Calculus

Problem 11201

Given $f(x)=x^{1/3}-x^{4/3}$ , find roots, intervals of increase/decrease, concavity, limits as $x \to \pm\infty$ , and sketch $y=f(x)$ .

See Solution

Problem 11202

Find $\lim _{x \rightarrow 1^{+}}(x-1)^{\ln x}$ using logarithms.

See Solution

Problem 11203

Find the $x$ -values for relative extrema of $f(x)=2x+7x^{2/7}$ and their values. Select A, B, C, or D.

See Solution

Problem 11204

Find the derivative $y^{\prime}$ for the function $y=\frac{9}{5 x^{2}}$ . What is $y^{\prime}=$ ?

See Solution

Problem 11205

Differentiate the function $f(x)=-5-2x+4e^{x}$ . Find $f^{\prime}(x)=$ .

See Solution

Problem 11206

Find the $x$ -values where $f(x)=10x+13x^{10/13}$ has relative extrema and their values. Choices: A, B, C, D.

See Solution

Problem 11207

If $F^{\prime}(x)=f(x)$ , then $f$ is the derivative of $F$ and $F$ is the antiderivative of $f$ .

See Solution

Problem 11208

Find the antiderivatives of $f(x)=1$ . Which is correct? A. $F(x)=x$ B. $F(x)=x^{2}$ C. $F(x)=x+C$ D. $F(x)=C x$

See Solution

Problem 11209

Find the derivative of the function $4x + 5$ . What is $\frac{d}{d x}(4 x+5)=\square$ ?

See Solution

Problem 11210

Find the derivative $f^{\prime}(x)$ for the function $f(x)=15 x e^{x}$ .

See Solution

Problem 11211

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

See Solution

Problem 11212

Find the derivative $f'(x)$ of the function $f(x) = e^{12x}$ .

See Solution

Problem 11213

Sketch a function graph that meets these conditions and check if $f$ is continuous at $x=-3$ : $f(-3)=-2$ , $\lim_{x \to -3^{-}} f(x)=2$ , $\lim_{x \to -3^{+}} f(x)=-2$ .

See Solution

Problem 11214

Find the missing expression ? to make this equation valid: $\frac{d}{d x}(4 x+7)^{3}=3(4 x+7)^{2} ?$

See Solution

Problem 11215

Why do two antiderivatives of a function differ by a constant? Choose the correct explanation about $F$ and $G$ .

See Solution

Problem 11216

Differentiate the function $y=(1+3 x^{3})^{9}$ . Find $y'=\square$ .

See Solution

Problem 11217

Find the derivative of the function $f(x) = -4 \ln x + 7x^2 - 7$ . What is $f'(x)$ ?

See Solution

Problem 11218

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{15}$ .

See Solution

Problem 11219

Find the derivative $f^{\prime}(x)$ for the function $f(x)=7 x^{5}(x^{4}-6)$ .

See Solution

Problem 11220

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{x^{2}+8}{3 x-7}$ .

See Solution

Problem 11221

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\ln x+2 e^{x}-3 x^{2}$ .

See Solution

Problem 11222

How long will \ $430 grow to \$510 at a continuous interest rate of$ 2.5\%$? Round to the nearest year.

See Solution

Problem 11223

A cup of water cools from $207^{\circ} \mathrm{F}$ to $182^{\circ} \mathrm{F}$ in 1.5 min. Find $k$ and temp after 5 min.

See Solution

Problem 11224

Find the derivative $f^{\prime}(t)$ for the function $f(t)=-6 t^{2}+2 t+7$ . What is $f^{\prime}(t)=\square$ ?

See Solution

Problem 11225

Find the derivative of the function $f(x)=4 e^{x}+5 x-\ln x$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 11226

Logan wants to invest for an account to reach \ $19,300 in 19 years at a continuous interest rate of$ 3.5\%$. How much?

See Solution

Problem 11227

Find the horizontal or oblique asymptotes for the function $f(x)=\frac{x^{2}-2}{x^{3}-8}$ .

See Solution

Problem 11228

Estimate $\Delta f=f(4.6)-f(5)$ for $f(x)=x-7 x^{2}$ using linear approximation. Actual change to two decimal places?

See Solution

Problem 11229

Find the derivative $f^{\prime}(x)$ for the function $f(x)=8 e^{-4 x}$ .

See Solution

Problem 11230

Find $f^{\prime}(x)$ for $f(x)=4 x^{3} \ln x$ and identify the correct product rule application.

See Solution

Problem 11231

Sketch a function graph and check if $f$ is continuous at $x=2$ given $f(2)=1$ and $\lim_{x \rightarrow 2} f(x)=1$ .

See Solution

Problem 11232

Find the cost function $C(x)=8+\sqrt{2x+48}$ for $0 \leq x \leq 50$ . Calculate $C'(x)$ and $C'(8)$ , $C'(26)$ . Interpret the results.

See Solution

Problem 11233

Find the number of bicycle helmets sold per week, $x=78 \sqrt{p+16}-400$ , for price $p$ between 20 and 100.
(A) Determine $\frac{d x}{d p}$ . $\frac{d x}{d p}=\square$
(B) Calculate supply and rate of change when $p=84$ . Supply is $\square$ helmets/week and rate is $\square$ helmets/dollar.

See Solution

Problem 11234

Estimate $\Delta f=f(9.03)-f(9)$ for $f(x)=x^{4}$ using Linear Approximation. Round to two decimal places.

See Solution

Problem 11235

Estimate $\Delta f=f(4.6)-f(5)$ for $f(x)=x-7 x^{2}$ using Linear Approximation. Find actual change and compute error.

See Solution

Problem 11236

Find $f^{\prime}(x)$ for $f(x)=\left(\tan \left(x^{8}\right)+6\right)^{6}$ . Choose from the options given.

See Solution

Problem 11237

Find $f'(x)$ for $f(x)=\ln(x)e^{x^2+2x+4}$ . Options include various expressions involving $e$ and $\ln$ .

See Solution

Problem 11238

Find the derivative $f'(x)$ of $f(x)=(x^{6}+\log_{3}(x^{2}))^{3}+x$ . Choose from the options provided.

See Solution

Problem 11239

Find $f^{\prime}(x)$ for $f(x)=e^{\left(\cos \left(x^{3}\right)\right)^{4}}$ . Choices: a) b) c) d) e) None.

See Solution

Problem 11240

Find $f^{\prime}(x)$ for $f(x)=\frac{\left(x^{5}-3^{x^{2}}+2 t\right)^{3}}{x}$ . Choose from options a) to d).

See Solution

Problem 11241

Differentiate $10 x^{2} \sin (2 y) + y \ln (x) = 6$ implicitly to find $y^{\prime}$ . Choose from options a) to e).

See Solution

Problem 11242

Find $f^{\prime}(x)$ for $f(x)=x^{2}(x^{2}+20)^{\frac{1}{4}}$ . Choose from the given options a) to e).

See Solution

Problem 11243

Find the rate of change of $P(t)=2+5 \tan^{-1}\left(\frac{t}{2}\right)$ when $P(t)=6$ . Choices: (A) -0.606 (B) 0.250 (C) 1.214 (D) 1.942

See Solution

Problem 11244

Find the rate of change of area $A(t)$ of spilled oil at $t=10$ . Which option represents this: (A) $A(10)$ , (B) $A(11)-A(9)$ , (C) $\frac{A(10)}{10}$ , (D) $A^{\prime}(10)$ ?

See Solution

Problem 11245

Estimate $\Delta f=f(4.9)-f(5)$ for $f(x)=x-4x^{2}$ using Linear Approximation. Find actual change and errors.

See Solution

Problem 11246

Find when the speed of the particle with position $y(t)=t^{3}-6 t^{2}+9 t$ is increasing for $0<t<4$ .

See Solution

Problem 11247

Estimate $\Delta f=f(4.9)-f(5)$ for $f(x)=x-4 x^{2}$ using Linear Approximation. Find actual $\Delta f$ and error.

See Solution

Problem 11248

Find the linear approximation of $y=(1+3x)^{-1/2}$ at $x=5$ .

See Solution

Problem 11249

Find $t$ where the linear approximation of $A(t)$ , given $A'(t)=2+9 e^{0.4 \sin t}$ and $A(1.2)=7.5$ , equals 15.

See Solution

Problem 11250

Given the piecewise function $f(x)=\left\{\begin{array}{ll} x^{2}-3 x & x \leq 8 \\ 5 x & x>8 \end{array}\right.$ , check continuity and differentiability at $x=8$ . Find $f'(8)$ or $f(8)$ .

See Solution

Problem 11251

Find the derivative of $g(x)=-8 x^{2}(9 x^{2})$ using the Product Rule and algebra.

See Solution

Problem 11252

The tangent line to $y=f(x)$ at $(7,37)$ is $y=6x-5$ . Find $f^{\prime}(7)$ and the instantaneous rate of change at $x=7$ .

See Solution

Problem 11253

Estimate $\Delta f=f(4.9)-f(5)$ for $f(x)=x-4x^{2}$ using Linear Approximation. Find actual $\Delta f$ and errors.

See Solution

Problem 11254

Estimate $\Delta f=f(4.9)-f(5)$ for $f(x)=x-4 x^{2}$ using Linear Approximation. Find actual change and compute error.

See Solution

Problem 11255

Find the derivative of $f(x)=\frac{3}{\sqrt[5]{x}}$ and evaluate at $x=1$ and $x=4$ .

See Solution

Problem 11256

Taylor takes 1,000mg of Cefalexin at 5am. How much is in his system at 4pm? Sam takes 500mg at 5am and 11am. How much is in his system at 4pm? Round to 4 decimal places.

See Solution

Problem 11257

Given the function $f(x)=\frac{6 x^{2}}{x^{2}-16}$ , find:
(A) First derivative $f^{\prime}(x)$ , critical numbers, intervals of increase/decrease, and local extrema.
(B) Left/right limits at vertical asymptotes $x=-4$ and $x=4$ , and limits at infinity.
(C) Second derivative $f^{\prime \prime}(x)$ , intervals of concavity, and inflection points.
(D) Determine the symmetry of $f$ .
(E) Domain, range, $y$ -intercept, and $x$ -intercepts of $f$ .

See Solution

Problem 11258

Find $f$ and $g$ such that $h=(2x + 7x^5 - 10x^9)(9 + 2x - 4x^2)$ and use the product rule for $h'$ .

See Solution

Problem 11259

Approximate the value of $e^{-0.33}$ using a calculator.

See Solution

Problem 11260

Estimate $\Delta y$ using differentials for $y=x^{1/2} e^{4x-1}$ at $a=1$ with $dx=0.1$ . Round to four decimal places.

See Solution

Problem 11261

Given $f^{\prime}(2)=2$ and $g^{\prime}(2)=-1$ , find $h^{\prime}(2)$ for each function:
(A) $h(x)=2 f(x)$ ; (B) $h(x)=-13 g(x)$ ; (C) $h(x)=6 f(x)+13 g(x)$ ; (D) $h(x)=10 g(x)-11 f(x)$ ; (E) $h(x)=2 f(x)+9 g(x)+6$ ; (F) $h(x)=-6 g(x)-13 f(x)-6 x$ .

See Solution

Problem 11262

A car on a highway goes past an intersection at 70 mph. How fast is it moving away from a farmhouse 8 miles away when 5 miles past? The rate is $\square$ mph.

See Solution

Problem 11263

Find the second derivative of the cubic function $f(x)=3 x^{3}+17 x^{2}+14 x-9$ . What is $f^{\prime \prime}(x)$ ?

See Solution

Problem 11264

Find the derivative $f^{\prime}(x)$ of $f(x)=x^{15} e^{x}$ and evaluate it at $x=1$ . What is $f^{\prime}(1)$ ?

See Solution

Problem 11265

Find the derivative of $f(x)=x^{15} e^{x}$ and evaluate it at $x=1$ : $f^{\prime}(1)=\,$

See Solution

Problem 11266

A balloon's volume grows at $2.5 \mathrm{ft}^{3} / \mathrm{min}$ . Find the diameter's growth rate when it's 1.1 feet.

See Solution

Problem 11267

Find the derivative of $f(x)=\frac{2 \sqrt{x}}{3 x^{2}+4 x-9}$ and evaluate it at $x=3$ .

See Solution

Problem 11268

Calculate the average value of $f(x)=\frac{1}{\sqrt{4 x-3}}$ from $x=3$ to $x=21$ , rounded to four decimal places.

See Solution

Problem 11269

Find the average of $f(x)=x^{3} \sqrt{1+x^{2}}$ from $0$ to $\sqrt{3}$ . Round to four decimal places.

See Solution

Problem 11270

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

See Solution

Problem 11271

Given $f(x)=5 x^{2}+2 x$ , find: (A) $f^{\prime}(x)$ , (B) slopes at $x=2$ and $x=3$ , (C) tangent line equations, (D) where tangent is horizontal.

See Solution

Problem 11272

Find the first derivative of $f(x)=\frac{5 x^{2}}{x^{2}-9}$ and determine critical numbers, increasing/decreasing intervals, and local extrema.

See Solution

Problem 11273

Estimate the change in stopping distance $\Delta F$ for $F(s)=1.1 s+0.054 s^{2}$ at speeds $s=55$ and $s=85$ mph. Give answers to two decimal places.

See Solution

Problem 11274

Find the $x$ -coordinates of inflection points for the function $g(x)=\frac{1+x}{1+x^{2}}$ with $g^{\prime \prime}(x)=\frac{2(x-1)(x^{2}+4x+1)}{(1+x^{2})^{3}}$ . Possible values: $-\sqrt{\frac{2}{3}}$ , 3, 1, $-2+\sqrt{3}$ , $5\sqrt{3}$ .

See Solution

Problem 11275

Estimate the area under $f(x)=x^{2}+6$ from $x=0$ to $x=8$ using left, midpoint, and right Riemann sums with $n=4$ .

See Solution

Problem 11276

Evaluate the iterated integral $\iiint_{W} f d V$ with limits: $A=0$ , $B=2 \pi$ , $C=0$ , $D=3$ , $E=-1$ , $F=2$ .

See Solution

Problem 11277

Find the limits as $x$ approaches 9 for the following: (a) $\lim _{x \rightarrow 9}[g(x)-8]^{2}$ (b) $\lim _{x \rightarrow 9}[1+\sqrt{x h(x)}]$ (c) $\lim _{x \rightarrow 9}\left[\frac{h(x)}{x+g(x)}\right]$ (d) $\lim _{x \rightarrow 9}\left[\frac{x}{g(x)+h(x)}\right]$

See Solution

Problem 11278

Sketch the area between $y=10 x^{2}$ and $y=10 \sqrt[3]{x}$ , then find the volume when this area rotates around the $y$ -axis.

See Solution

Problem 11279

Classify the limit types as IND, a fixed value, or DNE for the following: $\pi^{\infty}$ , $\infty^{\infty}$ , $\frac{1}{-\infty}$ , etc.

See Solution

Problem 11280

Differentiate the function $(1+9 x)^{-\frac{1}{2}}$ with respect to $x$ .

See Solution

Problem 11281

Given $f(x)=2 x(x-2)^{3}$ , determine the sign of $f^{\prime \prime}(x)$ in intervals: $(-\infty, 1)$ , $1$ , $(1,2)$ , $2$ , $(2, \infty)$ .

See Solution

Problem 11282

Find where $f(x)=\frac{7 x-6}{x+6}$ is concave up/down and locate its inflection points.

See Solution

Problem 11283

Find the volume of the solid formed by rotating the area between $y=3-(x-3)^{2}$ and $y=-1$ around $x=0$ using two methods.

See Solution

Problem 11284

Find the position function $s(t)$ from the velocity $v(t)=5 \sqrt[3]{t}$ with initial position $s(0)=1$ .

See Solution

Problem 11285

Solve the initial value problem: $g'(x)=6x^5-4x^3-6$ with $g(1)=-20$ . Find $g(x)=\square$ .

See Solution

Problem 11286

Find the indefinite integral and verify by differentiating: $\int 7 \sqrt[5]{x} d x=\square$

See Solution

Problem 11287

Find the antiderivatives of $f(y)=-\frac{11}{y^{12}}$ and verify by differentiating. Antiderivative: $F(y)=\square$

See Solution

Problem 11288

Evaluate the integral $2 \pi \int_{0}^{1}(2-x)(\sqrt{x}-x) d x$ .

See Solution

Problem 11289

Calculate the integral $\int_{-1}^{4}|2 x-1| d x$ .

See Solution

Problem 11290

Find when the speed of a particle with velocity $v(t)=9 t^{2}-72 t+63$ is increasing for $t \geq 0$ .

See Solution

Problem 11291

Find the antiderivative of $x^{p}$ and the values of $p$ for which it applies.

See Solution

Problem 11292

Find the position function for an object with acceleration $a(t)=-26$ , initial velocity $v(0)=22$ , and position $s(0)=0$ .

See Solution

Problem 11293

Calculate the average of $f(x)=x^{3} \sqrt{1+x^{2}}$ from $0$ to $\sqrt{3}$ , rounded to four decimal places.

See Solution

Problem 11294

Find the average of $f(x)=x^{3} \sqrt{1+x^{2}}$ from $0$ to $\sqrt{3}$ , rounded to four decimal places.

See Solution

Problem 11295

Find the time for coffee cooling from $210^{\circ} \mathrm{F}$ to $140^{\circ} \mathrm{F}$ in a $72^{\circ} \mathrm{F}$ room using $y(t)=72+\left(210-72\right)e^{-0.01457118 t}$ . Round to the nearest tenth of a minute.

See Solution

Problem 11296

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=x^{2}-7x+6$ .

See Solution

Problem 11297

Find the maximum error in the area $A$ of a square carpet with side $s = 4 \mathrm{ft}$ , accurate to 0.7 inches.

See Solution

Problem 11298

Approximate $\frac{1}{99}$ to four decimal places using $L(x)$ for $f(x)=\frac{1}{x}$ at $a=100$ and find percentage error.

See Solution

Problem 11299

Determine if the critical points of $f(x)=x^{4}-9 x^{3}+9 x^{2}-18$ are max, min, or neither using the second derivative test.

See Solution

Problem 11300

As $x \rightarrow \infty, y \rightarrow -45$ for the function $f(x)=\frac{1}{(x-70)^{2}}-45$ .

See Solution

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Calculus

Problem 11201

Given f(x)=x1/3−x4/3f(x)=x^{1/3}-x^{4/3}f(x)=x1/3−x4/3, find roots, intervals of increase/decrease, concavity, limits as x→±∞x \to \pm\inftyx→±∞, and sketch y=f(x)y=f(x)y=f(x).

Problem 11202

Find lim⁡x→1+(x−1)ln⁡x\lim _{x \rightarrow 1^{+}}(x-1)^{\ln x}limx→1+​(x−1)lnx using logarithms.

Problem 11203

Find the xxx-values for relative extrema of f(x)=2x+7x2/7f(x)=2x+7x^{2/7}f(x)=2x+7x2/7 and their values. Select A, B, C, or D.

Problem 11204

Find the derivative y′y^{\prime}y′ for the function y=95x2y=\frac{9}{5 x^{2}}y=5x29​. What is y′=y^{\prime}=y′=?

Problem 11205

Differentiate the function f(x)=−5−2x+4exf(x)=-5-2x+4e^{x}f(x)=−5−2x+4ex. Find f′(x)=f^{\prime}(x)=f′(x)=.

Problem 11206

Find the xxx-values where f(x)=10x+13x10/13f(x)=10x+13x^{10/13}f(x)=10x+13x10/13 has relative extrema and their values. Choices: A, B, C, D.

Problem 11207

If F′(x)=f(x)F^{\prime}(x)=f(x)F′(x)=f(x), then fff is the derivative of FFF and FFF is the antiderivative of fff.

Problem 11208

Find the antiderivatives of f(x)=1f(x)=1f(x)=1. Which is correct? A. F(x)=xF(x)=xF(x)=x B. F(x)=x2F(x)=x^{2}F(x)=x2 C. F(x)=x+CF(x)=x+CF(x)=x+C D. F(x)=CxF(x)=C xF(x)=Cx

Problem 11209

Find the derivative of the function 4x+54x + 54x+5. What is ddx(4x+5)=□\frac{d}{d x}(4 x+5)=\squaredxd​(4x+5)=□?

Problem 11210

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=15xexf(x)=15 x e^{x}f(x)=15xex.

Problem 11211

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=3ln⁡(1+7x2)f(x)=3 \ln(1+7 x^{2})f(x)=3ln(1+7x2).

Problem 11212

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=e12xf(x) = e^{12x}f(x)=e12x.

Problem 11213

Sketch a function graph that meets these conditions and check if fff is continuous at x=−3x=-3x=−3: f(−3)=−2f(-3)=-2f(−3)=−2, lim⁡x→−3−f(x)=2\lim_{x \to -3^{-}} f(x)=2limx→−3−​f(x)=2, lim⁡x→−3+f(x)=−2\lim_{x \to -3^{+}} f(x)=-2limx→−3+​f(x)=−2.

Problem 11214

Find the missing expression ? to make this equation valid: ddx(4x+7)3=3(4x+7)2?\frac{d}{d x}(4 x+7)^{3}=3(4 x+7)^{2} ?dxd​(4x+7)3=3(4x+7)2?

Problem 11215

Why do two antiderivatives of a function differ by a constant? Choose the correct explanation about FFF and GGG.

Problem 11216

Differentiate the function y=(1+3x3)9y=(1+3 x^{3})^{9}y=(1+3x3)9. Find y′=□y'=\squarey′=□.

Problem 11217

Find the derivative of the function f(x)=−4ln⁡x+7x2−7f(x) = -4 \ln x + 7x^2 - 7f(x)=−4lnx+7x2−7. What is f′(x)f'(x)f′(x)?

Problem 11218

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x15y=x^{15}y=x15.

Problem 11219

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=7x5(x4−6)f(x)=7 x^{5}(x^{4}-6)f(x)=7x5(x4−6).

Problem 11220

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=x2+83x−7f(x)=\frac{x^{2}+8}{3 x-7}f(x)=3x−7x2+8​.

Problem 11221

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=ln⁡x+2ex−3x2f(x)=\ln x+2 e^{x}-3 x^{2}f(x)=lnx+2ex−3x2.

Problem 11222

How long will \430growto$510atacontinuousinterestrateof430 grow to \$510 at a continuous interest rate of 430growto$510atacontinuousinterestrateof2.5\%$? Round to the nearest year.

Problem 11223

A cup of water cools from 207∘F207^{\circ} \mathrm{F}207∘F to 182∘F182^{\circ} \mathrm{F}182∘F in 1.5 min. Find kkk and temp after 5 min.

Problem 11224

Find the derivative f′(t)f^{\prime}(t)f′(t) for the function f(t)=−6t2+2t+7f(t)=-6 t^{2}+2 t+7f(t)=−6t2+2t+7. What is f′(t)=□f^{\prime}(t)=\squaref′(t)=□?

Problem 11225

Find the derivative of the function f(x)=4ex+5x−ln⁡xf(x)=4 e^{x}+5 x-\ln xf(x)=4ex+5x−lnx. What is f′(x)f^{\prime}(x)f′(x)?

Problem 11226

Logan wants to invest for an account to reach \19,300in19yearsatacontinuousinterestrateof19,300 in 19 years at a continuous interest rate of 19,300in19yearsatacontinuousinterestrateof3.5\%$. How much?

Problem 11227

Find the horizontal or oblique asymptotes for the function f(x)=x2−2x3−8f(x)=\frac{x^{2}-2}{x^{3}-8}f(x)=x3−8x2−2​.

Problem 11228

Estimate Δf=f(4.6)−f(5)\Delta f=f(4.6)-f(5)Δf=f(4.6)−f(5) for f(x)=x−7x2f(x)=x-7 x^{2}f(x)=x−7x2 using linear approximation. Actual change to two decimal places?

Problem 11229

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=8e−4xf(x)=8 e^{-4 x}f(x)=8e−4x.

Problem 11230

Find f′(x)f^{\prime}(x)f′(x) for f(x)=4x3ln⁡xf(x)=4 x^{3} \ln xf(x)=4x3lnx and identify the correct product rule application.

Problem 11231

Sketch a function graph and check if fff is continuous at x=2x=2x=2 given f(2)=1f(2)=1f(2)=1 and lim⁡x→2f(x)=1\lim_{x \rightarrow 2} f(x)=1limx→2​f(x)=1.

Problem 11232

Find the cost function C(x)=8+2x+48C(x)=8+\sqrt{2x+48}C(x)=8+2x+48​ for 0≤x≤500 \leq x \leq 500≤x≤50. Calculate C′(x)C'(x)C′(x) and C′(8)C'(8)C′(8), C′(26)C'(26)C′(26). Interpret the results.

Problem 11233

Problem 11234

Estimate Δf=f(9.03)−f(9)\Delta f=f(9.03)-f(9)Δf=f(9.03)−f(9) for f(x)=x4f(x)=x^{4}f(x)=x4 using Linear Approximation. Round to two decimal places.

Problem 11235

Estimate Δf=f(4.6)−f(5)\Delta f=f(4.6)-f(5)Δf=f(4.6)−f(5) for f(x)=x−7x2f(x)=x-7 x^{2}f(x)=x−7x2 using Linear Approximation. Find actual change and compute error.

Problem 11236

Find f′(x)f^{\prime}(x)f′(x) for f(x)=(tan⁡(x8)+6)6f(x)=\left(\tan \left(x^{8}\right)+6\right)^{6}f(x)=(tan(x8)+6)6. Choose from the options given.

Problem 11237

Find f′(x)f'(x)f′(x) for f(x)=ln⁡(x)ex2+2x+4f(x)=\ln(x)e^{x^2+2x+4}f(x)=ln(x)ex2+2x+4. Options include various expressions involving eee and ln⁡\lnln.

Problem 11238

Find the derivative f′(x)f'(x)f′(x) of f(x)=(x6+log⁡3(x2))3+xf(x)=(x^{6}+\log_{3}(x^{2}))^{3}+xf(x)=(x6+log3​(x2))3+x. Choose from the options provided.

Problem 11239

Find f′(x)f^{\prime}(x)f′(x) for f(x)=e(cos⁡(x3))4f(x)=e^{\left(\cos \left(x^{3}\right)\right)^{4}}f(x)=e(cos(x3))4. Choices: a) b) c) d) e) None.

Problem 11240

Find f′(x)f^{\prime}(x)f′(x) for f(x)=(x5−3x2+2t)3xf(x)=\frac{\left(x^{5}-3^{x^{2}}+2 t\right)^{3}}{x}f(x)=x(x5−3x2+2t)3​. Choose from options a) to d).

Given $f(x)=x^{1/3}-x^{4/3}$ , find roots, intervals of increase/decrease, concavity, limits as $x \to \pm\infty$ , and sketch $y=f(x)$ .

Find $\lim _{x \rightarrow 1^{+}}(x-1)^{\ln x}$ using logarithms.

Find the $x$ -values for relative extrema of $f(x)=2x+7x^{2/7}$ and their values. Select A, B, C, or D.

Find the derivative $y^{\prime}$ for the function $y=\frac{9}{5 x^{2}}$ . What is $y^{\prime}=$ ?

Differentiate the function $f(x)=-5-2x+4e^{x}$ . Find $f^{\prime}(x)=$ .

Find the $x$ -values where $f(x)=10x+13x^{10/13}$ has relative extrema and their values. Choices: A, B, C, D.

If $F^{\prime}(x)=f(x)$ , then $f$ is the derivative of $F$ and $F$ is the antiderivative of $f$ .

Find the antiderivatives of $f(x)=1$ . Which is correct? A. $F(x)=x$ B. $F(x)=x^{2}$ C. $F(x)=x+C$ D. $F(x)=C x$

Find the derivative of the function $4x + 5$ . What is $\frac{d}{d x}(4 x+5)=\square$ ?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=15 x e^{x}$ .

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

Find the derivative $f'(x)$ of the function $f(x) = e^{12x}$ .

Sketch a function graph that meets these conditions and check if $f$ is continuous at $x=-3$ : $f(-3)=-2$ , $\lim_{x \to -3^{-}} f(x)=2$ , $\lim_{x \to -3^{+}} f(x)=-2$ .

Find the missing expression ? to make this equation valid: $\frac{d}{d x}(4 x+7)^{3}=3(4 x+7)^{2} ?$

Why do two antiderivatives of a function differ by a constant? Choose the correct explanation about $F$ and $G$ .

Differentiate the function $y=(1+3 x^{3})^{9}$ . Find $y'=\square$ .

Find the derivative of the function $f(x) = -4 \ln x + 7x^2 - 7$ . What is $f'(x)$ ?

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{15}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=7 x^{5}(x^{4}-6)$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{x^{2}+8}{3 x-7}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\ln x+2 e^{x}-3 x^{2}$ .

How long will \ $430 grow to \$510 at a continuous interest rate of$ 2.5\%$? Round to the nearest year.

A cup of water cools from $207^{\circ} \mathrm{F}$ to $182^{\circ} \mathrm{F}$ in 1.5 min. Find $k$ and temp after 5 min.

Find the derivative $f^{\prime}(t)$ for the function $f(t)=-6 t^{2}+2 t+7$ . What is $f^{\prime}(t)=\square$ ?

Find the derivative of the function $f(x)=4 e^{x}+5 x-\ln x$ . What is $f^{\prime}(x)$ ?

Logan wants to invest for an account to reach \ $19,300 in 19 years at a continuous interest rate of$ 3.5\%$. How much?

Find the horizontal or oblique asymptotes for the function $f(x)=\frac{x^{2}-2}{x^{3}-8}$ .

Estimate $\Delta f=f(4.6)-f(5)$ for $f(x)=x-7 x^{2}$ using linear approximation. Actual change to two decimal places?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=8 e^{-4 x}$ .

Find $f^{\prime}(x)$ for $f(x)=4 x^{3} \ln x$ and identify the correct product rule application.

Sketch a function graph and check if $f$ is continuous at $x=2$ given $f(2)=1$ and $\lim_{x \rightarrow 2} f(x)=1$ .

Find the cost function $C(x)=8+\sqrt{2x+48}$ for $0 \leq x \leq 50$ . Calculate $C'(x)$ and $C'(8)$ , $C'(26)$ . Interpret the results.

Estimate $\Delta f=f(9.03)-f(9)$ for $f(x)=x^{4}$ using Linear Approximation. Round to two decimal places.

Estimate $\Delta f=f(4.6)-f(5)$ for $f(x)=x-7 x^{2}$ using Linear Approximation. Find actual change and compute error.

Find $f^{\prime}(x)$ for $f(x)=\left(\tan \left(x^{8}\right)+6\right)^{6}$ . Choose from the options given.

Find $f'(x)$ for $f(x)=\ln(x)e^{x^2+2x+4}$ . Options include various expressions involving $e$ and $\ln$ .

Find the derivative $f'(x)$ of $f(x)=(x^{6}+\log_{3}(x^{2}))^{3}+x$ . Choose from the options provided.

Find $f^{\prime}(x)$ for $f(x)=e^{\left(\cos \left(x^{3}\right)\right)^{4}}$ . Choices: a) b) c) d) e) None.

Find $f^{\prime}(x)$ for $f(x)=\frac{\left(x^{5}-3^{x^{2}}+2 t\right)^{3}}{x}$ . Choose from options a) to d).

Differentiate $10 x^{2} \sin (2 y) + y \ln (x) = 6$ implicitly to find $y^{\prime}$ . Choose from options a) to e).

Find $f^{\prime}(x)$ for $f(x)=x^{2}(x^{2}+20)^{\frac{1}{4}}$ . Choose from the given options a) to e).

Find the rate of change of $P(t)=2+5 \tan^{-1}\left(\frac{t}{2}\right)$ when $P(t)=6$ . Choices: (A) -0.606 (B) 0.250 (C) 1.214 (D) 1.942

Find the rate of change of area $A(t)$ of spilled oil at $t=10$ . Which option represents this: (A) $A(10)$ , (B) $A(11)-A(9)$ , (C) $\frac{A(10)}{10}$ , (D) $A^{\prime}(10)$ ?

Estimate $\Delta f=f(4.9)-f(5)$ for $f(x)=x-4x^{2}$ using Linear Approximation. Find actual change and errors.

Find when the speed of the particle with position $y(t)=t^{3}-6 t^{2}+9 t$ is increasing for $0<t<4$ .

Estimate $\Delta f=f(4.9)-f(5)$ for $f(x)=x-4 x^{2}$ using Linear Approximation. Find actual $\Delta f$ and error.

Find the linear approximation of $y=(1+3x)^{-1/2}$ at $x=5$ .

Find $t$ where the linear approximation of $A(t)$ , given $A'(t)=2+9 e^{0.4 \sin t}$ and $A(1.2)=7.5$ , equals 15.

Given the piecewise function $f(x)=\left\{\begin{array}{ll} x^{2}-3 x & x \leq 8 \\ 5 x & x>8 \end{array}\right.$ , check continuity and differentiability at $x=8$ . Find $f'(8)$ or $f(8)$ .

Find the derivative of $g(x)=-8 x^{2}(9 x^{2})$ using the Product Rule and algebra.

The tangent line to $y=f(x)$ at $(7,37)$ is $y=6x-5$ . Find $f^{\prime}(7)$ and the instantaneous rate of change at $x=7$ .

Estimate $\Delta f=f(4.9)-f(5)$ for $f(x)=x-4x^{2}$ using Linear Approximation. Find actual $\Delta f$ and errors.

Estimate $\Delta f=f(4.9)-f(5)$ for $f(x)=x-4 x^{2}$ using Linear Approximation. Find actual change and compute error.

Find the derivative of $f(x)=\frac{3}{\sqrt[5]{x}}$ and evaluate at $x=1$ and $x=4$ .

Find $f$ and $g$ such that $h=(2x + 7x^5 - 10x^9)(9 + 2x - 4x^2)$ and use the product rule for $h'$ .

Approximate the value of $e^{-0.33}$ using a calculator.

Estimate $\Delta y$ using differentials for $y=x^{1/2} e^{4x-1}$ at $a=1$ with $dx=0.1$ . Round to four decimal places.

A car on a highway goes past an intersection at 70 mph. How fast is it moving away from a farmhouse 8 miles away when 5 miles past? The rate is $\square$ mph.

Find the second derivative of the cubic function $f(x)=3 x^{3}+17 x^{2}+14 x-9$ . What is $f^{\prime \prime}(x)$ ?

Find the derivative $f^{\prime}(x)$ of $f(x)=x^{15} e^{x}$ and evaluate it at $x=1$ . What is $f^{\prime}(1)$ ?

Find the derivative of $f(x)=x^{15} e^{x}$ and evaluate it at $x=1$ : $f^{\prime}(1)=\,$

A balloon's volume grows at $2.5 \mathrm{ft}^{3} / \mathrm{min}$ . Find the diameter's growth rate when it's 1.1 feet.

Find the derivative of $f(x)=\frac{2 \sqrt{x}}{3 x^{2}+4 x-9}$ and evaluate it at $x=3$ .

Calculate the average value of $f(x)=\frac{1}{\sqrt{4 x-3}}$ from $x=3$ to $x=21$ , rounded to four decimal places.

Find the average of $f(x)=x^{3} \sqrt{1+x^{2}}$ from $0$ to $\sqrt{3}$ . Round to four decimal places.

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

Given $f(x)=5 x^{2}+2 x$ , find: (A) $f^{\prime}(x)$ , (B) slopes at $x=2$ and $x=3$ , (C) tangent line equations, (D) where tangent is horizontal.

Find the first derivative of $f(x)=\frac{5 x^{2}}{x^{2}-9}$ and determine critical numbers, increasing/decreasing intervals, and local extrema.

Estimate the change in stopping distance $\Delta F$ for $F(s)=1.1 s+0.054 s^{2}$ at speeds $s=55$ and $s=85$ mph. Give answers to two decimal places.