Calculus

Problem 22401

Estimate $f(0.4)$ and $f(0.5)$ using $f(x)=\sin(x) \approx x$ with $n=2$ . Find the maximum error for given $M$ values.

See Solution

Problem 22402

Find the object's velocity at $t=3$ for $s(t)=-3t+t^{2}$ . Options: a) $3 \mathrm{~m/s}$ b) None c) $0 \mathrm{~m/s}$ d) $6 \mathrm{~m/s}$ e) $-3 \mathrm{~m/s}$ .

See Solution

Problem 22403

Find the 3rd-order Taylor polynomial for $f(x)=2 \sin(5x)$ at $a=0$ . Answer: $\square$ .

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Problem 22404

Find the linear and quadratic approximations of $f(x)=16 x^{3/2}$ at $a=1$ and use them to estimate $16(1.2^{3/2})$ .

See Solution

Problem 22405

Find the point on the graph of $f(x)=3x^{2}+x$ where the tangent is parallel to $y=4x+1$ .

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Problem 22406

Find the linear and quadratic approximations of $f(x)=16x^{3/2}$ at $a=1$ and estimate $16(1.2^{3/2})$ .

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Problem 22407

Find the slope of the tangent to $y=\frac{4}{x}$ at the point $(1,4)$ . Options: a) None b) -4 c) 4 d) $\frac{1}{4}$ e) $-\frac{1}{4}$

See Solution

Problem 22408

Find $g^{\prime}(0)$ for $g(x)=(x^{2}+7x-1)^{4}$ . Options: a) 7 b) -4 c) -28 d) 0 e) None

See Solution

Problem 22409

Find $f^{\prime}(-1)$ for $f(x)=\frac{1-x}{x^{2}}$ . Options: a) 0 b) -1 c) None d) 3 e) 1

See Solution

Problem 22410

Invest \$4000 at 8.25\% interest, compounded continuously. Find the value after 3, 6, and 9 years (round to nearest cent).

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Problem 22411

Find the slope of the tangent to the curve $y=(x+1)^{2}(x-1)$ at the point $(2,9)$ .

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Problem 22412

Find the linear and quadratic approximating polynomials for $f(x)=32 x^{3/2}$ at $a=9$ and use them to estimate $32(9.1^{3/2})$ .

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Problem 22413

Calculate the midpoint Riemann sum for $f(x) = x + 3$ on $[3,7]$ using $n = 4$ .

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Problem 22414

Which of these does not cause a derivative to fail: a) discontinuity, b) horizontal tangent, c) cusp, d) None, e) vertical tangent?

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Problem 22415

Berechnen Sie die lokale Änderungsrate von $f$ an $a$ mit der h-Methode für die Funktionen a) bis f) und vergleichen Sie die Grenzwerte.

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Problem 22416

Evaluate the limits: $\lim_{x \rightarrow 4} (16x - 2x^2 - 25)$ and $\lim_{x \rightarrow 4} (x^2 - 8x + 23)$ to find $\lim_{x \rightarrow 4} g(x)$ using the Squeeze Theorem.

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Problem 22417

Find the linear and quadratic approximations for $f(x)=-\frac{1}{x}$ at $a=1$ and use them to estimate $-\frac{1}{1.03}$ .

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Problem 22418

Given the piecewise function $f(x)=\left\{\begin{array}{lll}\frac{8}{x+4} & \text { if } & x<-2 \\ -\frac{8}{x} & \text { if } & x>-2\end{array}\right.$ , find the limits and value at $x=-2$ .
Compute: $\lim _{x \rightarrow-2^{-}} f(x), \quad \lim _{x \rightarrow-2^{+}} f(x), \quad f(-2)$
Determine where $f$ is discontinuous.

See Solution

Problem 22419

Ein Stein wird senkrecht hochgeschleudert. Höhe $h(t)=-5 t^{2}+10 t$ .
a) Bestätigen Sie $v(t)=h'(t)=-10 t+10$ in $\frac{m}{s}$ . b) Bestimmen Sie die höchste Höhe und den Zeitpunkt. c) Finden Sie, wann $v=5 \frac{m}{s}$ ist. d) Bestimmen Sie den Zeitpunkt und die Geschwindigkeit beim Aufprall.

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Problem 22420

Evaluate the integral $\int \frac{dx}{x^{2}-2x+82}$ . Rewrite the denominator by completing the square: $\frac{1}{x^{2}-2x+82}=\square$ .

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Problem 22421

Bestimmen Sie die Ableitung $f^{\prime}(x)$ für $f(x)=4 x^{\frac{1}{2}}$ .

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Problem 22422

Find $f^{\prime}(x)$ and $f^{\prime}(4)$ for $f(x)=\int_{4}^{x^{2}} \frac{1}{1+t^{2}} d t$ .

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Problem 22423

Bestimme die Ableitungen $f^{\prime}(x)$ für die Funktionen a) $3 x^{2}$ , b) $5 x^{3}$ , c) $\frac{1}{2} x^{4}$ , d) $-2 x^{7}$ , e) $\frac{1}{2} x^{-2}$ , f) $-5 x^{-3}$ , g) $-0,2 x^{5}$ , h) $4 x^{\frac{1}{2}}$ .

See Solution

Problem 22424

Find a $\delta$ for $\varepsilon=0.01$ such that $\lim_{x \rightarrow 5} -1x - 6 = -11$ .

See Solution

Problem 22425

Evaluate the integral $\int x^{3} e^{2 x} d x$ using the reduction formula for $n=3$ .

See Solution

Problem 22426

Find the average slope of $f(x)=6 \sqrt{x}+8$ on $[3,6]$ and the unique $c$ in $(3,6)$ where $f^{\prime}(c)$ equals this slope.

See Solution

Problem 22427

Find the horizontal asymptotes by calculating these limits:
1. $\lim_{x \rightarrow \infty} \frac{-9 x}{11+2 x}=\square$
2. $\lim_{x \rightarrow -\infty} \frac{4 x-9}{x^{3}+8 x-4}=\square$
3. $\lim_{x \rightarrow \infty} \frac{x^{2}-13 x-15}{15-5 x^{2}}=\square$
4. $\lim_{x \rightarrow \infty} \frac{\sqrt{x^{2}+9 x}}{9-11 x}=\square$
5. $\lim_{x \rightarrow -\infty} \frac{\sqrt{x^{2}+9 x}}{9-11 x}=\square$

See Solution

Problem 22428

Approximate $\int_{2}^{9}(x^{2}+5) dx$ using 4 rectangles for left and right Riemann sums.

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Problem 22429

Prove that if $f$ has continuous derivatives on $[a, b]$ with $f^{\prime}(a)=f^{\prime}(b)=0$ , then:
$\int_{a}^{b} x f^{\prime \prime}(x) d x=f(a)-f(b)$
Use integration by parts with $u(x)=x$ and find $u^{\prime}(x)$ and $v(x)$ .

See Solution

Problem 22430

Find the limit: $\lim _{x \rightarrow \infty} \frac{\ln (8 x+7)}{x}$ .

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Problem 22431

Find the coordinates of the maximum point $A$ for $f(x)=(x-2)^{2} e^{3x}$ and the range of $k$ for two distinct roots.

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Problem 22432

Show that if $f$ has continuous derivatives on $[a, b]$ with $f'(a)=f'(b)=0$ , then $\int_{a}^{b} x f''(x) dx = f(a) - f(b)$ . Use integration by parts with $u(x)=x$ and $v'(x)=f''(x)$ . Find $u'(x)$ and $v(x)$ . $u'(x)=\nabla$ and $v(x)=\square$ .

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Problem 22433

Given the function $f(x)=\frac{3 x}{x^{2}-25}$ , find critical numbers, decreasing intervals, local maxima/minima, inflection points, concavity, and asymptotes.

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Problem 22434

Find the limit: $\lim _{x \rightarrow \infty} \frac{\ln (8 x+7)}{x}$ .

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Problem 22435

Find the absolute maximum and minimum of $f(x)=3 x^{2}-2 x+6$ for $0 \leq x \leq 8$ .

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Problem 22436

Invest \ $8,900 at 7\% interest compounded continuously. Find the function for account value after$ t$ years and APY.

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Problem 22437

Calculate the average rate of change of $h(x)=3 x^{2}+4$ between $x=5$ and $x=7$ . Simplify your answer.

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Problem 22438

Find the limit: $\lim _{x \rightarrow \infty}(8 x+7)^{\frac{1}{x}}$ . Identify the indeterminate form.

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Problem 22439

Given the function $f(x)=\frac{3 x}{x^{2}-25}$ , find critical numbers, decreasing intervals, local maxima/minima, inflection points, and concavity.

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Problem 22440

Find the interval of convergence for $G(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n 5^{n}} x^{n}$ , $G^{(100)}(0)$ , and $G(2)$ .

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Problem 22441

Given the function $f(x)=5(x^{2}-2)(4x+9)^{\frac{1}{2}}$ , show $f'(x)=\frac{k(5x^{2}+9x-2)}{(4x+9)^{\frac{1}{2}}}$ , find $k$ , $f'(x)=0$ values, local max $P$ , and range of $g(x)=2f(x)+4$ .

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Problem 22442

Find the tangent line equation for $y=\arctan \left(\frac{x}{2}\right)$ at the point $\left(2, \frac{\pi}{4}\right)$ .

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Problem 22443

1. Trouvez l'élasticité-prix de la demande avec $p=3$ dans $q=100 e^{-3 p^{2}+p}$ . Options: A) 48 B) 50 C) 51 D) 55 E) 63 F) aucune.
2. Trouvez $\frac{d y}{d x}$ au point $(1,1)$ pour $4 x y^{2}+3 x^{2} y=7$ . Options: A) $-\frac{10}{11}$ B) $-\frac{4}{7}$ C) $\frac{1}{13}$ D) $-\frac{4}{11}$ E) $\frac{10}{13}$ F) aucune.
3. Trouvez le coût marginal de $C(x)=400000+160 x+0,001 x^{2}$ à $x=10000$ . Options: A) 160 B) 180 C) 200 D) 220 E) 240 F) aucune.
4. Trouvez $f(8)$ si $f^{\prime}(x)=x^{\frac{2}{3}}$ et $f(1)=1$ . Options: A) $\frac{64}{5}$ B) $\frac{82}{5}$ C) $\frac{74}{5}$ D) $\frac{98}{5}$ E) $\frac{6}{5}$ F) aucune.
5. Calculez $\int_{0}^{1} x e^{2 x} d x$ . Options: A) $\mathrm{e}^{2}$ B) $\frac{\mathrm{e}^{2}}{2}$ C) $\frac{e-1}{2}$ D) $\frac{e^{2}+1}{4}$ E) $\frac{e+3}{2}$ F) aucune.

See Solution

Problem 22444

Given the function $f(x)=3 x^{2} \ln (x)$ for $x>0$ , find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

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Problem 22445

Find critical numbers of $f(x)=12x-4\ln(x)$ for $x>0$ . Indicate intervals where $f$ is increasing and decreasing. List local maxima and minima.

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Problem 22446

Determine the inflection points of $f(x)=4 \sin ^{2} x-1$ for $0 \leq x \leq \pi$ .

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Problem 22447

Find the average rate of change of $g(x)=4 x^{3}-7$ on the interval $[-3,3]$ .

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Problem 22448

Determine the intervals where $g(x)=x^{5}+x^{4}$ is decreasing. Choose from: a) $x<0$ , b) $-\frac{4}{5}<x<0$ , c) $x<-\frac{4}{5}$ , d) $x<-\frac{4}{5}$ and $x>0$ .

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Problem 22449

Trouvez la tangente à $\left(x^{3}+y^{3}\right)^{3}=6 x y+2$ au point $(1,1)$ , puis résolvez 3 autres problèmes.

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Problem 22450

Find the number $c$ for the Mean Value Theorem with $f(x)=e^{-7x}$ on the interval $[0,7]$ .

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Problem 22451

Given the function $f(x)=3 x^{2} \ln (x)$ for $x>0$ , find critical values, intervals of increase/decrease, local maxima/minima, and concavity.
(A) Critical values:
(B) Increasing: $(e^{-\frac{1}{2}}, INF)$
(C) Decreasing: $(0, e^{-\frac{1}{2}})$
(D) Local maxima: NONE
(E) Local minima: $e^{-\frac{1}{2}}$
(F) Concave up: $(e^{-\frac{3}{2}}, INF)$
(G) Concave down:
(H) Inflection points: NONE

See Solution

Problem 22452

Given the function $f(x)=3 x^{2} \ln (x)$ for $x>0$ , find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

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Problem 22453

Find the average slope of $f(x)=10 \sqrt{x}+5$ on $[3,9]$ : $\frac{f(9)-f(3)}{9-3}$ .

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Problem 22454

Find the number $c$ that satisfies the Mean Value Theorem for $f(x)=3x^3+3x+19$ on the interval $[0,2]$ .

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Problem 22455

Estimate the balance in a savings account after 5 years with daily deposits and \$19,000 at 8\% interest compounded continuously.

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Problem 22456

Which option helps find the slope of the tangent line to $\sin^{-1}(2x^2+y^2)=\frac{2}{x}+y^2$ ?

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Problem 22457

Given $f(x)=(7-5 x) e^{x}$ , find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

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Problem 22458

Graph $f(x)=x+4/x$ and the secant line through $(1,5)$ and $(8,8.5)$ . Find $c$ for the Mean Value Theorem on $[1,8]$ . $c=$

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Problem 22459

Check if the Mean Value Theorem applies to $f(x)=15 x^{2}+9 x+5$ on $[1,6]$ . If yes, find $c$ values. If no, enter DNE. $c=$

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Problem 22460

Which option does not need the chain rule to find $\frac{d y}{d x}$ ? (A) $y=\cos ^{-1}(10 x^{5}-x^{2})$ (B) $4 x^{10}+6 y^{3}=x^{2} y^{5}-2$ (C) $y=10 \sqrt{x}-\frac{4}{x^{3}}+x^{5}$ (D) $\sin(2 x-y)+e^{2 y}+\frac{x}{6}=0$

See Solution

Problem 22461

Check if the Mean Value Theorem applies to $f(x)=9 \ln (x)+3$ on $[1,10]$ . Find values of $c$ or enter DNE. $c=$

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Problem 22462

Find the average slope of $f(x)=10 \sqrt{x}+5$ on the interval $[3,9]$ : $\frac{f(9)-f(3)}{9-3}=\square$ .

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Problem 22463

Find local extrema and inflection points for $f(x)$ given $f^{\prime}(x)=(x+4)(10-x)(14-x)$ .

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Problem 22464

Find the volume of the solid formed by revolving $y=\sqrt{x}$ from $x=0$ to $x=16$ around the $\mathrm{x}$ -axis.

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Problem 22465

Check if the Mean Value Theorem applies to $f(x)=2 \sin^{-1} x$ on $[-1,1]$ and find values of $c$ in $[a,b]$ .

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Problem 22466

Estimate $\int_{-3}^{5} (x^{2}-2) \, dx$ using $R_{4}$ and $M_{4}$ sums. Sketch rectangles and label $x_{1}, x_{2}, \ldots$ .

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Problem 22467

Find the volume of the solid formed by revolving the area between $y=e^{x}$ , $y=0$ , $x=-1$ , and $x=7$ around the $x$ -axis.

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Problem 22468

Find $g(x)$ given that $g^{\prime \prime}(x)=2-6x-\sin(x)$ , $g^{\prime}(0)=4$ , and $g(0)=1$ .

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Problem 22469

Find the average weekly sales from week 1 to week 8 for $S(t) = 700 e^{t}$ .

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Problem 22470

Find the average rate of change of $f(x)=-6 \sqrt{x+5}-8 x$ from $x=-2$ to $x=5$ , rounded to the nearest tenth.

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Problem 22471

If $f$ and $g$ are continuous, find $\int_{1}^{4}\left[3 g(x) - 4 f(x) - \sqrt{9 - (x - 1)^{2}}\right] \, dx$ given $\int_{4}^{1} 3 f(x) \, dx = 36$ and $\int_{1}^{4} 5 g(x) \, dx = -21$ .

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Problem 22472

Evaluate the Riemann sum for $f(x)=4 x^{2}-3$ with endpoints, $a=0$ , $b=4$ , and $n=4$ . Then, find $\int_{0}^{4}(4 x^{2}-3) dx$ as $n \rightarrow \infty$ .

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Problem 22473

Evaluate the Riemann sum for $f(x)=4 x^{2}-3$ with endpoints $a=0$ , $b=4$ , and $n=4$ . Then find $\int_{0}^{4}(4 x^{2}-3) dx$ as $n \rightarrow \infty$ .

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Problem 22474

A forest fire covers 1300 acres at midnight and spreads at $f(t)=3 \sqrt{t}$ . What is the rate after 20 hours? $\square$ acres/hour.

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Problem 22475

Show that if $f$ has continuous derivatives on $[a, b]$ with $f^{\prime}(a)=f^{\prime}(b)=0$ , then
$\int_{a}^{b} x f^{\prime \prime}(x) d x=f(a)-f(b)$
by using integration by parts. Define $u(x)=x$ and find $u^{\prime}(x)$ and $v(x)$ .

See Solution

Problem 22476

Find the slope of the tangent to the curve $f(x)=(4x+7)^{\frac{1}{3}}$ at $x=5$ , rounded to 1 decimal place.

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Problem 22477

Show that if $f$ has continuous derivatives on $[a, b]$ with $f^{\prime}(a)=f^{\prime}(b)=0$ , then $\int_{a}^{b} x f^{\prime \prime}(x) d x=f(a)-f(b)$ . Use integration by parts with $u(x)=x$ and $v(x)=f^{\prime}(x)$ .

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Problem 22478

Find the slope of the tangent line to $\tan^{-1}(x-2y+2)=x^2-3y+\tan^{-1}(2)-1$ . Which equation applies?

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Problem 22479

Estimez l'intégrale $J=\int_{0}^{0.4} x^{2} e^{x} d x$ en utilisant les méthodes suivantes : (1) points milieux avec $n=4$ , (2) Simpson avec $n=4$ , (3) théorème fondamental du calcul. Arrondissez à 6 décimales.

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Problem 22480

Calculate the area between the curve $y=x^{2}(x-2)^{2}$ and the $x$ -axis for $0 \leq x \leq 2$ .

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Problem 22481

Find the tangent line equation for $y=\sqrt{x^{2}+16}$ at $x=3$ and compute $f^{\prime \prime}(x)$ for $f(x)=\frac{5}{3x+4}$ .

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Problem 22482

Find the intervals where the function $f(x)=\int_{0}^{x}\left(e^{-t^{3}+3 t}\right) d t$ is concave up and down.

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Problem 22483

Find the derivative of $y=\int_{0}^{x} \sqrt{4 t+9} \, dt$ .

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Problem 22484

Find the derivative of the integral: $\frac{d}{d t} \int_{0}^{\sin t} \frac{1}{1-u^{2}} d u$ .

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Problem 22485

If \ $8000 is invested at 8% interest compounded continuously, graph$ S = 8000e^{0.08t} $for$ 0 \leq t \leq 15$.

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Problem 22486

Find the average value of $g(x)=2 x^{2}-5 x+3$ over $[-1,5]$ . Provide your answer as an EXACT value.

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Problem 22487

Find all functions $f(x)$ such that $f'(x) = x^8$ and $f(0) = 5$ .

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Problem 22488

Evaluate the integral: $\int \frac{(3+\ln (x))^{2}(2-\ln (x))}{4 x} d x$ .

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Problem 22489

Evaluate the integral: $\int \frac{(3+\ln (x))^{2}(2-\ln (x))}{4 x} d x$

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Problem 22490

Calculez les dépenses annuelles moyennes de santé en Ontario de 1990 à 2020 avec $F^{\prime}(t)=296(1.08)^{t}$ .

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Problem 22491

Find the linear approximation $L$ of $f(x)=8-x^{2}$ at $a=-1$ . Graph $f$ and $L$ , then determine if $L$ underestimates or overestimates $f$ .

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Problem 22492

A ball is thrown from 640 ft with an initial speed of 96 ft/s. Find $s(t)$ , time to ground, and max height.

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Problem 22493

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

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Problem 22494

Let $f(x)=8-x^{2}$ and $a=-1$ . Are linear approximations near $a$ under or over estimates? Compute $f''(-1)$ .

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Problem 22495

Find the integral of $\frac{x}{k}$ with respect to $x$ , where $k$ is a non-zero constant.

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Problem 22496

Find the derivatives of the functions: 5. $h(x)=4 f(x)+\frac{g(x)}{7}$ and 6. $h(x)=x^{3} f(x)$ .

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Problem 22497

Find the derivatives of the following functions: 5. $h(x)=4 f(x)+\frac{e(x)}{9}$ , 6. $h(x)=x^{3} f(x)$ , 7. $f(x)=\frac{f(\operatorname{mog}(x))}{2}$ , 8. $h(x)=\frac{3 f(x)}{g(x)+2}$ .

See Solution

Problem 22498

Find the horizontal tangent lines for the graph of $(y^{3}+1)^{2}=x^{2}+4x+4$ .

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Problem 22499

Find the temperature of an object taken from a freezer at $-8^{\circ} \mathrm{C}$ , given $T^{\prime}(t)=10 e^{-0.5 t}$ .

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Problem 22500

Find the profit function for a tie shop with marginal profit $M P(x) = 1.95 + 0.08x - 0.0027x^2$ and loss of \ $.60 at$ x=0$.

See Solution

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Calculus

Problem 22401

Estimate f(0.4)f(0.4)f(0.4) and f(0.5)f(0.5)f(0.5) using f(x)=sin⁡(x)≈xf(x)=\sin(x) \approx xf(x)=sin(x)≈x with n=2n=2n=2. Find the maximum error for given MMM values.

Problem 22402

Find the object's velocity at t=3t=3t=3 for s(t)=−3t+t2s(t)=-3t+t^{2}s(t)=−3t+t2. Options: a) 3 m/s3 \mathrm{~m/s}3 m/s b) None c) 0 m/s0 \mathrm{~m/s}0 m/s d) 6 m/s6 \mathrm{~m/s}6 m/s e) −3 m/s-3 \mathrm{~m/s}−3 m/s.

Problem 22403

Find the 3rd-order Taylor polynomial for f(x)=2sin⁡(5x)f(x)=2 \sin(5x)f(x)=2sin(5x) at a=0a=0a=0. Answer: □\square□.

Problem 22404

Find the linear and quadratic approximations of f(x)=16x3/2f(x)=16 x^{3/2}f(x)=16x3/2 at a=1a=1a=1 and use them to estimate 16(1.23/2)16(1.2^{3/2})16(1.23/2).

Problem 22405

Find the point on the graph of f(x)=3x2+xf(x)=3x^{2}+xf(x)=3x2+x where the tangent is parallel to y=4x+1y=4x+1y=4x+1.

Problem 22406

Find the linear and quadratic approximations of f(x)=16x3/2f(x)=16x^{3/2}f(x)=16x3/2 at a=1a=1a=1 and estimate 16(1.23/2)16(1.2^{3/2})16(1.23/2).

Problem 22407

Find the slope of the tangent to y=4xy=\frac{4}{x}y=x4​ at the point (1,4)(1,4)(1,4). Options: a) None b) -4 c) 4 d) 14\frac{1}{4}41​ e) −14-\frac{1}{4}−41​

Problem 22408

Find g′(0)g^{\prime}(0)g′(0) for g(x)=(x2+7x−1)4g(x)=(x^{2}+7x-1)^{4}g(x)=(x2+7x−1)4. Options: a) 7 b) -4 c) -28 d) 0 e) None

Problem 22409

Find f′(−1)f^{\prime}(-1)f′(−1) for f(x)=1−xx2f(x)=\frac{1-x}{x^{2}}f(x)=x21−x​. Options: a) 0 b) -1 c) None d) 3 e) 1

Problem 22410

Invest \$4000 at 8.25\% interest, compounded continuously. Find the value after 3, 6, and 9 years (round to nearest cent).

Problem 22411

Find the slope of the tangent to the curve y=(x+1)2(x−1)y=(x+1)^{2}(x-1)y=(x+1)2(x−1) at the point (2,9)(2,9)(2,9).

Problem 22412

Find the linear and quadratic approximating polynomials for f(x)=32x3/2f(x)=32 x^{3/2}f(x)=32x3/2 at a=9a=9a=9 and use them to estimate 32(9.13/2)32(9.1^{3/2})32(9.13/2).

Problem 22413

Calculate the midpoint Riemann sum for f(x)=x+3f(x) = x + 3f(x)=x+3 on [3,7][3,7][3,7] using n=4n = 4n=4.

Problem 22414

Which of these does not cause a derivative to fail: a) discontinuity, b) horizontal tangent, c) cusp, d) None, e) vertical tangent?

Problem 22415

Berechnen Sie die lokale Änderungsrate von fff an aaa mit der h-Methode für die Funktionen a) bis f) und vergleichen Sie die Grenzwerte.

Problem 22416

Evaluate the limits: lim⁡x→4(16x−2x2−25)\lim_{x \rightarrow 4} (16x - 2x^2 - 25)limx→4​(16x−2x2−25) and lim⁡x→4(x2−8x+23)\lim_{x \rightarrow 4} (x^2 - 8x + 23)limx→4​(x2−8x+23) to find lim⁡x→4g(x)\lim_{x \rightarrow 4} g(x)limx→4​g(x) using the Squeeze Theorem.

Problem 22417

Find the linear and quadratic approximations for f(x)=−1xf(x)=-\frac{1}{x}f(x)=−x1​ at a=1a=1a=1 and use them to estimate −11.03-\frac{1}{1.03}−1.031​.

Problem 22418

Problem 22419

Problem 22420

Evaluate the integral ∫dxx2−2x+82\int \frac{dx}{x^{2}-2x+82}∫x2−2x+82dx​. Rewrite the denominator by completing the square: 1x2−2x+82=□\frac{1}{x^{2}-2x+82}=\squarex2−2x+821​=□.

Problem 22421

Bestimmen Sie die Ableitung f′(x)f^{\prime}(x)f′(x) für f(x)=4x12f(x)=4 x^{\frac{1}{2}}f(x)=4x21​.

Problem 22422

Find f′(x)f^{\prime}(x)f′(x) and f′(4)f^{\prime}(4)f′(4) for f(x)=∫4x211+t2dtf(x)=\int_{4}^{x^{2}} \frac{1}{1+t^{2}} d tf(x)=∫4x2​1+t21​dt.

Problem 22423

Problem 22424

Find a δ\deltaδ for ε=0.01\varepsilon=0.01ε=0.01 such that lim⁡x→5−1x−6=−11\lim_{x \rightarrow 5} -1x - 6 = -11limx→5​−1x−6=−11.

Problem 22425

Evaluate the integral ∫x3e2xdx\int x^{3} e^{2 x} d x∫x3e2xdx using the reduction formula for n=3n=3n=3.

Problem 22426

Find the average slope of f(x)=6x+8f(x)=6 \sqrt{x}+8f(x)=6x​+8 on [3,6][3,6][3,6] and the unique ccc in (3,6)(3,6)(3,6) where f′(c)f^{\prime}(c)f′(c) equals this slope.

Problem 22427

Problem 22428

Approximate ∫29(x2+5)dx\int_{2}^{9}(x^{2}+5) dx∫29​(x2+5)dx using 4 rectangles for left and right Riemann sums.

Problem 22429

Problem 22430

Find the limit: lim⁡x→∞ln⁡(8x+7)x\lim _{x \rightarrow \infty} \frac{\ln (8 x+7)}{x}limx→∞​xln(8x+7)​.

Problem 22431

Find the coordinates of the maximum point AAA for f(x)=(x−2)2e3xf(x)=(x-2)^{2} e^{3x}f(x)=(x−2)2e3x and the range of kkk for two distinct roots.

Problem 22432

Problem 22433

Given the function f(x)=3xx2−25f(x)=\frac{3 x}{x^{2}-25}f(x)=x2−253x​, find critical numbers, decreasing intervals, local maxima/minima, inflection points, concavity, and asymptotes.

Problem 22434

Find the limit: lim⁡x→∞ln⁡(8x+7)x\lim _{x \rightarrow \infty} \frac{\ln (8 x+7)}{x}limx→∞​xln(8x+7)​.

Problem 22435

Find the absolute maximum and minimum of f(x)=3x2−2x+6f(x)=3 x^{2}-2 x+6f(x)=3x2−2x+6 for 0≤x≤80 \leq x \leq 80≤x≤8.

Problem 22436

Invest \8,900at7%interestcompoundedcontinuously.Findthefunctionforaccountvalueafter8,900 at 7\% interest compounded continuously. Find the function for account value after 8,900at7%interestcompoundedcontinuously.Findthefunctionforaccountvalueaftert$ years and APY.

Problem 22437

Calculate the average rate of change of h(x)=3x2+4h(x)=3 x^{2}+4h(x)=3x2+4 between x=5x=5x=5 and x=7x=7x=7. Simplify your answer.

Problem 22438

Find the limit: lim⁡x→∞(8x+7)1x\lim _{x \rightarrow \infty}(8 x+7)^{\frac{1}{x}}limx→∞​(8x+7)x1​. Identify the indeterminate form.

Problem 22439

Given the function f(x)=3xx2−25f(x)=\frac{3 x}{x^{2}-25}f(x)=x2−253x​, find critical numbers, decreasing intervals, local maxima/minima, inflection points, and concavity.

Problem 22440

Find the interval of convergence for G(x)=∑n=1∞(−1)n+1n5nxnG(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n 5^{n}} x^{n}G(x)=∑n=1∞​n5n(−1)n+1​xn, G(100)(0)G^{(100)}(0)G(100)(0), and G(2)G(2)G(2).

Problem 22441

Problem 22442

Find the tangent line equation for y=arctan⁡(x2)y=\arctan \left(\frac{x}{2}\right)y=arctan(2x​) at the point (2,π4)\left(2, \frac{\pi}{4}\right)(2,4π​).

Problem 22443

Problem 22444

Estimate $f(0.4)$ and $f(0.5)$ using $f(x)=\sin(x) \approx x$ with $n=2$ . Find the maximum error for given $M$ values.

Find the object's velocity at $t=3$ for $s(t)=-3t+t^{2}$ . Options: a) $3 \mathrm{~m/s}$ b) None c) $0 \mathrm{~m/s}$ d) $6 \mathrm{~m/s}$ e) $-3 \mathrm{~m/s}$ .

Find the 3rd-order Taylor polynomial for $f(x)=2 \sin(5x)$ at $a=0$ . Answer: $\square$ .

Find the linear and quadratic approximations of $f(x)=16 x^{3/2}$ at $a=1$ and use them to estimate $16(1.2^{3/2})$ .

Find the point on the graph of $f(x)=3x^{2}+x$ where the tangent is parallel to $y=4x+1$ .

Find the linear and quadratic approximations of $f(x)=16x^{3/2}$ at $a=1$ and estimate $16(1.2^{3/2})$ .

Find the slope of the tangent to $y=\frac{4}{x}$ at the point $(1,4)$ . Options: a) None b) -4 c) 4 d) $\frac{1}{4}$ e) $-\frac{1}{4}$

Find $g^{\prime}(0)$ for $g(x)=(x^{2}+7x-1)^{4}$ . Options: a) 7 b) -4 c) -28 d) 0 e) None

Find $f^{\prime}(-1)$ for $f(x)=\frac{1-x}{x^{2}}$ . Options: a) 0 b) -1 c) None d) 3 e) 1

Find the slope of the tangent to the curve $y=(x+1)^{2}(x-1)$ at the point $(2,9)$ .

Find the linear and quadratic approximating polynomials for $f(x)=32 x^{3/2}$ at $a=9$ and use them to estimate $32(9.1^{3/2})$ .

Calculate the midpoint Riemann sum for $f(x) = x + 3$ on $[3,7]$ using $n = 4$ .

Berechnen Sie die lokale Änderungsrate von $f$ an $a$ mit der h-Methode für die Funktionen a) bis f) und vergleichen Sie die Grenzwerte.

Evaluate the limits: $\lim_{x \rightarrow 4} (16x - 2x^2 - 25)$ and $\lim_{x \rightarrow 4} (x^2 - 8x + 23)$ to find $\lim_{x \rightarrow 4} g(x)$ using the Squeeze Theorem.

Find the linear and quadratic approximations for $f(x)=-\frac{1}{x}$ at $a=1$ and use them to estimate $-\frac{1}{1.03}$ .

Evaluate the integral $\int \frac{dx}{x^{2}-2x+82}$ . Rewrite the denominator by completing the square: $\frac{1}{x^{2}-2x+82}=\square$ .

Bestimmen Sie die Ableitung $f^{\prime}(x)$ für $f(x)=4 x^{\frac{1}{2}}$ .

Find $f^{\prime}(x)$ and $f^{\prime}(4)$ for $f(x)=\int_{4}^{x^{2}} \frac{1}{1+t^{2}} d t$ .

Find a $\delta$ for $\varepsilon=0.01$ such that $\lim_{x \rightarrow 5} -1x - 6 = -11$ .

Evaluate the integral $\int x^{3} e^{2 x} d x$ using the reduction formula for $n=3$ .

Find the average slope of $f(x)=6 \sqrt{x}+8$ on $[3,6]$ and the unique $c$ in $(3,6)$ where $f^{\prime}(c)$ equals this slope.

Approximate $\int_{2}^{9}(x^{2}+5) dx$ using 4 rectangles for left and right Riemann sums.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\ln (8 x+7)}{x}$ .

Find the coordinates of the maximum point $A$ for $f(x)=(x-2)^{2} e^{3x}$ and the range of $k$ for two distinct roots.

Given the function $f(x)=\frac{3 x}{x^{2}-25}$ , find critical numbers, decreasing intervals, local maxima/minima, inflection points, concavity, and asymptotes.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\ln (8 x+7)}{x}$ .

Find the absolute maximum and minimum of $f(x)=3 x^{2}-2 x+6$ for $0 \leq x \leq 8$ .

Invest \ $8,900 at 7\% interest compounded continuously. Find the function for account value after$ t$ years and APY.

Calculate the average rate of change of $h(x)=3 x^{2}+4$ between $x=5$ and $x=7$ . Simplify your answer.

Find the limit: $\lim _{x \rightarrow \infty}(8 x+7)^{\frac{1}{x}}$ . Identify the indeterminate form.

Given the function $f(x)=\frac{3 x}{x^{2}-25}$ , find critical numbers, decreasing intervals, local maxima/minima, inflection points, and concavity.

Find the interval of convergence for $G(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n 5^{n}} x^{n}$ , $G^{(100)}(0)$ , and $G(2)$ .

Find the tangent line equation for $y=\arctan \left(\frac{x}{2}\right)$ at the point $\left(2, \frac{\pi}{4}\right)$ .

Given the function $f(x)=3 x^{2} \ln (x)$ for $x>0$ , find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

Find critical numbers of $f(x)=12x-4\ln(x)$ for $x>0$ . Indicate intervals where $f$ is increasing and decreasing. List local maxima and minima.

Determine the inflection points of $f(x)=4 \sin ^{2} x-1$ for $0 \leq x \leq \pi$ .

Find the average rate of change of $g(x)=4 x^{3}-7$ on the interval $[-3,3]$ .

Determine the intervals where $g(x)=x^{5}+x^{4}$ is decreasing. Choose from: a) $x<0$ , b) $-\frac{4}{5}<x<0$ , c) $x<-\frac{4}{5}$ , d) $x<-\frac{4}{5}$ and $x>0$ .

Trouvez la tangente à $\left(x^{3}+y^{3}\right)^{3}=6 x y+2$ au point $(1,1)$ , puis résolvez 3 autres problèmes.

Find the number $c$ for the Mean Value Theorem with $f(x)=e^{-7x}$ on the interval $[0,7]$ .

Given the function $f(x)=3 x^{2} \ln (x)$ for $x>0$ , find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

Find the average slope of $f(x)=10 \sqrt{x}+5$ on $[3,9]$ : $\frac{f(9)-f(3)}{9-3}$ .

Find the number $c$ that satisfies the Mean Value Theorem for $f(x)=3x^3+3x+19$ on the interval $[0,2]$ .

Which option helps find the slope of the tangent line to $\sin^{-1}(2x^2+y^2)=\frac{2}{x}+y^2$ ?

Given $f(x)=(7-5 x) e^{x}$ , find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

Graph $f(x)=x+4/x$ and the secant line through $(1,5)$ and $(8,8.5)$ . Find $c$ for the Mean Value Theorem on $[1,8]$ . $c=$

Check if the Mean Value Theorem applies to $f(x)=15 x^{2}+9 x+5$ on $[1,6]$ . If yes, find $c$ values. If no, enter DNE. $c=$

Check if the Mean Value Theorem applies to $f(x)=9 \ln (x)+3$ on $[1,10]$ . Find values of $c$ or enter DNE. $c=$

Find the average slope of $f(x)=10 \sqrt{x}+5$ on the interval $[3,9]$ : $\frac{f(9)-f(3)}{9-3}=\square$ .

Find local extrema and inflection points for $f(x)$ given $f^{\prime}(x)=(x+4)(10-x)(14-x)$ .

Find the volume of the solid formed by revolving $y=\sqrt{x}$ from $x=0$ to $x=16$ around the $\mathrm{x}$ -axis.

Check if the Mean Value Theorem applies to $f(x)=2 \sin^{-1} x$ on $[-1,1]$ and find values of $c$ in $[a,b]$ .

Estimate $\int_{-3}^{5} (x^{2}-2) \, dx$ using $R_{4}$ and $M_{4}$ sums. Sketch rectangles and label $x_{1}, x_{2}, \ldots$ .

Find the volume of the solid formed by revolving the area between $y=e^{x}$ , $y=0$ , $x=-1$ , and $x=7$ around the $x$ -axis.

Find $g(x)$ given that $g^{\prime \prime}(x)=2-6x-\sin(x)$ , $g^{\prime}(0)=4$ , and $g(0)=1$ .

Find the average weekly sales from week 1 to week 8 for $S(t) = 700 e^{t}$ .

Find the average rate of change of $f(x)=-6 \sqrt{x+5}-8 x$ from $x=-2$ to $x=5$ , rounded to the nearest tenth.

Evaluate the Riemann sum for $f(x)=4 x^{2}-3$ with endpoints, $a=0$ , $b=4$ , and $n=4$ . Then, find $\int_{0}^{4}(4 x^{2}-3) dx$ as $n \rightarrow \infty$ .

Evaluate the Riemann sum for $f(x)=4 x^{2}-3$ with endpoints $a=0$ , $b=4$ , and $n=4$ . Then find $\int_{0}^{4}(4 x^{2}-3) dx$ as $n \rightarrow \infty$ .

A forest fire covers 1300 acres at midnight and spreads at $f(t)=3 \sqrt{t}$ . What is the rate after 20 hours? $\square$ acres/hour.