Calculus

Problem 11901

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

See Solution

Problem 11902

Find the derivative of $y = x^{6}+10x^{3}+25$ and $u = x^{3}+5$ using the chain rule.

See Solution

Problem 11903

Find $f^{\prime}(4)$ if $f(x)=g(x) \cdot x^{1/2}$ , $g(4)=8$ , and $g^{\prime}(4)=2$ .

See Solution

Problem 11904

Find $\frac{dy}{dx}$ for $y=x^{6}+10x^{3}+25$ using $u=x^{3}+5$ and the chain rule.

See Solution

Problem 11905

Find the velocity function of the object given $s(t)=\cos(t^{2}+8)$ for $t \geq 0$ .

See Solution

Problem 11906

Find the tangent line equation for $f(x)=\frac{x^{2}}{x+1}$ at $x=1$ . Choices: $y=0.5$ , $y=0.75 x$ , $y=0.75 x-0.25$ , $y=2 x$ , $y=0.75 x-1$ .

See Solution

Problem 11907

Find the derivative of $f(x)=\frac{x}{1+x}$ . What is it?

See Solution

Problem 11908

Find the derivative of $y=\cos^{3} x$ using the chain rule with $u=\cos x$ .

See Solution

Problem 11909

Find the derivative of $f(t)=5 \ln (t)-3 e^{-2 t}$ . What is $f^{\prime}(t) ?$

See Solution

Problem 11910

Solve the initial value problem:
$\dot{x}= 2 x + 3 y, \quad \dot{y}= -3 x + 2 y$
with $x(0)=1$ and $y(0)=-2$ .

See Solution

Problem 11911

Find the equation of the tangent line to $f(x)=(e^{3 x}+5 x^{2})^{3}$ at $x=0$ . Choose from the options.

See Solution

Problem 11912

Find the tangent line equation to $f(x)=x(x+2)^{3}$ at $x=0$ .

See Solution

Problem 11913

Solve the initial value problem: $\dot{x} = -2x + y$ , $\dot{y} = -2y$ , with $x(0)=4$ , $y(0)=-1$ .

See Solution

Problem 11914

How long for \$ 25,000 to grow to \$ 100,000 at 6.5\% continuous compounding?

See Solution

Problem 11915

Find the limit: $\lim _{x \rightarrow 5} \frac{\sqrt{2 x+6}-4}{15-3 x}$ .

See Solution

Problem 11916

Find $\frac{d y}{d x}$ for the equation $x^{3}+e^{2 y}=9$ .

See Solution

Problem 11917

Find $\frac{d y}{d x}$ for the equation $x^{3}+e^{2 y}=9$ .

See Solution

Problem 11918

Check if the function $f(x)=\left\{\begin{array}{l}\sin (\pi x) \text { for } x \leq 2 \\ \pi(x-2) \text { for } x>2\end{array}\right.$ is continuous or differentiable at $x=3$ .

See Solution

Problem 11919

Find the limit of the sequence $\left(1+\frac{1}{n}\right)^{n}$ as $n \to \infty$ .

See Solution

Problem 11920

Find the area between the curve $y=x(x-3)$ and the lines $x=0$ and $x=5$ .

See Solution

Problem 11921

Find points on the curve $x^{3}+e^{2 y}=9$ where the tangent line is horizontal.

See Solution

Problem 11922

Find points on the curve $x^{3}+e^{2 y}=9$ where the tangent line is horizontal.

See Solution

Problem 11923

Find points on the curve $x^{3}+e^{2 y}=9$ where the tangent line is vertical.

See Solution

Problem 11924

A stream with insecticide at $9 \mathrm{~g} / \mathrm{m}^{3}$ flows into a pond of volume $2000 \mathrm{~m}^{3}$ . Solve for $y(t)$ , the insecticide amount, and find the time to reach $7 \mathrm{~g} / \mathrm{m}^{3}$ .

See Solution

Problem 11925

Determine if the function $f(x)=\left\{\begin{array}{l}\sin (\pi x) \text { for } x \leq 2 \\ \pi(x-2) \text { for } x>2\end{array}\right.$ is continuous, differentiable, or both. Find points with vertical tangent lines.

See Solution

Problem 11926

A polluted stream flows into a pond. Find the insecticide concentration over time and when it reaches $7 \mathrm{~g} / \mathrm{m}^{3}$ .

See Solution

Problem 11927

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ of $x^{3}+e^{2 y}=9$ at the point $(2,0)$ .

See Solution

Problem 11928

A 15-ft ladder leans against a wall. If it slides down at 2 ft/sec, how fast is the angle $\theta$ changing when $\theta = \frac{\pi}{4}$ ?

See Solution

Problem 11929

Die Kostenfunktion $K(x) = 2x^{3}-147x^{2}+3792x+3375$ hat keine Extremstellen. Erklären Sie die Bedeutung für den Graphen.

See Solution

Problem 11930

Calculate the integral: $\int_{0}^{1} e^{2 x}+\frac{1}{2-x} d x$ .

See Solution

Problem 11931

Calculate the integral $\int_{0}^{\frac{\pi}{2}} \sin 2\left(x+\frac{\pi}{4}\right) d x$ .

See Solution

Problem 11932

A stone falls from rest. Its velocity $v$ at time $t$ satisfies $5 \frac{d v}{d t}+v=50$ . Find acceleration at $t=0$ .

See Solution

Problem 11933

Find the limit as $x$ approaches 0 for the expression $(2 x_{0}+h)$ .

See Solution

Problem 11934

Evaluate the limit: $\lim _{n \rightarrow \infty} \frac{\sqrt{n^{2}+5 n+1}-5 n-7}{2 n-6}$ .

See Solution

Problem 11935

Find the volume change rate $V$ of a ball with radius $r$ at $r=1 \mathrm{~m}$ , where $V=\frac{4}{3} \pi r^{3}$ .

See Solution

Problem 11936

Find the marginal-cost function for $C(x)=0.04 x^{3}+0.7 x^{2}+60 x+130$ and calculate it at $x=800$ .

See Solution

Problem 11937

Find the marginal-cost function for $C(x)=0.04 x^{3}+0.7 x^{2}+50 x+90$ and calculate it at $x=800$ .

See Solution

Problem 11938

Find the derivative of the following functions:
1) $f(x)=\frac{2x+1}{3x+2}$ 2) $f(x)=3x^2-\frac{1}{x}-\cos x$ 3) $f(x)=e^{\sqrt{x}}$ 4) $f(x)=\frac{\ln x}{x}$ 5) $f(x)=\frac{4}{\sqrt{x}}$ 6) $f(x)=\frac{2x^3+3x^2-4}{x^2}$

See Solution

Problem 11939

Find the marginal revenue function for $r=245 q+36 q^{2}-3 q^{3}$ and calculate it at $q=5$ , $q=10$ , and $q=15$ .

See Solution

Problem 11940

Find the marginal-cost function for $C(x)=0.05 x^{3}+0.2 x^{2}+40 x+140$ and calculate it at $x=1000$ .

See Solution

Problem 11941

Find the limit: $\lim _{x \rightarrow-1^{+}} \frac{-x^{6}+2 x^{5}+x^{4}-x^{3}+x+2}{x+1}$ .

See Solution

Problem 11942

Find the marginal cost function for $c(q)=18 q^{2}-11 q+3$ and calculate it at $q=4$ , $q=7$ , and $q=11$ .

See Solution

Problem 11943

Find the limit as $n$ approaches infinity: $\lim _{n \rightarrow \infty}\left(\frac{5 n^{4}}{2+2 n^{3}+5 n^{4}}+\frac{8 n^{3}}{5-7 n^{2}+5 n^{3}}\right)$ .

See Solution

Problem 11944

A 67 kg snowboarder starts at a 22 m hill with a speed of 15 m/s. Find: (a) total mechanical energy, (b) speed midway and bottom, (c) energy transformation.

See Solution

Problem 11945

Find the partial derivative of $3 \ln x_{1} + 3 \ln x_{2}$ with respect to $x_{1}$ .

See Solution

Problem 11946

Find the derivative of $y=6 x^{3}+15 x^{2}$ and evaluate it at $x=9.15$ , rounding to two decimal places.

See Solution

Problem 11947

Differentiate $y=4 x(2 x+4)^{3 / 2}$ and find the value at $x=2$ . Round to two decimal places.

See Solution

Problem 11948

Find the marginal profit if $q=25,000$ given $R(q)=2q-(1/25000)q^2$ and $C(q)=2100+0.25q$ .

See Solution

Problem 11949

Find the rate of change of $A$ with respect to $r$ for $A=511(1+(r / 1200))^{120}$ after 10 years. Round to two decimal places.

See Solution

Problem 11950

Find the point on the curve $f(x)=5 x^{2}+2 x-6$ where the tangent slope is -4. Provide $x$ and $y$ rounded to two decimal places.

See Solution

Problem 11951

A soccer player kicks a $0.43 \mathrm{~kg}$ ball down an $18 \mathrm{~m}$ hill. Find its speed at the bottom. Then, if kicked up at $4.2 \mathrm{~m/s}$ , find the speed when it returns down.

See Solution

Problem 11952

Find the marginal profit when $q=25,000$ for the revenue function $R(q)=2q-\frac{1}{25000}q^2$ and cost function $C(q)=2100+0.25q$ . Round to two decimal places.

See Solution

Problem 11953

Determine if increasing advertising spending from \ $12,000 ($ x=12 $) raises profit by finding the vertex of$ P(x)=1000+32x-2x^{2}$.

See Solution

Problem 11954

Calculate the heat needed to raise 1 mole of oxygen from $300 \mathrm{~K}$ to $1300 \mathrm{~K}$ at constant pressure using: $C_P=6.095+3.253 \times 10^{-3} \mathrm{~T}-1.017 \times 10^{-6} \mathrm{~T}^{2}$ .

See Solution

Problem 11955

Find the derivative $f'(x)$ for the function $f(x)=8x^{4}(x^{3}-2)$ .

See Solution

Problem 11956

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{x}{x-49}$ .

See Solution

Problem 11957

Find the derivative of the function $f(x)=\frac{x}{x-49}$ . What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 11958

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(2x+3)(5x-6)$ .

See Solution

Problem 11959

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{2 x-1}{6 x+5}$ .

See Solution

Problem 11960

Find the derivative $f^{\prime}(x)$ for the function $f(x)=9 x e^{x}$ .

See Solution

Problem 11961

Differentiate the function $y=x^{2} e^{x}$ . Find $\frac{d y}{d x}=\square$ .

See Solution

Problem 11962

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

See Solution

Problem 11963

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

See Solution

Problem 11964

A skater descends a frictionless ramp from a height of $5.0 \mathrm{~m}$ . What is his speed at the bottom? A. $25 \mathrm{~m/s}$ B. $7.0 \mathrm{~m/s}$ C. $9.9 \mathrm{~m/s}$ D. cannot be determined

See Solution

Problem 11965

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(x^{2}-6)(x^{2}+2)$ .

See Solution

Problem 11966

Find the derivative $f'(x)$ of the function $f(x)=\frac{x^{2}+6}{5x-7}$ .

See Solution

Problem 11967

Find the derivative $\frac{d y}{d t}$ for $y=(19+e^{t}) \ln t$ . What is $\frac{d y}{d t}=\square$ ?

See Solution

Problem 11968

Find the derivative $f^{\prime}(x)$ for $f(x)=\frac{\ln x}{23+x}$ . Answer: $f^{\prime}(x)=\square$ .

See Solution

Problem 11969

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{2}}$

See Solution

Problem 11970

Find $f^{\prime}(x)$ for $f(x)=x^{5}(x^{7}-6)$ using the product rule. Which option shows the correct result?

See Solution

Problem 11971

Find $f^{\prime}(x)$ using the quotient rule for $f(x)=\frac{x^{6}+4}{x^{6}}$ . Which option shows the correct result?

See Solution

Problem 11972

Find the derivative $\frac{d y}{d t}$ for $y=\frac{6 t \ln t}{e^{t}}$ . Simplify your answer.

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Problem 11973

Differentiate $e^{x}+e^{y}+4 x=0$ implicitly to prove $e^{2 y} \frac{d^{2} y}{d x^{2}}+\left(4+e^{x}\right)^{2}+e^{x+y}=0$ .

See Solution

Problem 11974

Find the derivative $S^{\prime}(t)$ of the sales function $S(t)=\frac{70 t^{2}}{t^{2}+100}$ .

See Solution

Problem 11975

Reihe konvergiert für $n \rightarrow \infty$ : a) $\sum_{i=1}^{n}-i$ ?

See Solution

Problem 11976

Find $S(20)$ and $S^{\prime}(20)$ using $S(t)=\frac{70 t^{2}}{t^{2}+100}$ . What do they indicate?

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Problem 11977

Untersuche die Konvergenz der folgenden Reihen für $n \rightarrow \infty$ : a) $\sum_{i=1}^{n}-i$ , b) $\sum_{i=0}^{n}\left(\frac{1}{2}\right)^{i}$ , c) $\sum_{i=0}^{n} 3 \cdot 2^{2 i} \cdot 5^{1-i}$ , d) $\sum_{i=0}^{n} 2 \cdot 7^{i} \cdot 5^{3-i}$ , e) $\sum_{i=1}^{n} 3^{2 i} \cdot 11^{2-i}$ .

See Solution

Problem 11978

Sand leaks from a bag modeled by $S(t)=K\left(1-\frac{t^{2}}{\mu}\right)^{3}$ .
1. (a) How much sand is in the bag at $t=0$ ? (b) What is the leak rate at time $t$ ? (c) For $\mu=6$ , is the leak rate speeding up or slowing down at $t=2$ ? Explain.
2. (a) When does the bag empty? (b) Verify $\lim _{t \rightarrow T^{-}} \frac{d S}{d t}=0$ . Does this make sense?
3. For $K=80$ and $\mu=8$ , find $\left(S^{-1}\right)^{\prime}(10)$ .
4. Laila studies $S_{1}(\mu)=K\left(1-\frac{1}{\mu}\right)^{3}$ at $t=1$ . (a) Find $\frac{d S_{1}}{d \mu}$ . (b) Is Laila correct that more sand leaks out for larger $\mu$ ? Explain.

See Solution

Problem 11979

Find the derivative of $y=x^{4}+12x$ .

See Solution

Problem 11980

Find the second derivative of $G(x)=x f\left(x^{2}\right)$ , i.e., calculate $\frac{d^{2} G(x)}{d x^{2}}$ .

See Solution

Problem 11981

Find the second derivative of $G(x) = x f(x^2)$ .

See Solution

Problem 11982

Find $F^{\prime \prime}(2)$ for $F(x)=[f(x)]^{2}$ given $f(2)=-1, f^{\prime}(2)=-2, f^{\prime \prime}(2)=3$ .

See Solution

Problem 11983

Find $x \geq 3$ to maximize profit from $P(x)=-x^{3}+15 x^{2}-48 x+450$ . Verify using a number line or second derivative test.

See Solution

Problem 11984

What is the long-term behavior of the stray cat population given by $p(t)=\frac{350 t}{2 t+7}$ as $t \to \infty$ ?

See Solution

Problem 11985

Find the absolute minimum and maximum of $f(x)=x^{4}-18 x^{2}+7$ on the interval $-2 \leq x \leq 7$ .

See Solution

Problem 11986

Find the rate of distance change between two planes: one at 36 km and the other at 16 km from the airport, flying at $163 \mathrm{~km/hr}$ and $257 \mathrm{~km/hr}$ .

See Solution

Problem 11987

Find the limit as $x$ approaches infinity for the expression $\frac{4x + 8}{x}$ .

See Solution

Problem 11988

Determine if the following sequences converge: A. $\left\{\frac{(-1)^{n}}{n}\right\}_{n=1}^{\infty}$ B. $\left\{\frac{n^{2}+1}{n}\right\}_{n=1}^{\infty}$ .

See Solution

Problem 11989

Determine the limits of these sequences: A. $\frac{(-1)^{n}}{n}$ , B. $\frac{n^{2}+1}{n}$ .

See Solution

Problem 11990

Indicate T (true) or F (false) for each statement about functions on given intervals. You need all correct for credit.
1. Every differentiable function on $[1,3]$ has a minimum.
2. Every function on $[0,2]$ has max and min.
3. Every function on $(-3,1]$ has max and min.
4. Every continuous function on $(2,5]$ has a max.

See Solution

Problem 11991

Given the function $f(x)=-2 x^{3}+33 x^{2}-144 x+5$ , find the critical points $A$ and $B$ . Determine if $f(x)$ is INC or DEC on the intervals $(-\infty, A], [A, B], [B, \infty)$ .

See Solution

Problem 11992

Given that $f(x)$ is positive and concave up on $I$ , answer these:
a.) $f^{\prime \prime}(x)>\square$ on $I$ . b.) $g^{\prime \prime}(x)=2(A^{2}+B f^{\prime \prime}(x))$ , where $A=\square$ and $B=\square$ . c.) $g^{\prime \prime}(x)>\square$ on $I$ . d.) $g(x)$ is $\square$ on $I$ .
Choose from: C U, C D, $f(x)$ , $f^{\prime \prime}(x)$ , 0, or 1.

See Solution

Problem 11993

Find the limit of the sequence $\left\{\frac{4 n}{n+1}\right\}_{n=0}^{\infty}$ .

See Solution

Problem 11994

Berechnen Sie Nullstelle, Extrem- und Wendepunkt von $f(x)=(1-x) \cdot e^{2 x}$ und untersuchen Sie $f$ für $x \rightarrow \pm \infty$ . Zeigen Sie, dass $F(x)=\left(-\frac{1}{2} x+\frac{3}{4}\right) \cdot e^{2 x}$ eine Stammfunktion ist.

See Solution

Problem 11995

Find the linearization $L(x)$ of $f(x)=x^{3}-2x+3$ at $x=2$ .

See Solution

Problem 11996

Bestimmen Sie die Ableitung der Funktion $f(x)=3x+1$ an der Stelle $x_0$ mit dem Differentialquotienten.

See Solution

Problem 11997

Find the linear approximation of $f(x)=\sqrt{1+x}$ at $x_{0}=8$ to estimate $\sqrt{8.9}$ and $\sqrt{9.1}$ .

See Solution

Problem 11998

Bestimmen Sie die Ableitung von $f(x)=3 x+1$ an der Stelle $x_{0}$ mit dem Differentialquotienten.

See Solution

Problem 11999

Find the linear approximation of $f(x)=\sqrt{1+x}$ at $x_{0}=8$ to estimate $\sqrt{9.1}$ .

See Solution

Problem 12000

Estimate $\ln(1.02)$ using a linear approximation.

See Solution

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Calculus

Problem 11901

Find the limit: lim⁡x→−∞xln⁡(1−1x)\lim _{x \rightarrow-\infty} x \ln \left(1-\frac{1}{x}\right)limx→−∞​xln(1−x1​).

Problem 11902

Find the derivative of y=x6+10x3+25y = x^{6}+10x^{3}+25y=x6+10x3+25 and u=x3+5u = x^{3}+5u=x3+5 using the chain rule.

Problem 11903

Find f′(4)f^{\prime}(4)f′(4) if f(x)=g(x)⋅x1/2f(x)=g(x) \cdot x^{1/2}f(x)=g(x)⋅x1/2, g(4)=8g(4)=8g(4)=8, and g′(4)=2g^{\prime}(4)=2g′(4)=2.

Problem 11904

Find dydx\frac{dy}{dx}dxdy​ for y=x6+10x3+25y=x^{6}+10x^{3}+25y=x6+10x3+25 using u=x3+5u=x^{3}+5u=x3+5 and the chain rule.

Problem 11905

Find the velocity function of the object given s(t)=cos⁡(t2+8)s(t)=\cos(t^{2}+8)s(t)=cos(t2+8) for t≥0t \geq 0t≥0.

Problem 11906

Find the tangent line equation for f(x)=x2x+1f(x)=\frac{x^{2}}{x+1}f(x)=x+1x2​ at x=1x=1x=1. Choices: y=0.5y=0.5y=0.5, y=0.75xy=0.75 xy=0.75x, y=0.75x−0.25y=0.75 x-0.25y=0.75x−0.25, y=2xy=2 xy=2x, y=0.75x−1y=0.75 x-1y=0.75x−1.

Problem 11907

Find the derivative of f(x)=x1+xf(x)=\frac{x}{1+x}f(x)=1+xx​. What is it?

Problem 11908

Find the derivative of y=cos⁡3xy=\cos^{3} xy=cos3x using the chain rule with u=cos⁡xu=\cos xu=cosx.

Problem 11909

Find the derivative of f(t)=5ln⁡(t)−3e−2tf(t)=5 \ln (t)-3 e^{-2 t}f(t)=5ln(t)−3e−2t. What is f′(t)?f^{\prime}(t) ?f′(t)?

Problem 11910

Solve the initial value problem: x˙=2x+3y,y˙=−3x+2y \dot{x}= 2 x + 3 y, \quad \dot{y}= -3 x + 2 y x˙=2x+3y,y˙​=−3x+2y with x(0)=1x(0)=1x(0)=1 and y(0)=−2y(0)=-2y(0)=−2.

Problem 11911

Find the equation of the tangent line to f(x)=(e3x+5x2)3f(x)=(e^{3 x}+5 x^{2})^{3}f(x)=(e3x+5x2)3 at x=0x=0x=0. Choose from the options.

Problem 11912

Find the tangent line equation to f(x)=x(x+2)3f(x)=x(x+2)^{3}f(x)=x(x+2)3 at x=0x=0x=0.

Problem 11913

Solve the initial value problem: x˙=−2x+y\dot{x} = -2x + yx˙=−2x+y, y˙=−2y\dot{y} = -2yy˙​=−2y, with x(0)=4x(0)=4x(0)=4, y(0)=−1y(0)=-1y(0)=−1.

Problem 11914

How long for \$ 25,000 to grow to \$ 100,000 at 6.5\% continuous compounding?

Problem 11915

Find the limit: lim⁡x→52x+6−415−3x\lim _{x \rightarrow 5} \frac{\sqrt{2 x+6}-4}{15-3 x}limx→5​15−3x2x+6​−4​.

Problem 11916

Find dydx\frac{d y}{d x}dxdy​ for the equation x3+e2y=9x^{3}+e^{2 y}=9x3+e2y=9.

Problem 11917

Find dydx\frac{d y}{d x}dxdy​ for the equation x3+e2y=9x^{3}+e^{2 y}=9x3+e2y=9.

Problem 11918

Check if the function f(x)={sin⁡(πx) for x≤2π(x−2) for x>2f(x)=\left\{\begin{array}{l}\sin (\pi x) \text { for } x \leq 2 \\ \pi(x-2) \text { for } x>2\end{array}\right.f(x)={sin(πx) for x≤2π(x−2) for x>2​ is continuous or differentiable at x=3x=3x=3.

Problem 11919

Find the limit of the sequence (1+1n)n\left(1+\frac{1}{n}\right)^{n}(1+n1​)n as n→∞n \to \inftyn→∞.

Problem 11920

Find the area between the curve y=x(x−3)y=x(x-3)y=x(x−3) and the lines x=0x=0x=0 and x=5x=5x=5.

Problem 11921

Find points on the curve x3+e2y=9x^{3}+e^{2 y}=9x3+e2y=9 where the tangent line is horizontal.

Problem 11922

Find points on the curve x3+e2y=9x^{3}+e^{2 y}=9x3+e2y=9 where the tangent line is horizontal.

Problem 11923

Find points on the curve x3+e2y=9x^{3}+e^{2 y}=9x3+e2y=9 where the tangent line is vertical.

Problem 11924

A stream with insecticide at 9 g/m39 \mathrm{~g} / \mathrm{m}^{3}9 g/m3 flows into a pond of volume 2000 m32000 \mathrm{~m}^{3}2000 m3. Solve for y(t)y(t)y(t), the insecticide amount, and find the time to reach 7 g/m37 \mathrm{~g} / \mathrm{m}^{3}7 g/m3.

Problem 11925

Problem 11926

A polluted stream flows into a pond. Find the insecticide concentration over time and when it reaches 7 g/m37 \mathrm{~g} / \mathrm{m}^{3}7 g/m3.

Problem 11927

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ of x3+e2y=9x^{3}+e^{2 y}=9x3+e2y=9 at the point (2,0)(2,0)(2,0).

Problem 11928

A 15-ft ladder leans against a wall. If it slides down at 2 ft/sec, how fast is the angle θ\thetaθ changing when θ=π4\theta = \frac{\pi}{4}θ=4π​?

Problem 11929

Die Kostenfunktion K(x)=2x3−147x2+3792x+3375K(x) = 2x^{3}-147x^{2}+3792x+3375K(x)=2x3−147x2+3792x+3375 hat keine Extremstellen. Erklären Sie die Bedeutung für den Graphen.

Problem 11930

Calculate the integral: ∫01e2x+12−xdx\int_{0}^{1} e^{2 x}+\frac{1}{2-x} d x∫01​e2x+2−x1​dx.

Problem 11931

Calculate the integral ∫0π2sin⁡2(x+π4)dx\int_{0}^{\frac{\pi}{2}} \sin 2\left(x+\frac{\pi}{4}\right) d x∫02π​​sin2(x+4π​)dx.

Problem 11932

A stone falls from rest. Its velocity vvv at time ttt satisfies 5dvdt+v=505 \frac{d v}{d t}+v=505dtdv​+v=50. Find acceleration at t=0t=0t=0.

Problem 11933

Find the limit as xxx approaches 0 for the expression (2x0+h)(2 x_{0}+h)(2x0​+h).

Problem 11934

Evaluate the limit: lim⁡n→∞n2+5n+1−5n−72n−6\lim _{n \rightarrow \infty} \frac{\sqrt{n^{2}+5 n+1}-5 n-7}{2 n-6}limn→∞​2n−6n2+5n+1​−5n−7​.

Problem 11935

Find the volume change rate VVV of a ball with radius rrr at r=1 mr=1 \mathrm{~m}r=1 m, where V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3.

Problem 11936

Find the marginal-cost function for C(x)=0.04x3+0.7x2+60x+130C(x)=0.04 x^{3}+0.7 x^{2}+60 x+130C(x)=0.04x3+0.7x2+60x+130 and calculate it at x=800x=800x=800.

Problem 11937

Find the marginal-cost function for C(x)=0.04x3+0.7x2+50x+90C(x)=0.04 x^{3}+0.7 x^{2}+50 x+90C(x)=0.04x3+0.7x2+50x+90 and calculate it at x=800x=800x=800.

Problem 11938

Problem 11939

Find the marginal revenue function for r=245q+36q2−3q3r=245 q+36 q^{2}-3 q^{3}r=245q+36q2−3q3 and calculate it at q=5q=5q=5, q=10q=10q=10, and q=15q=15q=15.

Problem 11940

Find the marginal-cost function for C(x)=0.05x3+0.2x2+40x+140C(x)=0.05 x^{3}+0.2 x^{2}+40 x+140C(x)=0.05x3+0.2x2+40x+140 and calculate it at x=1000x=1000x=1000.

Problem 11941

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

Find the derivative of $y = x^{6}+10x^{3}+25$ and $u = x^{3}+5$ using the chain rule.

Find $f^{\prime}(4)$ if $f(x)=g(x) \cdot x^{1/2}$ , $g(4)=8$ , and $g^{\prime}(4)=2$ .

Find $\frac{dy}{dx}$ for $y=x^{6}+10x^{3}+25$ using $u=x^{3}+5$ and the chain rule.

Find the velocity function of the object given $s(t)=\cos(t^{2}+8)$ for $t \geq 0$ .

Find the tangent line equation for $f(x)=\frac{x^{2}}{x+1}$ at $x=1$ . Choices: $y=0.5$ , $y=0.75 x$ , $y=0.75 x-0.25$ , $y=2 x$ , $y=0.75 x-1$ .

Find the derivative of $f(x)=\frac{x}{1+x}$ . What is it?

Find the derivative of $y=\cos^{3} x$ using the chain rule with $u=\cos x$ .

Find the derivative of $f(t)=5 \ln (t)-3 e^{-2 t}$ . What is $f^{\prime}(t) ?$

Solve the initial value problem:
$\dot{x}= 2 x + 3 y, \quad \dot{y}= -3 x + 2 y$
with $x(0)=1$ and $y(0)=-2$ .

Find the equation of the tangent line to $f(x)=(e^{3 x}+5 x^{2})^{3}$ at $x=0$ . Choose from the options.

Find the tangent line equation to $f(x)=x(x+2)^{3}$ at $x=0$ .

Solve the initial value problem: $\dot{x} = -2x + y$ , $\dot{y} = -2y$ , with $x(0)=4$ , $y(0)=-1$ .

Find the limit: $\lim _{x \rightarrow 5} \frac{\sqrt{2 x+6}-4}{15-3 x}$ .

Find $\frac{d y}{d x}$ for the equation $x^{3}+e^{2 y}=9$ .

Find $\frac{d y}{d x}$ for the equation $x^{3}+e^{2 y}=9$ .

Check if the function $f(x)=\left\{\begin{array}{l}\sin (\pi x) \text { for } x \leq 2 \\ \pi(x-2) \text { for } x>2\end{array}\right.$ is continuous or differentiable at $x=3$ .

Find the limit of the sequence $\left(1+\frac{1}{n}\right)^{n}$ as $n \to \infty$ .

Find the area between the curve $y=x(x-3)$ and the lines $x=0$ and $x=5$ .

Find points on the curve $x^{3}+e^{2 y}=9$ where the tangent line is horizontal.

Find points on the curve $x^{3}+e^{2 y}=9$ where the tangent line is horizontal.

Find points on the curve $x^{3}+e^{2 y}=9$ where the tangent line is vertical.

A stream with insecticide at $9 \mathrm{~g} / \mathrm{m}^{3}$ flows into a pond of volume $2000 \mathrm{~m}^{3}$ . Solve for $y(t)$ , the insecticide amount, and find the time to reach $7 \mathrm{~g} / \mathrm{m}^{3}$ .

A polluted stream flows into a pond. Find the insecticide concentration over time and when it reaches $7 \mathrm{~g} / \mathrm{m}^{3}$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ of $x^{3}+e^{2 y}=9$ at the point $(2,0)$ .

A 15-ft ladder leans against a wall. If it slides down at 2 ft/sec, how fast is the angle $\theta$ changing when $\theta = \frac{\pi}{4}$ ?

Die Kostenfunktion $K(x) = 2x^{3}-147x^{2}+3792x+3375$ hat keine Extremstellen. Erklären Sie die Bedeutung für den Graphen.

Calculate the integral: $\int_{0}^{1} e^{2 x}+\frac{1}{2-x} d x$ .

Calculate the integral $\int_{0}^{\frac{\pi}{2}} \sin 2\left(x+\frac{\pi}{4}\right) d x$ .

A stone falls from rest. Its velocity $v$ at time $t$ satisfies $5 \frac{d v}{d t}+v=50$ . Find acceleration at $t=0$ .

Find the limit as $x$ approaches 0 for the expression $(2 x_{0}+h)$ .

Evaluate the limit: $\lim _{n \rightarrow \infty} \frac{\sqrt{n^{2}+5 n+1}-5 n-7}{2 n-6}$ .

Find the volume change rate $V$ of a ball with radius $r$ at $r=1 \mathrm{~m}$ , where $V=\frac{4}{3} \pi r^{3}$ .

Find the marginal-cost function for $C(x)=0.04 x^{3}+0.7 x^{2}+60 x+130$ and calculate it at $x=800$ .

Find the marginal-cost function for $C(x)=0.04 x^{3}+0.7 x^{2}+50 x+90$ and calculate it at $x=800$ .

Find the marginal revenue function for $r=245 q+36 q^{2}-3 q^{3}$ and calculate it at $q=5$ , $q=10$ , and $q=15$ .

Find the marginal-cost function for $C(x)=0.05 x^{3}+0.2 x^{2}+40 x+140$ and calculate it at $x=1000$ .

Find the limit: $\lim _{x \rightarrow-1^{+}} \frac{-x^{6}+2 x^{5}+x^{4}-x^{3}+x+2}{x+1}$ .

Find the marginal cost function for $c(q)=18 q^{2}-11 q+3$ and calculate it at $q=4$ , $q=7$ , and $q=11$ .

Find the limit as $n$ approaches infinity: $\lim _{n \rightarrow \infty}\left(\frac{5 n^{4}}{2+2 n^{3}+5 n^{4}}+\frac{8 n^{3}}{5-7 n^{2}+5 n^{3}}\right)$ .

Find the partial derivative of $3 \ln x_{1} + 3 \ln x_{2}$ with respect to $x_{1}$ .

Find the derivative of $y=6 x^{3}+15 x^{2}$ and evaluate it at $x=9.15$ , rounding to two decimal places.

Differentiate $y=4 x(2 x+4)^{3 / 2}$ and find the value at $x=2$ . Round to two decimal places.

Find the marginal profit if $q=25,000$ given $R(q)=2q-(1/25000)q^2$ and $C(q)=2100+0.25q$ .

Find the rate of change of $A$ with respect to $r$ for $A=511(1+(r / 1200))^{120}$ after 10 years. Round to two decimal places.

Find the point on the curve $f(x)=5 x^{2}+2 x-6$ where the tangent slope is -4. Provide $x$ and $y$ rounded to two decimal places.

A soccer player kicks a $0.43 \mathrm{~kg}$ ball down an $18 \mathrm{~m}$ hill. Find its speed at the bottom. Then, if kicked up at $4.2 \mathrm{~m/s}$ , find the speed when it returns down.

Find the marginal profit when $q=25,000$ for the revenue function $R(q)=2q-\frac{1}{25000}q^2$ and cost function $C(q)=2100+0.25q$ . Round to two decimal places.

Determine if increasing advertising spending from \ $12,000 ($ x=12 $) raises profit by finding the vertex of$ P(x)=1000+32x-2x^{2}$.

Find the derivative $f'(x)$ for the function $f(x)=8x^{4}(x^{3}-2)$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{x}{x-49}$ .

Find the derivative of the function $f(x)=\frac{x}{x-49}$ . What is $f^{\prime}(x)=\square$ ?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(2x+3)(5x-6)$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{2 x-1}{6 x+5}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=9 x e^{x}$ .

Differentiate the function $y=x^{2} e^{x}$ . Find $\frac{d y}{d x}=\square$ .

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

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

A skater descends a frictionless ramp from a height of $5.0 \mathrm{~m}$ . What is his speed at the bottom? A. $25 \mathrm{~m/s}$ B. $7.0 \mathrm{~m/s}$ C. $9.9 \mathrm{~m/s}$ D. cannot be determined

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(x^{2}-6)(x^{2}+2)$ .

Find the derivative $f'(x)$ of the function $f(x)=\frac{x^{2}+6}{5x-7}$ .

Find the derivative $\frac{d y}{d t}$ for $y=(19+e^{t}) \ln t$ . What is $\frac{d y}{d t}=\square$ ?

Find the derivative $f^{\prime}(x)$ for $f(x)=\frac{\ln x}{23+x}$ . Answer: $f^{\prime}(x)=\square$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{2}}$

Find $f^{\prime}(x)$ for $f(x)=x^{5}(x^{7}-6)$ using the product rule. Which option shows the correct result?

Find $f^{\prime}(x)$ using the quotient rule for $f(x)=\frac{x^{6}+4}{x^{6}}$ . Which option shows the correct result?

Find the derivative $\frac{d y}{d t}$ for $y=\frac{6 t \ln t}{e^{t}}$ . Simplify your answer.

Differentiate $e^{x}+e^{y}+4 x=0$ implicitly to prove $e^{2 y} \frac{d^{2} y}{d x^{2}}+\left(4+e^{x}\right)^{2}+e^{x+y}=0$ .

Find the derivative $S^{\prime}(t)$ of the sales function $S(t)=\frac{70 t^{2}}{t^{2}+100}$ .