Calculus

Problem 22801

Find where the tangent line is vertical and horizontal for the curve $x=1-\frac{1}{a} \cos a t, \quad y=t-\frac{1}{a} \sin a t$ ( $a>0$ ).

See Solution

Problem 22802

Find values of $p$ for which the series $\sum_{n=1}^{\infty}\left(\frac{p}{n+1} \frac{1}{n+3}\right)$ converges.

See Solution

Problem 22803

Given $x y=3$ and $\frac{d y}{d t}=2$ , find $\frac{d x}{d t}$ when $x=3$ .

See Solution

Problem 22804

Analyze the Leonard-Jones potential $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ as $r \rightarrow 0$ . What happens to $V(r)$ ?

See Solution

Problem 22805

If $p(x)$ and $q(x)$ are polynomials with $q$ monic and $\operatorname{deg}(p)<3$ , is the integral $\int \frac{p(x)}{q(x)} dx$ always, sometimes, or never equal to the given expression?

See Solution

Problem 22806

1. Pour quelles valeurs de $a$ et $b$ la fonction $f(x)$ est-elle continue sur $\mathbb{R}$ ? A) $a=1$ , $b=3$ B) $a=-2$ , $b=2$ C) $a=1$ , $b=2$ D) $a=3$ , $b=5$ E) $a=-3$ , $b=6$
2. Quelle est la valeur de $f_{xy}(1,1)$ pour $f(x,y)=e^{x^{2}}+2xy-2y$ ? A) 2 B) $2e$ C) $3e$ D) $4e$ E) $6e$ F) aucune de ses réponses
3. Combien de bureaux de chaque modèle peut-on fabriquer avec 530 unités de bois, 66,9 de contreplaqué et 31,8 de panneau particule ?
4. Quelles quantités supplémentaires de matériaux sont nécessaires pour 29 bureaux du modèle 1, 55 du modèle 2 et 43 du modèle 3 ?
5. Calculez les matrices suivantes : 1) $BC + A$ 2) $B^T$ 3) $3C^T A B^T$
6. Calculez les intégrales : a) $\int \frac{x e^{x^{2}}}{\sqrt{e^{x^{2}+2}}} dx$ b) $\int x^{2} \ln(x) dx$

See Solution

Problem 22807

Find the point on the graph of $y=e^{-x}$ where the tangent line's slope is -4. Provide the point as an ordered pair.

See Solution

Problem 22808

Find the volume of solid $S$ with a triangular cross-section. Base bounded by $x=a^{2}-y^{2}$ , $y=\frac{6}{a} x-4 a$ , and $x$ -axis.

See Solution

Problem 22809

Find the rate of change of $y$ with respect to $x$ in the equation $2x + y = 6$ .

See Solution

Problem 22810

Find the power series for $f(x)=\frac{x^{2}}{x^{4}+16}$ and its interval of convergence.

See Solution

Problem 22811

Maximize profit $P(x, y)=24x-x^2-xy-2y^2+33y-43$ for goods $x$ and $y$ . Find optimal $x$ and $y$ values.

See Solution

Problem 22812

Express the limit $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} c_{k}^{2} \Delta x$ as a definite integral over $[0,2]$ .

See Solution

Problem 22813

Question 7 : Trouvez $f_{xy}(1,1)$ pour $f(x, y)=e^{x^{2}}+2xy-2y$ .
Question 8 : Avec 530 bois, 66,9 contreplaqué, 31,8 panneau, combien de bureaux de chaque modèle peuvent être fabriqués ?
Question 9 : Pour les matrices $A$ , $B$ , et $C$ , calculez : 1) $BC+A$ 2) $B^{T}$ 3) $3C^{T}AB^{T}$
Question 10 : Calculez les intégrales : a) $\int \frac{xe^{x^{2}}}{\sqrt{e^{x^{2}+2}}} dx$ b) $\int x^{2} \ln(x) dx$

See Solution

Problem 22814

Show that for $n=1$ , the function $f(P)=\frac{P^{n}}{P^{n}+30^{n}}$ has no inflection points.

See Solution

Problem 22815

Find inflection points of $f(x)=8 e^{-7 x^{2}}$ for $x \geq 0$ and provide their coordinates as ordered pairs.

See Solution

Problem 22816

Find the tangent line equation for $y=f(x)$ at $x=4$ given $f(4)=-2$ and the slope $f'(4)$ . What is the equation?

See Solution

Problem 22817

Find where the tangent line is vertical and horizontal for the curve $x=1-\frac{1}{a} \cos a t$ , $y=t-\frac{1}{a} \sin a t$ ( $a>0$ ).

See Solution

Problem 22818

Show that for $n=3$ , the function $f(P)=\frac{P^{3}}{P^{3}+30^{3}}$ is increasing and approaches 1 as $P \rightarrow \infty$ .

See Solution

Problem 22819

Hill's equation models blood oxygen saturation: $f(P)=\frac{P^{n}}{P^{n}+30^{n}}$ . For $n=1$ , show $f(P)$ is increasing and $f(P) \rightarrow 1$ as $P \rightarrow \infty$ .

See Solution

Problem 22820

Deposit \ $5000 at$ 5.5\%$ interest compounded continuously. Find:
(a) Formula for $A(t)$ , (b) differential equation, (c) balance after 6 years, (d) time to reach \$8000, (e) growth rate at \$8000.

See Solution

Problem 22821

What is the average amount of radium in a vault after 1100 years if starting with 100 grams and a half-life of 1690 years? Average: $\square$ grams.

See Solution

Problem 22822

Partition the interval $[0,4]$ into 8 equal parts. Approximate the volume $S_{8}$ of cylinders at left endpoints. Overestimate? Why?

See Solution

Problem 22823

What is the average amount of radium (in grams) left in a vault after 1000 years if starting with 200 grams and a half-life of 1690 years? There will be an average of $\square$ grams of radium in the vault during the next 1000 years.

See Solution

Problem 22824

Find the limits: a. $\lim _{x \rightarrow \frac{7}{6}^{+}} \frac{15 x}{7-6 x}=$ b. $\lim _{x \rightarrow \frac{7}{6}^{-}} \frac{15 x}{7-6 x}=$

See Solution

Problem 22825

Find the average rate of change of rainfall from day 8 to day 14 for $f(t)$ , and answer parts B, C, and D regarding $f$ .

See Solution

Problem 22826

Calculate $\int_{4}^{1} f^{\prime}(x) \, dx$ using the function values from the table provided.

See Solution

Problem 22827

Evaluate the integral $\int_{-\pi}^{\pi} f(x) dx$ where $f(x)=\begin{cases}2 x^{4}, & -\pi \leq x<0 \\ 5 \sin (x), & 0 \leq x \leq \pi\end{cases}$ .

See Solution

Problem 22828

Find the limit: $\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \ln \left(2+t^{2}\right) d t}{x}$ .

See Solution

Problem 22829

Given the data for $C(t)$ at $t = 0, 10, 20, 40$ days, estimate $C^{\prime}(15)$ . Interpret $C^{\prime}(15)$ . For $D(t)=0.357(1.14)^{t}$ , find $D^{\prime}(15)$ .

See Solution

Problem 22830

A cup of tea cools from 120°F in a room at 70°F. After 3 min it's 100°F. Find its temp after 5 min, rounded to one decimal.

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Problem 22831

Find the population after 12 years of continuous growth at $4\%$ per year starting from 230000. Round down.

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Problem 22832

Evaluate the integral $\int \sin x \cos (\cos x) \, dx$ .

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Problem 22833

Verify the Intermediate Value Theorem for $f(x)=x^{2}+7x+1$ on $[0,9]$ and find $c$ where $f(c)=19$ .

See Solution

Problem 22834

Show that the function $f(x)=x^{2}-6-\cos x$ has at least one zero in the interval $[0, \pi]$ .

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Problem 22835

The bacteria count is given by $n(t)=930 e^{0.1 t}$ . Find the growth rate, initial population, and count at $t=5$ .

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Problem 22836

Calculate the area between the curves $y=7 x^{2}-x^{3}+x$ and $y=x^{2}+9 x$ . The area is 4.

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Problem 22837

Check if the Intermediate Value Theorem applies to $f(x)=x^{2}+7x+1$ on $[0,9]$ and find $c$ where $f(c)=19$ .

See Solution

Problem 22838

Find the area between the curves $y=x^{3}-12 x^{2}+35 x$ and $y=-x^{3}+12 x^{2}-35 x$ .

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Problem 22839

Analyze the function $f(x)=\frac{1}{x-8}$ as $x$ approaches 8 from both sides. Does $f(x)$ go to $\infty$ or $-\infty$ ?

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Problem 22840

Analyze the function $f(x)=7\left|\frac{x}{x^{2}-4}\right|$ as $x$ approaches 2 from the left and right. Does it go to $\infty$ or $-\infty$ ?

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Problem 22841

Given the function $f(x)=\frac{1}{x^{2}-16}$ , complete the table for $f(x)$ and analyze limits as $x$ approaches -4. What are:
$\lim _{x \rightarrow-4^{-}} f(x)=$ $\lim _{x \rightarrow-4^{+}} f(x)=$

See Solution

Problem 22842

Find the one-sided limit: $\lim _{x \rightarrow \pi^{+}} \frac{\sqrt{x}}{\csc x}$ (enter DNE if it doesn't exist).

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Problem 22843

Find or estimate $f(x)$ for $x=0,2,4,6$ given $f(0)=80$ and the derivative values: $f^{\prime}(0)=5$ , $f^{\prime}(2)=17$ , $f^{\prime}(4)=29$ , $f^{\prime}(6)=38$ .

See Solution

Problem 22844

Find the general antiderivative of $h(t)=\frac{-\sin(t)}{\cos^2(t)}$ for $-\frac{\pi}{2}<t<\frac{\pi}{2}$ .

See Solution

Problem 22845

Determine the limits: (a) $\lim _{x \rightarrow c}[f(x)+g(x)]$ (b) $\lim _{x \rightarrow c}[f(x) g(x)]$ (c) $\lim _{x \rightarrow c} \frac{g(x)}{f(x)}$ Given: $\lim _{x \rightarrow c} f(x)=\infty$ and $\lim _{x \rightarrow c} g(x)=-2$ .

See Solution

Problem 22846

Find the one-sided limit: $\lim _{x \rightarrow 0^{-}}\left(4+\frac{3}{x}\right)$ . If it doesn't exist, state DNE.

See Solution

Problem 22847

Find the tangent line to $f(x)=x^{3}$ that is parallel to $3x-y+8=0$ . Provide both $y$ -intercepts.

See Solution

Problem 22848

Find the limit of mass $m=\frac{m_{0}}{\sqrt{1-\left(v^{2} / c^{2}\right)}}$ as $v$ approaches $c$ from the left.

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Problem 22849

Find $f(2)$ for the function with $f^{\prime \prime}(x)=6x+4\sin(x)$ , given $f(0)=2$ and $f^{\prime}(0)=2$ .

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Problem 22850

Find $\frac{d R}{d r}$ for resistors $r$ and $s$ in parallel, and determine if $R$ is increasing, decreasing, or neither. Also, find global max and min for $R$ in $a \leq r \leq b$ .

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Problem 22851

Find the function $f(x)$ and the number $c$ such that the limit equals $f^{\prime}(c)$ :
$\lim_{x \rightarrow 4} \frac{6 \sqrt{x}-12}{x-4}$
$f(x)=\square$ , $c=\square$ .

See Solution

Problem 22852

Find the value(s) of $x$ where the slope of the tangent line to $y=1+250 x^{3}-3 x^{5}$ is maximized. Answer: $x=$

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Problem 22853

Find the derivative of the integral $\frac{d}{d t} \int_{0}^{t^{10}} \sqrt{u^{7}} du$ using two methods.

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Problem 22854

Given $h(x)=x f(x)+(g(x))^{3}$ , find $h^{\prime}(3)$ using provided values of $f(x)$ and $g(x)$ .

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Problem 22855

Calculate the area between the $x$ -axis and $y=-x^{2}-3x$ for $-8 \leq x \leq 3$ .

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Problem 22856

Find the $x$ -values where the function $y=\frac{x^{2}}{x^{2}-16}$ is differentiable. Use interval notation for your answer.

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Problem 22857

Find the derivative of $f(x)=x^{3}+2x^{2}+5$ at $x=-2$ . If it doesn't exist, write UNDEFINED.

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Problem 22858

Find the function $f(x)$ given that the limit equals $f^{\prime}(c)$ where $c=4$ : $\lim _{x \rightarrow 4} \frac{6 \sqrt{x}-12}{x-4}$

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Problem 22859

Find the derivative $\frac{d y}{d x}$ for $y=\int_{0}^{\cot x} \frac{d t}{1+t^{2}}$ .

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Problem 22860

Evaluate the integral: $\frac{d}{d t} \int_{0}^{t^{10}} \sqrt{u^{7}} du=\frac{d}{d t}(\square)$

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Problem 22861

Given the function $f(x)=|x-6|$ , find the left and right derivatives at $x=6$ . Is it differentiable there? Enter Yes or No.

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Problem 22862

Given the function $f(x)=3 x^{6}-7 x^{5}$ , find critical numbers, intervals of increase/decrease, local maxima/minima, concavity, inflection points, and asymptotes.

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Problem 22863

Find the derivative of $y=\frac{9}{7 x^{4}}$ and simplify the result.

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Problem 22864

Calculate the area between the $\mathrm{x}$ -axis and $y=2x^{3}+12x^{2}+16x$ for $-4 \leq x \leq 0$ .

See Solution

Problem 22865

Find the slope of the function $y=(6x+1)^{2}$ at the point $(0,1)$ using its derivative.

See Solution

Problem 22866

Find the tangent line equation for $f(x)=-5 x^{4}+9 x^{2}-4$ at the point (1,0). $y=$

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Problem 22867

Find the derivative of $f(t)=7 t^{2/3}-5 t^{1/3}+8$ . What is $f'(t)$ ?

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Problem 22868

Find the derivative of the function $y=\frac{\pi}{3} \sin (\theta)$ . What is $y^{\prime}$ ?

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Problem 22869

Given the function $f(x)=9 x-6 \ln (x)$ for $x>0$ , find critical numbers, increasing/decreasing intervals, local maxima/minima, concavity, and inflection points.

See Solution

Problem 22870

Find the point(s) where the function $y=x^{2}+5$ has a horizontal tangent line. If none, enter NONE.

See Solution

Problem 22871

Find the absolute min and max of $f(t)=16 \cos (t)+8 \sin (2 t)$ on $[0, \frac{\pi}{2}]$ .

See Solution

Problem 22872

Calculate the average rate of change of $f(t)=6 t^{2}-1$ over $[3,3.1]$ and compare it with rates at the endpoints.

See Solution

Problem 22873

A projectile is shot upward at 133 m/s. Find its velocity after 4s and 10s using $s(t)=-4.9t^2+v_0t+s_0$ .

See Solution

Problem 22874

Given the function $f(x)=4 x^{2} \ln (x)$ for $x>0$ , find critical values, intervals of increase/decrease, local max/min, concavity, inflection points, and sketch the graph.

See Solution

Problem 22875

Differentiate the function using the Product Rule: $f(x)=x^{7} \cos (x)$ .

See Solution

Problem 22876

Determine the absolute min and max of $f(t)=t \sqrt{9-t^{2}}$ on the interval $[-1,3]$ .

See Solution

Problem 22877

Find the derivative of $f(x)=\frac{x}{x-9}$ using the Quotient Rule: $f^{\prime}(x)=$

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Problem 22878

Find the critical numbers of the function $h(t)=t^{3/4}-9t^{1/4}$ . Enter answers as a list or DNE if none exist.

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Problem 22879

Find the critical numbers of the function $f(x)=2 x^{3}+x^{2}+4 x$ . Enter answers as a comma-separated list or DNE.

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Problem 22880

Given $f(x)=(8-4 x) e^{x}$ , find critical values, intervals of increase/decrease, local max/min, concavity, inflection points, and sketch the graph.

See Solution

Problem 22881

Find the critical numbers of the function $f(x)=2x^3-3x^2-36x$ .

See Solution

Problem 22882

Find the limit: $\lim_{x \to \infty} x^{e^{-x}}$ .

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Problem 22883

Find the derivative of $y=\frac{5}{2 x^{7}}$ without the Quotient Rule. What is $y'$ ?

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Problem 22884

Find the derivative $f^{\prime}(x)$ and evaluate it at $c=0$ for the function $f(x)=(x^{5}+5x)(3x^{4}+4x-3)$ .

See Solution

Problem 22885

Sketch the graph of $f(x)=\frac{6}{x}$ for $x \geq 6$ and find its absolute and local max/min values.

See Solution

Problem 22886

Evaluate the integral: $\frac{d}{d x} \int_{1}^{\cos x} 6 t^{5} d t=\frac{d}{d x}(\square)$

See Solution

Problem 22887

Test the series $\sum_{n=1}^{\infty} 5(-1)^{n} e^{-n}$ for convergence using the Alternating Series Test. Identify $b_{n}$ and evaluate $\lim _{n \rightarrow \infty} b_{n}$ .

See Solution

Problem 22888

Find the derivative of the function $f(x)=x^{3}\left(1-\frac{1}{x+5}\right)$ . What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 22889

Find the radius of convergence, $R$ , and interval of convergence, $I$ , for the series $\sum_{n=1}^{\infty} \frac{n}{7^{n}}(x+3)^{n}$ .

See Solution

Problem 22890

Find the derivatives $f^{\prime}(x)$ and $f^{\prime}(0)$ for the function $f(x)=(x^{5}+5x)(3x^{4}+4x-3)$ .

See Solution

Problem 22891

Find local extrema and inflection points for $f(x)$ given $f^{\prime}(x)=(x+7)(8-x)(14-x)$ .

See Solution

Problem 22892

Find the derivative of the function $y=-\csc (x)-\cos (x)$ ; answer: $y'=\square$ .

See Solution

Problem 22893

Find the min and max values of $f(x)=\frac{x}{x^{2}+1}$ on the interval $[0,4]$ .

See Solution

Problem 22894

Find where the function $f(x)=\frac{x^{2}}{x-8}$ has a horizontal tangent line and give the points $(x, y)$ .

See Solution

Problem 22895

Find the radius of convergence, $R$ , and the interval of convergence, $I$ , for the series $\sum_{n=1}^{\infty} n !(8 x-1)^{n}$ .

See Solution

Problem 22896

Find the growth rate of a bacteria population modeled by $P(t)=430\left(1+\frac{3t}{76+t^{2}}\right)$ at $t=2$ .

See Solution

Problem 22897

Find the fourth derivative of $f^{(3)}(x)=\sqrt[3]{x^{2}}$ . What is $f^{(4)}(x)$ ?

See Solution

Problem 22898

Find the second derivative of $f(x)=\sec x$ , so compute $f^{\prime \prime}(x)$ .

See Solution

Problem 22899

Find the tangent line equation at the point $(4,-2 \sqrt{3})$ for the curve $x^{2} y^{2}-9 x^{2}-4 y^{2}=0$ .

See Solution

Problem 22900

Gravel is dumped at $40 \mathrm{ft}^{3} / \mathrm{min}$ . Find the height increase rate when the pile is $9 \mathrm{ft}$ high.

See Solution

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Calculus

Problem 22801

Find where the tangent line is vertical and horizontal for the curve x=1−1acos⁡at,y=t−1asin⁡atx=1-\frac{1}{a} \cos a t, \quad y=t-\frac{1}{a} \sin a tx=1−a1​cosat,y=t−a1​sinat (a>0a>0a>0).

Problem 22802

Find values of ppp for which the series ∑n=1∞(pn+11n+3)\sum_{n=1}^{\infty}\left(\frac{p}{n+1} \frac{1}{n+3}\right)∑n=1∞​(n+1p​n+31​) converges.

Problem 22803

Given xy=3x y=3xy=3 and dydt=2\frac{d y}{d t}=2dtdy​=2, find dxdt\frac{d x}{d t}dtdx​ when x=3x=3x=3.

Problem 22804

Analyze the Leonard-Jones potential V(r)=1r6−Ar3V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}V(r)=r61​−r3A​ as r→0r \rightarrow 0r→0. What happens to V(r)V(r)V(r)?

Problem 22805

If p(x)p(x)p(x) and q(x)q(x)q(x) are polynomials with qqq monic and deg⁡(p)<3\operatorname{deg}(p)<3deg(p)<3, is the integral ∫p(x)q(x)dx\int \frac{p(x)}{q(x)} dx∫q(x)p(x)​dx always, sometimes, or never equal to the given expression?

Problem 22806

Problem 22807

Find the point on the graph of y=e−xy=e^{-x}y=e−x where the tangent line's slope is -4. Provide the point as an ordered pair.

Problem 22808

Find the volume of solid SSS with a triangular cross-section. Base bounded by x=a2−y2x=a^{2}-y^{2}x=a2−y2, y=6ax−4ay=\frac{6}{a} x-4 ay=a6​x−4a, and xxx-axis.

Problem 22809

Find the rate of change of yyy with respect to xxx in the equation 2x+y=62x + y = 62x+y=6.

Problem 22810

Find the power series for f(x)=x2x4+16f(x)=\frac{x^{2}}{x^{4}+16}f(x)=x4+16x2​ and its interval of convergence.

Problem 22811

Maximize profit P(x,y)=24x−x2−xy−2y2+33y−43P(x, y)=24x-x^2-xy-2y^2+33y-43P(x,y)=24x−x2−xy−2y2+33y−43 for goods xxx and yyy. Find optimal xxx and yyy values.

Problem 22812

Express the limit lim⁡n→∞∑k=1nck2Δx\lim _{n \rightarrow \infty} \sum_{k=1}^{n} c_{k}^{2} \Delta xlimn→∞​∑k=1n​ck2​Δx as a definite integral over [0,2][0,2][0,2].

Problem 22813

Problem 22814

Show that for n=1n=1n=1, the function f(P)=PnPn+30nf(P)=\frac{P^{n}}{P^{n}+30^{n}}f(P)=Pn+30nPn​ has no inflection points.

Problem 22815

Find inflection points of f(x)=8e−7x2f(x)=8 e^{-7 x^{2}}f(x)=8e−7x2 for x≥0x \geq 0x≥0 and provide their coordinates as ordered pairs.

Problem 22816

Find the tangent line equation for y=f(x)y=f(x)y=f(x) at x=4x=4x=4 given f(4)=−2f(4)=-2f(4)=−2 and the slope f′(4)f'(4)f′(4). What is the equation?

Problem 22817

Find where the tangent line is vertical and horizontal for the curve x=1−1acos⁡atx=1-\frac{1}{a} \cos a tx=1−a1​cosat, y=t−1asin⁡aty=t-\frac{1}{a} \sin a ty=t−a1​sinat (a>0a>0a>0).

Problem 22818

Show that for n=3n=3n=3, the function f(P)=P3P3+303f(P)=\frac{P^{3}}{P^{3}+30^{3}}f(P)=P3+303P3​ is increasing and approaches 1 as P→∞P \rightarrow \inftyP→∞.

Problem 22819

Hill's equation models blood oxygen saturation: f(P)=PnPn+30nf(P)=\frac{P^{n}}{P^{n}+30^{n}}f(P)=Pn+30nPn​. For n=1n=1n=1, show f(P)f(P)f(P) is increasing and f(P)→1f(P) \rightarrow 1f(P)→1 as P→∞P \rightarrow \inftyP→∞.

Problem 22820

Deposit \5000at5000 at 5000at5.5\%$ interest compounded continuously. Find:(a) Formula for A(t)A(t)A(t), (b) differential equation, (c) balance after 6 years, (d) time to reach \$8000, (e) growth rate at \$8000.

Problem 22821

What is the average amount of radium in a vault after 1100 years if starting with 100 grams and a half-life of 1690 years? Average: □\square□ grams.

Problem 22822

Partition the interval [0,4][0,4][0,4] into 8 equal parts. Approximate the volume S8S_{8}S8​ of cylinders at left endpoints. Overestimate? Why?

Problem 22823

What is the average amount of radium (in grams) left in a vault after 1000 years if starting with 200 grams and a half-life of 1690 years? There will be an average of □\square□ grams of radium in the vault during the next 1000 years.

Problem 22824

Find the limits: a. lim⁡x→76+15x7−6x=\lim _{x \rightarrow \frac{7}{6}^{+}} \frac{15 x}{7-6 x}=limx→67​+​7−6x15x​= b. lim⁡x→76−15x7−6x=\lim _{x \rightarrow \frac{7}{6}^{-}} \frac{15 x}{7-6 x}=limx→67​−​7−6x15x​=

Problem 22825

Find the average rate of change of rainfall from day 8 to day 14 for f(t)f(t)f(t), and answer parts B, C, and D regarding fff.

Problem 22826

Calculate ∫41f′(x) dx\int_{4}^{1} f^{\prime}(x) \, dx∫41​f′(x)dx using the function values from the table provided.

Problem 22827

Evaluate the integral ∫−ππf(x)dx\int_{-\pi}^{\pi} f(x) dx∫−ππ​f(x)dx where f(x)={2x4,−π≤x<05sin⁡(x),0≤x≤πf(x)=\begin{cases}2 x^{4}, & -\pi \leq x<0 \\ 5 \sin (x), & 0 \leq x \leq \pi\end{cases}f(x)={2x4,5sin(x),​−π≤x<00≤x≤π​.

Problem 22828

Find the limit: lim⁡x→0∫x0ln⁡(2+t2)dtx\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \ln \left(2+t^{2}\right) d t}{x}limx→0​x∫x0​ln(2+t2)dt​.

Problem 22829

Given the data for C(t)C(t)C(t) at t=0,10,20,40t = 0, 10, 20, 40t=0,10,20,40 days, estimate C′(15)C^{\prime}(15)C′(15). Interpret C′(15)C^{\prime}(15)C′(15). For D(t)=0.357(1.14)tD(t)=0.357(1.14)^{t}D(t)=0.357(1.14)t, find D′(15)D^{\prime}(15)D′(15).

Problem 22830

A cup of tea cools from 120°F in a room at 70°F. After 3 min it's 100°F. Find its temp after 5 min, rounded to one decimal.

Problem 22831

Find the population after 12 years of continuous growth at 4%4\%4% per year starting from 230000. Round down.

Problem 22832

Evaluate the integral ∫sin⁡xcos⁡(cos⁡x) dx\int \sin x \cos (\cos x) \, dx∫sinxcos(cosx)dx.

Problem 22833

Verify the Intermediate Value Theorem for f(x)=x2+7x+1f(x)=x^{2}+7x+1f(x)=x2+7x+1 on [0,9][0,9][0,9] and find ccc where f(c)=19f(c)=19f(c)=19.

Problem 22834

Show that the function f(x)=x2−6−cos⁡xf(x)=x^{2}-6-\cos xf(x)=x2−6−cosx has at least one zero in the interval [0,π][0, \pi][0,π].

Problem 22835

The bacteria count is given by n(t)=930e0.1tn(t)=930 e^{0.1 t}n(t)=930e0.1t. Find the growth rate, initial population, and count at t=5t=5t=5.

Problem 22836

Calculate the area between the curves y=7x2−x3+xy=7 x^{2}-x^{3}+xy=7x2−x3+x and y=x2+9xy=x^{2}+9 xy=x2+9x. The area is 4.

Problem 22837

Check if the Intermediate Value Theorem applies to f(x)=x2+7x+1f(x)=x^{2}+7x+1f(x)=x2+7x+1 on [0,9][0,9][0,9] and find ccc where f(c)=19f(c)=19f(c)=19.

Problem 22838

Find the area between the curves y=x3−12x2+35xy=x^{3}-12 x^{2}+35 xy=x3−12x2+35x and y=−x3+12x2−35xy=-x^{3}+12 x^{2}-35 xy=−x3+12x2−35x.

Problem 22839

Analyze the function f(x)=1x−8f(x)=\frac{1}{x-8}f(x)=x−81​ as xxx approaches 8 from both sides. Does f(x)f(x)f(x) go to ∞\infty∞ or −∞-\infty−∞?

Problem 22840

Analyze the function f(x)=7∣xx2−4∣f(x)=7\left|\frac{x}{x^{2}-4}\right|f(x)=7∣∣​x2−4x​∣∣​ as xxx approaches 2 from the left and right. Does it go to ∞\infty∞ or −∞-\infty−∞?

Problem 22841

Find where the tangent line is vertical and horizontal for the curve $x=1-\frac{1}{a} \cos a t, \quad y=t-\frac{1}{a} \sin a t$ ( $a>0$ ).

Find values of $p$ for which the series $\sum_{n=1}^{\infty}\left(\frac{p}{n+1} \frac{1}{n+3}\right)$ converges.

Given $x y=3$ and $\frac{d y}{d t}=2$ , find $\frac{d x}{d t}$ when $x=3$ .

Analyze the Leonard-Jones potential $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ as $r \rightarrow 0$ . What happens to $V(r)$ ?

If $p(x)$ and $q(x)$ are polynomials with $q$ monic and $\operatorname{deg}(p)<3$ , is the integral $\int \frac{p(x)}{q(x)} dx$ always, sometimes, or never equal to the given expression?

Find the point on the graph of $y=e^{-x}$ where the tangent line's slope is -4. Provide the point as an ordered pair.

Find the volume of solid $S$ with a triangular cross-section. Base bounded by $x=a^{2}-y^{2}$ , $y=\frac{6}{a} x-4 a$ , and $x$ -axis.

Find the rate of change of $y$ with respect to $x$ in the equation $2x + y = 6$ .

Find the power series for $f(x)=\frac{x^{2}}{x^{4}+16}$ and its interval of convergence.

Maximize profit $P(x, y)=24x-x^2-xy-2y^2+33y-43$ for goods $x$ and $y$ . Find optimal $x$ and $y$ values.

Express the limit $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} c_{k}^{2} \Delta x$ as a definite integral over $[0,2]$ .

Show that for $n=1$ , the function $f(P)=\frac{P^{n}}{P^{n}+30^{n}}$ has no inflection points.

Find inflection points of $f(x)=8 e^{-7 x^{2}}$ for $x \geq 0$ and provide their coordinates as ordered pairs.

Find the tangent line equation for $y=f(x)$ at $x=4$ given $f(4)=-2$ and the slope $f'(4)$ . What is the equation?

Find where the tangent line is vertical and horizontal for the curve $x=1-\frac{1}{a} \cos a t$ , $y=t-\frac{1}{a} \sin a t$ ( $a>0$ ).

Show that for $n=3$ , the function $f(P)=\frac{P^{3}}{P^{3}+30^{3}}$ is increasing and approaches 1 as $P \rightarrow \infty$ .

Hill's equation models blood oxygen saturation: $f(P)=\frac{P^{n}}{P^{n}+30^{n}}$ . For $n=1$ , show $f(P)$ is increasing and $f(P) \rightarrow 1$ as $P \rightarrow \infty$ .

Deposit \ $5000 at$ 5.5\%$ interest compounded continuously. Find:
(a) Formula for $A(t)$ , (b) differential equation, (c) balance after 6 years, (d) time to reach \$8000, (e) growth rate at \$8000.

What is the average amount of radium in a vault after 1100 years if starting with 100 grams and a half-life of 1690 years? Average: $\square$ grams.

Partition the interval $[0,4]$ into 8 equal parts. Approximate the volume $S_{8}$ of cylinders at left endpoints. Overestimate? Why?

What is the average amount of radium (in grams) left in a vault after 1000 years if starting with 200 grams and a half-life of 1690 years? There will be an average of $\square$ grams of radium in the vault during the next 1000 years.

Find the limits: a. $\lim _{x \rightarrow \frac{7}{6}^{+}} \frac{15 x}{7-6 x}=$ b. $\lim _{x \rightarrow \frac{7}{6}^{-}} \frac{15 x}{7-6 x}=$

Find the average rate of change of rainfall from day 8 to day 14 for $f(t)$ , and answer parts B, C, and D regarding $f$ .

Calculate $\int_{4}^{1} f^{\prime}(x) \, dx$ using the function values from the table provided.

Evaluate the integral $\int_{-\pi}^{\pi} f(x) dx$ where $f(x)=\begin{cases}2 x^{4}, & -\pi \leq x<0 \\ 5 \sin (x), & 0 \leq x \leq \pi\end{cases}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \ln \left(2+t^{2}\right) d t}{x}$ .

Given the data for $C(t)$ at $t = 0, 10, 20, 40$ days, estimate $C^{\prime}(15)$ . Interpret $C^{\prime}(15)$ . For $D(t)=0.357(1.14)^{t}$ , find $D^{\prime}(15)$ .

Find the population after 12 years of continuous growth at $4\%$ per year starting from 230000. Round down.

Evaluate the integral $\int \sin x \cos (\cos x) \, dx$ .

Verify the Intermediate Value Theorem for $f(x)=x^{2}+7x+1$ on $[0,9]$ and find $c$ where $f(c)=19$ .

Show that the function $f(x)=x^{2}-6-\cos x$ has at least one zero in the interval $[0, \pi]$ .

The bacteria count is given by $n(t)=930 e^{0.1 t}$ . Find the growth rate, initial population, and count at $t=5$ .

Calculate the area between the curves $y=7 x^{2}-x^{3}+x$ and $y=x^{2}+9 x$ . The area is 4.

Check if the Intermediate Value Theorem applies to $f(x)=x^{2}+7x+1$ on $[0,9]$ and find $c$ where $f(c)=19$ .

Find the area between the curves $y=x^{3}-12 x^{2}+35 x$ and $y=-x^{3}+12 x^{2}-35 x$ .

Analyze the function $f(x)=\frac{1}{x-8}$ as $x$ approaches 8 from both sides. Does $f(x)$ go to $\infty$ or $-\infty$ ?

Analyze the function $f(x)=7\left|\frac{x}{x^{2}-4}\right|$ as $x$ approaches 2 from the left and right. Does it go to $\infty$ or $-\infty$ ?

Find the one-sided limit: $\lim _{x \rightarrow \pi^{+}} \frac{\sqrt{x}}{\csc x}$ (enter DNE if it doesn't exist).

Find or estimate $f(x)$ for $x=0,2,4,6$ given $f(0)=80$ and the derivative values: $f^{\prime}(0)=5$ , $f^{\prime}(2)=17$ , $f^{\prime}(4)=29$ , $f^{\prime}(6)=38$ .

Find the general antiderivative of $h(t)=\frac{-\sin(t)}{\cos^2(t)}$ for $-\frac{\pi}{2}<t<\frac{\pi}{2}$ .

Find the one-sided limit: $\lim _{x \rightarrow 0^{-}}\left(4+\frac{3}{x}\right)$ . If it doesn't exist, state DNE.

Find the tangent line to $f(x)=x^{3}$ that is parallel to $3x-y+8=0$ . Provide both $y$ -intercepts.

Find the limit of mass $m=\frac{m_{0}}{\sqrt{1-\left(v^{2} / c^{2}\right)}}$ as $v$ approaches $c$ from the left.

Find $f(2)$ for the function with $f^{\prime \prime}(x)=6x+4\sin(x)$ , given $f(0)=2$ and $f^{\prime}(0)=2$ .

Find $\frac{d R}{d r}$ for resistors $r$ and $s$ in parallel, and determine if $R$ is increasing, decreasing, or neither. Also, find global max and min for $R$ in $a \leq r \leq b$ .

Find the function $f(x)$ and the number $c$ such that the limit equals $f^{\prime}(c)$ :
$\lim_{x \rightarrow 4} \frac{6 \sqrt{x}-12}{x-4}$
$f(x)=\square$ , $c=\square$ .

Find the value(s) of $x$ where the slope of the tangent line to $y=1+250 x^{3}-3 x^{5}$ is maximized. Answer: $x=$

Find the derivative of the integral $\frac{d}{d t} \int_{0}^{t^{10}} \sqrt{u^{7}} du$ using two methods.

Given $h(x)=x f(x)+(g(x))^{3}$ , find $h^{\prime}(3)$ using provided values of $f(x)$ and $g(x)$ .

Calculate the area between the $x$ -axis and $y=-x^{2}-3x$ for $-8 \leq x \leq 3$ .

Find the $x$ -values where the function $y=\frac{x^{2}}{x^{2}-16}$ is differentiable. Use interval notation for your answer.

Find the derivative of $f(x)=x^{3}+2x^{2}+5$ at $x=-2$ . If it doesn't exist, write UNDEFINED.

Find the function $f(x)$ given that the limit equals $f^{\prime}(c)$ where $c=4$ : $\lim _{x \rightarrow 4} \frac{6 \sqrt{x}-12}{x-4}$

Find the derivative $\frac{d y}{d x}$ for $y=\int_{0}^{\cot x} \frac{d t}{1+t^{2}}$ .

Evaluate the integral: $\frac{d}{d t} \int_{0}^{t^{10}} \sqrt{u^{7}} du=\frac{d}{d t}(\square)$

Given the function $f(x)=|x-6|$ , find the left and right derivatives at $x=6$ . Is it differentiable there? Enter Yes or No.

Given the function $f(x)=3 x^{6}-7 x^{5}$ , find critical numbers, intervals of increase/decrease, local maxima/minima, concavity, inflection points, and asymptotes.

Find the derivative of $y=\frac{9}{7 x^{4}}$ and simplify the result.

Calculate the area between the $\mathrm{x}$ -axis and $y=2x^{3}+12x^{2}+16x$ for $-4 \leq x \leq 0$ .

Find the slope of the function $y=(6x+1)^{2}$ at the point $(0,1)$ using its derivative.

Find the tangent line equation for $f(x)=-5 x^{4}+9 x^{2}-4$ at the point (1,0). $y=$

Find the derivative of $f(t)=7 t^{2/3}-5 t^{1/3}+8$ . What is $f'(t)$ ?

Find the derivative of the function $y=\frac{\pi}{3} \sin (\theta)$ . What is $y^{\prime}$ ?

Given the function $f(x)=9 x-6 \ln (x)$ for $x>0$ , find critical numbers, increasing/decreasing intervals, local maxima/minima, concavity, and inflection points.

Find the point(s) where the function $y=x^{2}+5$ has a horizontal tangent line. If none, enter NONE.

Find the absolute min and max of $f(t)=16 \cos (t)+8 \sin (2 t)$ on $[0, \frac{\pi}{2}]$ .

Calculate the average rate of change of $f(t)=6 t^{2}-1$ over $[3,3.1]$ and compare it with rates at the endpoints.

A projectile is shot upward at 133 m/s. Find its velocity after 4s and 10s using $s(t)=-4.9t^2+v_0t+s_0$ .

Given the function $f(x)=4 x^{2} \ln (x)$ for $x>0$ , find critical values, intervals of increase/decrease, local max/min, concavity, inflection points, and sketch the graph.