Calculus

Problem 1201

Identify the interval where the graph of $f$ is concave down based on its rate of change:
(A) $0<x<1$ (B) $1<x<2$ (C) $2<x<3$ (D) $3<x<4$

See Solution

Problem 1202

Find the vertical asymptotes of the function $y=\frac{x^{2}+2 x+1}{x+x^{2}}$ .

See Solution

Problem 1203

Evaluate the limit: $\lim _{x \rightarrow 1^{+}} \frac{-4}{1-x}$

See Solution

Problem 1204

Find $\lim _{x \rightarrow 2} g(x)$ given $2x + 1 \leq g(x) \leq x^2 - 2x + 5$ .

See Solution

Problem 1205

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{2}+2x-3}{x^{2}-1}$ .

See Solution

Problem 1206

Zairia freefalls for $3.2 \mathrm{~s}$ . How far does she fall in that time? Round to the nearest tenth.

See Solution

Problem 1207

Calculate John's velocity just before hitting the water after falling for 7.5 seconds. Round to the nearest tenth.

See Solution

Problem 1208

A tennis ball drops from $5.1 \mathrm{~m}$ . Find the time to hit the ground, rounded to the nearest hundredth.

See Solution

Problem 1209

Find the number of thousand mp3 players to produce for minimum marginal cost given $C(x)=x^{2}-100x+8200$ . What is the minimum cost?

See Solution

Problem 1210

An object moves as per the table. What's true about the rate of change of distance over time? A, B, C, or D?

See Solution

Problem 1211

Evaluate the expression $\left[\frac{8}{11} x^{11 / 8}\right]_{0}^{1}$ .

See Solution

Problem 1212

Find $f(1+h)$ for $f(x)=\frac{x^{2}}{8}$ and $h=1, 0.1, 0.01, 0.001, 0.0001$ , rounding to seven decimal places.

See Solution

Problem 1213

Find the instantaneous rate of change of $R(t)=240+30 t^{3}$ at $t=1$ . Round to one decimal place.

See Solution

Problem 1214

Find the limit as $h$ approaches 0 for $\frac{\frac{5}{-3+h}+\frac{5}{3}}{h}$ .

See Solution

Problem 1215

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{5}{-4+h}+\frac{5}{4}}{h}$ .

See Solution

Problem 1216

Find $a$ such that $\lim _{x \rightarrow 3} \frac{2 x^{2}+a x+a-2}{x^{2}+x-12}$ exists.

See Solution

Problem 1217

Find the non-negative value of $b$ such that the limit $\lim _{x \rightarrow 0}\left\{\frac{\sqrt{8 x+b}-4}{x}\right\}$ exists.

See Solution

Problem 1218

Find if the limit $\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$ exists for $f(x)=x^{2}+4x$ and calculate its value.

See Solution

Problem 1219

Which statements must be true by the Intermediate Value Theorem? Check if roots exist in $[2,4]$ for the given functions.

See Solution

Problem 1220

Which statements are true due to the Intermediate Value Theorem (IVT) regarding the functions and intervals given?

See Solution

Problem 1221

Find the volume of a solid with a circular base radius 4 and square cross-sections perpendicular to the $x$ -axis.

See Solution

Problem 1222

Evaluate the integral using substitution and partial fractions: $\int \frac{1}{x \sqrt{x+1}} d x$ .

See Solution

Problem 1223

Find the 2 positive terms in the integral of $\cos ^{4} x \sin ^{2} x \, dx$ .

See Solution

Problem 1224

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

See Solution

Problem 1225

Find the derivative $f^{\prime}(x)$ for $f(x)=3x-2$ using the limit definition: $\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 1226

Find the derivative $f^{\prime}(x)$ for $f(x)=-2 x^{2}+x-19$ using the limit definition: $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 1227

Determine if breakfast cereal consumption was changing faster in 2001 or 2011 using the model $c(t)=-0.0032 t^{3}+0.097 t^{2}-0.358 t+12.4$ .

See Solution

Problem 1228

Determine if cereal consumption was changing faster in 2005 or 2012 using $C(t)=-0.0035 t^{3}+0.09 t^{2}-0.347 t+12.25$ . Calculate $C'(15)$ and $C'(22)$ .

See Solution

Problem 1229

Complete the tables for the behavior of $f(x)=\frac{1}{x-2}$ , $f(x)=\frac{1}{x+5}$ , and $f(x)=\frac{1}{x+8}$ .

See Solution

Problem 1230

Given position $s=f(t)=4 t^{3}+2 t+9$ , find velocity $v(t)$ , $v(3)$ , acceleration $a(t)$ , and $a(3)$ .

See Solution

Problem 1231

Find the derivatives of these functions using formulas $3, 3, 1-3, 3.6$ :
1. $f(x)=2 x^{2}+3 x-7$
2. $f(x)=x^{5}+2 x^{4}+5 x^{3}+4 x^{2}+x+1$
3. $y=5 x^{2}-4 x+9-\frac{2}{3 x^{3}}+\frac{4}{x^{5}}$
4. $y=(x+7)(5 x-2)$
5. $g(x)=\frac{5 x+3}{7 x+2}$
6. $y=\frac{3-2 x}{4 x-3}$

See Solution

Problem 1232

Find the rate of change of cost in the function $y+2.75=1.5(x-1)$ for a taxi ride per mile. Choices: a) \$1.50 b) \$3.25 c) \$2.75 d) \$4.25.

See Solution

Problem 1233

Find the derivatives of these functions:
1. $h(x)=(x^{2}-3x+2)(x^{2}+5x+1)$
2. $h(x)=(5x^{2}+4x+8)(9-x-3x^{2})$
3. $y=\frac{3x^{2}+5x+4}{2x^{2}+6x+3}$
4. $y=\frac{x^{2}+2x+3}{x^{2}-4x+6}$

See Solution

Problem 1234

Find the derivatives of these functions using formulas 3.3.1-3.3.6:
1. $h(x)=(x^{2}-3 x+2)(x^{2}+5 x+1)$
2. $h(x)=(5 x^{2}+4 x+8)(9-x-3 x^{2})$
3. $y=\frac{3 x^{2}+5 x+4}{2 x^{2}+6 x+3}$
4. $y=\frac{x^{2}+2 x+3}{x^{2}-4 x+6}$

See Solution

Problem 1235

Find the derivatives of these functions:
1. $h(x)=(x^{2}-3x+2)(x^{2}+5x+1)$
2. $h(x)=(5x^{2}+4x+8)(9-x-3x^{2})$
3. $y=\frac{3x^{2}+5x+4}{2x^{2}+6x+3}$
4. $y=\frac{x^{2}+2x+3}{x^{2}-4x+6}$
Show final answers.

See Solution

Problem 1236

Find the limit as $h$ approaches 0 for $\frac{f(-6+h)-f(-6)}{h}$ where $f(x)=-6x^2-7x-1$ .

See Solution

Problem 1237

If $\lim _{x \rightarrow t} f(x)=y$ , does $f$ need to be defined at $x=t$ ?

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Problem 1238

Find the limit: $\lim _{x \rightarrow 5}\left(\frac{x^{2}-25}{x^{2}-8 x+15}\right)$ .

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Problem 1239

Find the limit as $x$ approaches -17 from the left for the piecewise function $f(x)$ .

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Problem 1240

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}}\left(\frac{x}{-3 x^{4}-11 x}\right)$ .

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Problem 1241

Find the limit: $\lim _{x \rightarrow 49}\left(\frac{\sqrt{x}-7}{x-49}\right)$ .

See Solution

Problem 1242

Find the limit: $\lim _{x \rightarrow 144}\left(\frac{x-144}{\sqrt{x}-12}\right)$ .

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Problem 1243

Calculate the limit as $x$ approaches 4 of the expression $\frac{4 x^{2}+9 x+8}{6 x^{2}-3 x+7}$ .

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Problem 1244

Identify the discontinuity points for the function $f(x)=\frac{x}{x^{2}-1}$ .

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Problem 1245

Considérez la fonction $f(x)=\frac{7}{2}-\frac{1}{2}(e^{x}+e^{-x})$ . Trouvez la limite de $f$ à $+\infty$ , montrez qu'elle est décroissante sur $[0;+\infty[$ et que $f(x)=0$ a une unique solution $\alpha$ sur cet intervalle. Justifiez que $f(x)=0$ a deux solutions opposées dans $\mathbb{R}$ .

See Solution

Problem 1246

Find the marginal revenue when 3 units are sold, given the demand function $p=72-7.5q$ .

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Problem 1247

Find the instantaneous rate of change of $R(t)=240+30 t^{3}$ at $t=1$ . Answer: $R^{\prime}(1)=90$ dollars/day.

See Solution

Problem 1248

Evaluate the integral: $\int_{0}^{1} 6 \frac{\ln (x)}{\sqrt[3]{x}} d x$

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Problem 1249

Find the derivative of $(x+1) \sin x$ .

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Problem 1250

Estimate the soccer ball's instantaneous velocity at $t=2$ s using the average velocities given. Round to two decimal places.

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Problem 1251

Find the distance the ball travels from $t=4$ to $t=4.5$ using $s(t)=4.9 t^{2}$ .

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Problem 1252

Find the average velocity from $t=4$ to $t=4.5$ for $s(t) = 4.9t^2$ . Use decimal notation.

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Problem 1253

Estimate the ball's instantaneous velocity at $t=4$ using average velocities over $[4,4.1]$ , $[4,4.01]$ , and $[4,4.001]$ . Round to one decimal place.

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Problem 1254

Find the instantaneous velocity of a wrench at $t=6$ seconds, given $s(t)=4.9 t^{2} \mathrm{~m}$ .

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Problem 1255

Prove that $\lim _{x \rightarrow 6} 4 x=24$ using the precise definition of a limit.

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Problem 1256

Find the stone's average velocity over the interval $[0.5,3]$ given its height function $h=30t-4.9t^2$ .

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Problem 1257

Find the average velocity of a rock thrown upward on Mars, described by $y=10t-1.9t^2$ , over the interval $[2,3]$ .

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Problem 1258

Find the average velocity $\bar{v}$ of a stone with height $h(t)=30t-4.9t^2$ over intervals around $t=3$ .

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Problem 1259

Find the slope of the tangent line at $x=7$ for $f(x)=\ln(x)$ . Round your answer to three decimal places.

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Problem 1260

Estimate the slope of the tangent line for $y(t)=\sqrt{2 t+1}$ at $t=3$ . Round your answer to three decimal places.

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Problem 1261

Find the slope of the tangent line at $t=-7$ for the function $f(t)=2t-3$ . Provide your answer as a whole number.

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Problem 1262

Find the slope of the tangent line for $f(x)=3 e^{x}$ at $x=e$ . Round your answer to two decimal places.

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Problem 1263

Estimate the slope of the tangent line for $P(x)=5 x^{2}+7$ at $x=4$ . Provide a whole number answer.

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Problem 1264

Find the slope of the tangent line for $f(x)=3 e^{x}$ at $x=e$ . Round your answer to two decimal places.

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Problem 1265

Find the average velocity of $s(t)=t^{3}+2t$ over $[2,8]$ and estimate the instantaneous velocity at $t=2$ .

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Problem 1266

Find the average rate of change of $y=f(x)$ from $x=4$ to $x=7$ using the given values. Round to two decimal places.

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Problem 1267

Evaluate the function $f(x)$ and determine which statements about its limit and continuity at $x=1$ are true. Options: A. Only I, B. Only II, C. I and II, D. None, E. All.

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Problem 1268

Find the stone's average velocity over the interval $[0.5,3]$ using the height function $h=35 t-4.9 t^{2}$ .

See Solution

Problem 1269

Estimate the instantaneous velocity of a wrench at $t=5$ seconds, given $s(t)=4.9 t^{2} \mathrm{~m}$ .

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Problem 1270

Find the average velocity of $s(t)=t^{3}+2t$ over $[2,8]$ and estimate the instantaneous velocity at $t=2$ .

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Problem 1271

Estimate the slope of the tangent line for $y(t)=\sqrt{4t+1}$ at $t=1$ . Round your answer to three decimal places.

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Problem 1272

Find the slope of the tangent line at $t=-7$ for the function $f(t)=2t-3$ . Provide your answer as a whole number.

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Problem 1273

Estimate the slope of the tangent line for $P(x)=6 x^{2}+8$ at $x=6$ . Provide your answer as a whole number.

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Problem 1274

Estimate the slope of the tangent line at $x=11$ for $f(x)=\ln (x)$ . Round your answer to three decimal places.

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Problem 1275

Find the slope of the tangent line for $f(x)=2 e^{x}$ at $x=e$ . Round your answer to two decimal places.

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Problem 1276

Evaluate the integral from 0 to 1: $\int_{0}^{1} \sqrt{x^{2}+1} \, dx$ .

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Problem 1277

Find the average velocity $\bar{v}$ of a stone tossed with $h(t)=35t-4.9t^2$ over intervals $[1,1.01], [1,1.001], [1,1.0001], [0.9999,1], [0.999,1], [0.99,1]$ . Round to three decimal places.

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Problem 1278

Find $f^{\prime}(x)$ using the Fundamental Theorem of Calculus for $f(x)=\int_{-4}^{x^{3}} \sqrt{t^{2}+9} dt$ .

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Problem 1279

Determine the interval(s) where the curve $y=\int_{0}^{x} \frac{t^{2}}{t^{2}-3 t+5} d t$ is concave upward.

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Problem 1280

Find a function $f$ and a number $a$ such that $2+\int_{a}^{x} \frac{f(t)}{t^{5}} dt=6 x^{-1}$ . What is $f(x)$ ?

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Problem 1281

Evaluate the integral from 9 to 15 of $-(4x + 5)$ using area formulas. What is $\int_{9}^{15}(-(4x+5)) dx$ ?

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Problem 1282

Estimate the integrals using Riemann sums: (a) left-endpoint $\int_{0}^{10} f(x) dx \approx$ and (b) right-endpoint $\int_{0}^{10} f(x) dx \approx$ .

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Problem 1283

Find a function $f$ and a number $a$ such that $2 + \int_{a}^{x} \frac{f(t)}{t^{5}} dt = 6 x^{-1}$ .

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Problem 1284

Use the table of the increasing function $f$ to find lower and upper estimates for $\int_{0}^{25} f(x) dx$ . Lower estimate = , Upper estimate = .

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Problem 1285

Compute the following integrals using given values:
(a) $\int_{0}^{4} 4 f(x) d x=$ (b) $\int_{0}^{4}(-1 f(x)-3 g(x)) d x=$ (c) $\int_{0}^{4}(3-5 h(x)) d x=$

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Problem 1286

Compute the integrals given:
(a) $\int_{0}^{4} 4 f(x) d x = 62$
(b) $\int_{0}^{4}(-1 f(x)-3 g(x)) d x=$ ?

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Problem 1287

Given the integrals:
1. $\int_{2}^{6} f(x) dx = -10$ , $\int_{6}^{8} f(x) dx = -10$
2. $\int_{2}^{6} g(x) dx = -4$ , $\int_{6}^{8} g(x) dx = 4$
Find:
(a) $\int_{2}^{8}(f(x)+g(x)) dx=$ (number)
(b) $\int_{2}^{8}(f(x)-g(x)) dx=$ (number)
(c) $\int_{2}^{8}(7 f(x)-3 g(x)) dx=$ (number)

See Solution

Problem 1288

Find the average value $f_{\text{ave}}$ of $f(x)=32-|4x|$ from $x=-8$ to $x=8$ and points $c$ where $f(c)=f_{\text{ave}}$ .

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Problem 1289

Find the average value $f_{\text {ave }}$ of $f(x)=32-|4 x|$ for $x$ in the range $[-8, 8]$ . What is $f_{\text {ave }}$ ?

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Problem 1290

Evaluate the integral from 3 to 19 of $7 - |x - 11|$ using area formulas. What is $\int_{3}^{19}(7 - |x - 11|) d x$ ?

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Problem 1291

Find the right endpoint Riemann sum for $f(x)=\frac{15}{x}$ on $[2,6]$ using 8 rectangles of width 0.5. What is the sum?

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Problem 1292

Calculate the left endpoint Riemann sum for $f(x)=\frac{15}{x}$ over $[2,6]$ using 8 rectangles of width 0.5.

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Problem 1293

Calculate the left endpoint Riemann sum for $f(x)=\frac{x^{2}}{12}$ on $[2,6]$ and explain why it's an underestimate.

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Problem 1294

Find the indefinite integral: $\int \frac{1}{(x-1)(x+3)} dx = \sqrt{2} + C$ .

See Solution

Problem 1295

Evaluate the integral using partial fractions: $\int -\frac{2}{(x-6)^{2}(x+6)} \, dx = C$

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Problem 1296

Evaluate the expression: $-\frac{1}{18} \int \frac{1}{x-6} dx + \frac{1}{108} \int \frac{1}{(x-6)^{2}} dx - \frac{1}{216} \int \frac{1}{x+6} dx$ .

See Solution

Problem 1297

Evaluate the integral: $\int \frac{7 x+25}{(7-x)\left(x^{2}+25\right)} d x=$

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Problem 1298

Find the distance $d$ traveled by a particle with velocity $v(t)=\frac{7 t^{2}}{t^{2}+1}$ after $t=3$ sec.

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Problem 1299

A rumor spreads in a school. Given $y^{\prime}(t)=k y(1-y)$ , find when $90\%$ of 1000 students hear it.

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Problem 1300

Solve the logistic equation $\frac{d y}{d x}=5 y\left(1-\frac{y}{10}\right)$ for $y$ and find $y(0)=6$ .

See Solution

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Calculus

Problem 1201

Identify the interval where the graph of fff is concave down based on its rate of change: (A) 0<x<10<x<10<x<1 (B) 1<x<21<x<21<x<2 (C) 2<x<32<x<32<x<3 (D) 3<x<43<x<43<x<4

Problem 1202

Find the vertical asymptotes of the function y=x2+2x+1x+x2y=\frac{x^{2}+2 x+1}{x+x^{2}}y=x+x2x2+2x+1​.

Problem 1203

Evaluate the limit: lim⁡x→1+−41−x\lim _{x \rightarrow 1^{+}} \frac{-4}{1-x}limx→1+​1−x−4​

Problem 1204

Find lim⁡x→2g(x)\lim _{x \rightarrow 2} g(x)limx→2​g(x) given 2x+1≤g(x)≤x2−2x+52x + 1 \leq g(x) \leq x^2 - 2x + 52x+1≤g(x)≤x2−2x+5.

Problem 1205

Find the limit: lim⁡x→1x2+2x−3x2−1\lim _{x \rightarrow 1} \frac{x^{2}+2x-3}{x^{2}-1}limx→1​x2−1x2+2x−3​.

Problem 1206

Zairia freefalls for 3.2 s3.2 \mathrm{~s}3.2 s. How far does she fall in that time? Round to the nearest tenth.

Problem 1207

Calculate John's velocity just before hitting the water after falling for 7.5 seconds. Round to the nearest tenth.

Problem 1208

A tennis ball drops from 5.1 m5.1 \mathrm{~m}5.1 m. Find the time to hit the ground, rounded to the nearest hundredth.

Problem 1209

Find the number of thousand mp3 players to produce for minimum marginal cost given C(x)=x2−100x+8200C(x)=x^{2}-100x+8200C(x)=x2−100x+8200. What is the minimum cost?

Problem 1210

An object moves as per the table. What's true about the rate of change of distance over time? A, B, C, or D?

Problem 1211

Evaluate the expression [811x11/8]01\left[\frac{8}{11} x^{11 / 8}\right]_{0}^{1}[118​x11/8]01​.

Problem 1212

Find f(1+h)f(1+h)f(1+h) for f(x)=x28f(x)=\frac{x^{2}}{8}f(x)=8x2​ and h=1,0.1,0.01,0.001,0.0001h=1, 0.1, 0.01, 0.001, 0.0001h=1,0.1,0.01,0.001,0.0001, rounding to seven decimal places.

Problem 1213

Find the instantaneous rate of change of R(t)=240+30t3R(t)=240+30 t^{3}R(t)=240+30t3 at t=1t=1t=1. Round to one decimal place.

Problem 1214

Find the limit as hhh approaches 0 for 5−3+h+53h\frac{\frac{5}{-3+h}+\frac{5}{3}}{h}h−3+h5​+35​​.

Problem 1215

Find the limit: lim⁡h→05−4+h+54h\lim _{h \rightarrow 0} \frac{\frac{5}{-4+h}+\frac{5}{4}}{h}limh→0​h−4+h5​+45​​.

Problem 1216

Find aaa such that lim⁡x→32x2+ax+a−2x2+x−12\lim _{x \rightarrow 3} \frac{2 x^{2}+a x+a-2}{x^{2}+x-12}limx→3​x2+x−122x2+ax+a−2​ exists.

Problem 1217

Find the non-negative value of bbb such that the limit lim⁡x→0{8x+b−4x}\lim _{x \rightarrow 0}\left\{\frac{\sqrt{8 x+b}-4}{x}\right\}limx→0​{x8x+b​−4​} exists.

Problem 1218

Find if the limit lim⁡h→0f(1+h)−f(1)h\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}h→0lim​hf(1+h)−f(1)​ exists for f(x)=x2+4xf(x)=x^{2}+4xf(x)=x2+4x and calculate its value.

Problem 1219

Which statements must be true by the Intermediate Value Theorem? Check if roots exist in [2,4][2,4][2,4] for the given functions.

Problem 1220

Which statements are true due to the Intermediate Value Theorem (IVT) regarding the functions and intervals given?

Problem 1221

Find the volume of a solid with a circular base radius 4 and square cross-sections perpendicular to the xxx-axis.

Problem 1222

Evaluate the integral using substitution and partial fractions: ∫1xx+1dx\int \frac{1}{x \sqrt{x+1}} d x∫xx+1​1​dx.

Problem 1223

Find the 2 positive terms in the integral of cos⁡4xsin⁡2x dx\cos ^{4} x \sin ^{2} x \, dxcos4xsin2xdx.

Problem 1224

Find the limit: lim⁡x→+∞sin⁡−1(x1−2x)\lim _{x \rightarrow+\infty} \sin ^{-1}\left(\frac{x}{1-2 x}\right)limx→+∞​sin−1(1−2xx​).

Problem 1225

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=3x−2f(x)=3x-2f(x)=3x−2 using the limit definition: lim⁡h→0f(x+h)−f(x)h\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}limh→0​hf(x+h)−f(x)​.

Problem 1226

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=−2x2+x−19f(x)=-2 x^{2}+x-19f(x)=−2x2+x−19 using the limit definition: lim⁡h→0f(x+h)−f(x)h\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}limh→0​hf(x+h)−f(x)​.

Problem 1227

Determine if breakfast cereal consumption was changing faster in 2001 or 2011 using the model c(t)=−0.0032t3+0.097t2−0.358t+12.4c(t)=-0.0032 t^{3}+0.097 t^{2}-0.358 t+12.4c(t)=−0.0032t3+0.097t2−0.358t+12.4.

Problem 1228

Determine if cereal consumption was changing faster in 2005 or 2012 using C(t)=−0.0035t3+0.09t2−0.347t+12.25C(t)=-0.0035 t^{3}+0.09 t^{2}-0.347 t+12.25C(t)=−0.0035t3+0.09t2−0.347t+12.25. Calculate C′(15)C'(15)C′(15) and C′(22)C'(22)C′(22).

Problem 1229

Complete the tables for the behavior of f(x)=1x−2f(x)=\frac{1}{x-2}f(x)=x−21​, f(x)=1x+5f(x)=\frac{1}{x+5}f(x)=x+51​, and f(x)=1x+8f(x)=\frac{1}{x+8}f(x)=x+81​.

Problem 1230

Given position s=f(t)=4t3+2t+9s=f(t)=4 t^{3}+2 t+9s=f(t)=4t3+2t+9, find velocity v(t)v(t)v(t), v(3)v(3)v(3), acceleration a(t)a(t)a(t), and a(3)a(3)a(3).

Problem 1231

Problem 1232

Find the rate of change of cost in the function y+2.75=1.5(x−1)y+2.75=1.5(x-1)y+2.75=1.5(x−1) for a taxi ride per mile. Choices: a) \$1.50 b) \$3.25 c) \$2.75 d) \$4.25.

Problem 1233

Problem 1234

Problem 1235

Problem 1236

Find the limit as hhh approaches 0 for f(−6+h)−f(−6)h\frac{f(-6+h)-f(-6)}{h}hf(−6+h)−f(−6)​ where f(x)=−6x2−7x−1f(x)=-6x^2-7x-1f(x)=−6x2−7x−1.

Problem 1237

If lim⁡x→tf(x)=y\lim _{x \rightarrow t} f(x)=ylimx→t​f(x)=y, does fff need to be defined at x=tx=tx=t?

Problem 1238

Find the limit: lim⁡x→5(x2−25x2−8x+15)\lim _{x \rightarrow 5}\left(\frac{x^{2}-25}{x^{2}-8 x+15}\right)limx→5​(x2−8x+15x2−25​).

Problem 1239

Find the limit as xxx approaches -17 from the left for the piecewise function f(x)f(x)f(x).

Problem 1240

Find the limit as xxx approaches 0 from the right: lim⁡x→0+(x−3x4−11x)\lim _{x \rightarrow 0^{+}}\left(\frac{x}{-3 x^{4}-11 x}\right)limx→0+​(−3x4−11xx​).

Problem 1241

Find the limit: lim⁡x→49(x−7x−49)\lim _{x \rightarrow 49}\left(\frac{\sqrt{x}-7}{x-49}\right)limx→49​(x−49x​−7​).

Problem 1242

Identify the interval where the graph of $f$ is concave down based on its rate of change:
(A) $0<x<1$ (B) $1<x<2$ (C) $2<x<3$ (D) $3<x<4$

Find the vertical asymptotes of the function $y=\frac{x^{2}+2 x+1}{x+x^{2}}$ .

Evaluate the limit: $\lim _{x \rightarrow 1^{+}} \frac{-4}{1-x}$

Find $\lim _{x \rightarrow 2} g(x)$ given $2x + 1 \leq g(x) \leq x^2 - 2x + 5$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{2}+2x-3}{x^{2}-1}$ .

Zairia freefalls for $3.2 \mathrm{~s}$ . How far does she fall in that time? Round to the nearest tenth.

A tennis ball drops from $5.1 \mathrm{~m}$ . Find the time to hit the ground, rounded to the nearest hundredth.

Find the number of thousand mp3 players to produce for minimum marginal cost given $C(x)=x^{2}-100x+8200$ . What is the minimum cost?

Evaluate the expression $\left[\frac{8}{11} x^{11 / 8}\right]_{0}^{1}$ .

Find $f(1+h)$ for $f(x)=\frac{x^{2}}{8}$ and $h=1, 0.1, 0.01, 0.001, 0.0001$ , rounding to seven decimal places.

Find the instantaneous rate of change of $R(t)=240+30 t^{3}$ at $t=1$ . Round to one decimal place.

Find the limit as $h$ approaches 0 for $\frac{\frac{5}{-3+h}+\frac{5}{3}}{h}$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{5}{-4+h}+\frac{5}{4}}{h}$ .

Find $a$ such that $\lim _{x \rightarrow 3} \frac{2 x^{2}+a x+a-2}{x^{2}+x-12}$ exists.

Find the non-negative value of $b$ such that the limit $\lim _{x \rightarrow 0}\left\{\frac{\sqrt{8 x+b}-4}{x}\right\}$ exists.

Find if the limit $\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$ exists for $f(x)=x^{2}+4x$ and calculate its value.

Which statements must be true by the Intermediate Value Theorem? Check if roots exist in $[2,4]$ for the given functions.

Find the volume of a solid with a circular base radius 4 and square cross-sections perpendicular to the $x$ -axis.

Evaluate the integral using substitution and partial fractions: $\int \frac{1}{x \sqrt{x+1}} d x$ .

Find the 2 positive terms in the integral of $\cos ^{4} x \sin ^{2} x \, dx$ .

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

Find the derivative $f^{\prime}(x)$ for $f(x)=3x-2$ using the limit definition: $\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

Find the derivative $f^{\prime}(x)$ for $f(x)=-2 x^{2}+x-19$ using the limit definition: $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

Determine if breakfast cereal consumption was changing faster in 2001 or 2011 using the model $c(t)=-0.0032 t^{3}+0.097 t^{2}-0.358 t+12.4$ .

Determine if cereal consumption was changing faster in 2005 or 2012 using $C(t)=-0.0035 t^{3}+0.09 t^{2}-0.347 t+12.25$ . Calculate $C'(15)$ and $C'(22)$ .

Complete the tables for the behavior of $f(x)=\frac{1}{x-2}$ , $f(x)=\frac{1}{x+5}$ , and $f(x)=\frac{1}{x+8}$ .

Given position $s=f(t)=4 t^{3}+2 t+9$ , find velocity $v(t)$ , $v(3)$ , acceleration $a(t)$ , and $a(3)$ .

Find the rate of change of cost in the function $y+2.75=1.5(x-1)$ for a taxi ride per mile. Choices: a) \$1.50 b) \$3.25 c) \$2.75 d) \$4.25.

Find the limit as $h$ approaches 0 for $\frac{f(-6+h)-f(-6)}{h}$ where $f(x)=-6x^2-7x-1$ .

If $\lim _{x \rightarrow t} f(x)=y$ , does $f$ need to be defined at $x=t$ ?

Find the limit: $\lim _{x \rightarrow 5}\left(\frac{x^{2}-25}{x^{2}-8 x+15}\right)$ .

Find the limit as $x$ approaches -17 from the left for the piecewise function $f(x)$ .

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}}\left(\frac{x}{-3 x^{4}-11 x}\right)$ .

Find the limit: $\lim _{x \rightarrow 49}\left(\frac{\sqrt{x}-7}{x-49}\right)$ .

Find the limit: $\lim _{x \rightarrow 144}\left(\frac{x-144}{\sqrt{x}-12}\right)$ .

Calculate the limit as $x$ approaches 4 of the expression $\frac{4 x^{2}+9 x+8}{6 x^{2}-3 x+7}$ .

Identify the discontinuity points for the function $f(x)=\frac{x}{x^{2}-1}$ .

Find the marginal revenue when 3 units are sold, given the demand function $p=72-7.5q$ .

Find the instantaneous rate of change of $R(t)=240+30 t^{3}$ at $t=1$ . Answer: $R^{\prime}(1)=90$ dollars/day.

Evaluate the integral: $\int_{0}^{1} 6 \frac{\ln (x)}{\sqrt[3]{x}} d x$

Find the derivative of $(x+1) \sin x$ .

Estimate the soccer ball's instantaneous velocity at $t=2$ s using the average velocities given. Round to two decimal places.

Find the distance the ball travels from $t=4$ to $t=4.5$ using $s(t)=4.9 t^{2}$ .

Find the average velocity from $t=4$ to $t=4.5$ for $s(t) = 4.9t^2$ . Use decimal notation.

Estimate the ball's instantaneous velocity at $t=4$ using average velocities over $[4,4.1]$ , $[4,4.01]$ , and $[4,4.001]$ . Round to one decimal place.

Find the instantaneous velocity of a wrench at $t=6$ seconds, given $s(t)=4.9 t^{2} \mathrm{~m}$ .

Prove that $\lim _{x \rightarrow 6} 4 x=24$ using the precise definition of a limit.

Find the stone's average velocity over the interval $[0.5,3]$ given its height function $h=30t-4.9t^2$ .

Find the average velocity of a rock thrown upward on Mars, described by $y=10t-1.9t^2$ , over the interval $[2,3]$ .

Find the average velocity $\bar{v}$ of a stone with height $h(t)=30t-4.9t^2$ over intervals around $t=3$ .

Find the slope of the tangent line at $x=7$ for $f(x)=\ln(x)$ . Round your answer to three decimal places.

Estimate the slope of the tangent line for $y(t)=\sqrt{2 t+1}$ at $t=3$ . Round your answer to three decimal places.

Find the slope of the tangent line at $t=-7$ for the function $f(t)=2t-3$ . Provide your answer as a whole number.

Find the slope of the tangent line for $f(x)=3 e^{x}$ at $x=e$ . Round your answer to two decimal places.

Estimate the slope of the tangent line for $P(x)=5 x^{2}+7$ at $x=4$ . Provide a whole number answer.

Find the slope of the tangent line for $f(x)=3 e^{x}$ at $x=e$ . Round your answer to two decimal places.

Find the average velocity of $s(t)=t^{3}+2t$ over $[2,8]$ and estimate the instantaneous velocity at $t=2$ .

Find the average rate of change of $y=f(x)$ from $x=4$ to $x=7$ using the given values. Round to two decimal places.

Evaluate the function $f(x)$ and determine which statements about its limit and continuity at $x=1$ are true. Options: A. Only I, B. Only II, C. I and II, D. None, E. All.

Find the stone's average velocity over the interval $[0.5,3]$ using the height function $h=35 t-4.9 t^{2}$ .

Estimate the instantaneous velocity of a wrench at $t=5$ seconds, given $s(t)=4.9 t^{2} \mathrm{~m}$ .

Find the average velocity of $s(t)=t^{3}+2t$ over $[2,8]$ and estimate the instantaneous velocity at $t=2$ .

Estimate the slope of the tangent line for $y(t)=\sqrt{4t+1}$ at $t=1$ . Round your answer to three decimal places.

Find the slope of the tangent line at $t=-7$ for the function $f(t)=2t-3$ . Provide your answer as a whole number.

Estimate the slope of the tangent line for $P(x)=6 x^{2}+8$ at $x=6$ . Provide your answer as a whole number.

Estimate the slope of the tangent line at $x=11$ for $f(x)=\ln (x)$ . Round your answer to three decimal places.