$ , $B$ , $b$ , and $C$ .

Find if $f(x)$ goes to $\infty$ or $-\infty$ as $x$ approaches 4 from the left and right for $f(x)=\frac{1}{x-4}$ .

Show that the function $f(x)=6 x^{3}-7$ has a zero in the interval $[1,2]$ using the Intermediate Value Theorem.

Find the derivative of $f(x)=6 x^{4}+3 x^{3}+20 x^{1/4}+7 x^{-5}+4 x+21$ and determine coefficients $a$ through $k$ .

Approximate the integral $\int_{0}^{\pi} x^{2} \sin (x) d x$ using the Midpoint Rule with $n=6$ . Round to six decimal places.

Approximate the integral $\int_{0}^{\pi} x^{2} \sin (x) d x$ using the Midpoint Rule with $n=6$ . Round to six decimal places. $M_{6}=$

Find the derivative of $g(f(x))$ at $x=12$ where $f(x)=4(x-8)^{2}-60$ and $g(y)=3y^{4}+3y$ .

Differentiate $\ln \left(\frac{\sqrt{x^{2}+3 x}}{(x^{4}-4 x^{3})^{5}}\right)$ with respect to $x$ .

Find the derivative of the function $f(x)=7 e^{5 x}+3 \ln (x^{5}+3 x^{4}(x-2)^{2}+\frac{4}{x}+30)$ and identify coefficients $a$ to $n$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{2-7 x}{x+6}$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{2 h+h^{2}}{h}$ .

Find the change in marginal revenue from selling 20 to 30 units for $R(s)=s^{2}-2s-100$ .

Find the derivative of the function $\frac{\ln x}{\cos 2x}$ .

Find the derivative of the function $f(x)=e^{-1 / x^{2}}$ .

Find the derivative of $\frac{\ln x}{\cos 2 x}$ . Choose from the options provided.

Calculate the average rate of change of $f(x)=-3 x^{2}+4$ for these intervals: (a) 2 to 4, (b) 3 to 5, (c) 0 to 3.

Find the limit: $\lim _{t \rightarrow 9} \frac{t-9}{\sqrt{t}-3}$ .