Calculus

Problem 4701

Find $f^{\prime}(-1)$ for the piecewise function: $f(x) = -3x + 5$ (if $x < -1$ ) and $f(x) = -x^2 + 3$ (if $x \geq -1$ ). Options: A. -3 B. -2 C. 2 D. 3 E. nonexistent.

See Solution

Problem 4702

Find the limit as $x$ approaches infinity for the expression $\frac{1}{x}$ .

See Solution

Problem 4703

Gegeben ist die Funktion $f(x)=-2 x^{2} \cdot x^{x}$ für $x \in[-7 ; 0]$ .
1. a) Berechne $f(-7)$ und erkläre die Bedeutung.
b) Finde den tiefsten Punkt des Flussbettes.
c) Prüfe, ob der Winkel von $45^{\circ}$ überschritten wird, ohne die Ränder zu betrachten.

See Solution

Problem 4704

Bestimme die Extremstellen der Funktionen: a) $f(x)=x^{3}-x^{2}+1$ , b) $f(x)=\frac{1}{4} x^{4}$ , c) $f(x)=\frac{1}{4} x^{4}-2 x^{2}-2$ , d) $f(x)=\frac{4}{5} x^{5}-2 x^{4}$ , e) $f(x)=\sin (x)$ für $x \in[0.2 \pi]$ .

See Solution

Problem 4705

Find $h'(1)$ for $h(x)=f(g(x))$ given $f(1)=2$ , $f'(1)=3$ , $f'(2)=-4$ , $g(1)=2$ , $g'(1)=-3$ , $g'(2)=5$ .

See Solution

Problem 4706

Find the tangent line equation for $g(x)=x \cdot f(x)$ at $x=2$ given $f(2)=6$ and $f'$ has horizontal tangents at $x=1,3$ .

See Solution

Problem 4707

Find the limit: $\lim _{x \rightarrow-\infty} f(x)$ for the function $f(x)=2 x e^{2 x}$ .

See Solution

Problem 4708

Find $\lim_{x \rightarrow -\infty} f(x)$ for the function $f(x) = 2x e^{2x}$ .

See Solution

Problem 4709

Find the limit: $\lim _{x \rightarrow-\infty} f(x)$ where $f(x)=2 x e^{2 x}$ .

See Solution

Problem 4710

Find the curve equation through $(0,2)$ with slope $m = \frac{x}{y}$ .

See Solution

Problem 4711

Solve the differential equation: $\frac{d z}{d t}+e^{t+z}=0$ .

See Solution

Problem 4712

Solve $2 y' = x + y$ using the substitution $u = x + y$ .

See Solution

Problem 4713

Solve $\frac{d y}{d x}=y \tan x$ , with $y=1$ at $x=0$ . Also, show $\frac{d N}{d t}=-\frac{k}{\sqrt{N}}$ for cockroaches.

See Solution

Problem 4714

Bestimmen Sie die Werte von $t \in \mathbb{R}$ , für die $f_{t}(x)=\frac{1}{2} x^{3}+t x^{2}+6 x-2$ a) keine und b) genau einen waagerechten Tangentenpunkt hat. Zeigen Sie, dass dieser Punkt kein Extrempunkt ist.

See Solution

Problem 4715

Find the oscillation rate of the spring at $t=5$ s for $s(t)=3 \sin t$ . Round to four decimal places.

See Solution

Problem 4716

Find the first three derivatives of $f(x)=2x-3+e^{-x}$ and identify its local extrema. Show your work.

See Solution

Problem 4717

Find $x$ in $f(x)=5x-8\cos x$ for $0<x<2\pi$ where the tangent line slope is 9. $x=$

See Solution

Problem 4718

Find the 100th, 101st, and 102nd derivatives of the function $f(x) = \sum_{n=0}^{\infty}(x-2)^{4n}$ at $x=2$ .

See Solution

Problem 4719

A patient receives 50 mg of medicine that decreases by $10\%$ each hour. How much remains after 6 hours using $A(t)=l e^{r t}$ ?

See Solution

Problem 4720

Find the first and second derivatives of $f(x)=(x+1) \cdot e^{-0.5 x}$ .

See Solution

Problem 4721

Determine the horizontal asymptotes of the function $f(x) = \frac{(3x-7)(x+2)}{3(x-10)(x+2)}$ .

See Solution

Problem 4722

Determine the horizontal asymptotes for the function $f(x)=\frac{2 x+2}{3 x^{2}-x-4}$ .

See Solution

Problem 4723

Determine the horizontal asymptotes of the function $f(x)=\frac{(2 x+1)(x-2)}{x(2 x+1)}$ .

See Solution

Problem 4724

Determine the horizontal asymptotes of the function $f(x)=\frac{2 x-6}{5 x^{2}-16 x+3}$ .

See Solution

Problem 4725

Determine the horizontal asymptotes of the function $f(x)=\frac{2(x-8)}{(2 x-3)(x-8)}$ .

See Solution

Problem 4726

Determine the horizontal asymptotes of the function $f(x)=\frac{2 x^{2}-8}{2 x^{2}+x-45}$ .

See Solution

Problem 4727

Evaluate the integral $\iiint_{E} x y \, dV$ for the tetrahedron with vertices $(0,0,0),(4,0,0),(0,1,0),(0,0,7)$ .

See Solution

Problem 4728

Determine the horizontal asymptotes for the function $f(x)=\frac{3(x+4)(x-4)}{x(2 x-1)}$ .

See Solution

Problem 4729

Determine the horizontal asymptotes of the function $f(x)=\frac{3 x-2}{2 x^{2}+2 x-112}$ .

See Solution

Problem 4730

Find the first and second derivatives of $f(x)=x \cdot e^{-0.5 x}$ .

See Solution

Problem 4731

Evaluate the integral $\int \frac{y^{2} d y}{\left(y^{2}+16\right)^{3 / 2}}$ .

See Solution

Problem 4732

Differentiate $g(n)=\frac{a n^{2}+c n}{a}-(a n-c)^{6}$ with respect to $n$ . Result: $2 a n+c$ .

See Solution

Problem 4733

Differentiate $g(x)=\frac{x}{e}+e^{2} x+e x^{-1}+e \sqrt{3 x-4}$ .

See Solution

Problem 4734

Find the $x$ -coordinates where the graph of $f(x)=8 x^{6}-3 x^{4}+17$ has horizontal tangents. $\frac{d f(x)}{d x}=48 x^{5}-12 x^{3}$

See Solution

Problem 4735

Differentiate $g(n)=\frac{a n^{2}+c n}{a}-(a n-c)^{6}$ with respect to $n$ .

See Solution

Problem 4736

Zeichne den Graphen von $h(t)=\frac{1}{8} t^{2}-2 t+8$ . Bestimme, wann der Behälter leer und halb voll ist und die Sinkgeschwindigkeit.

See Solution

Problem 4737

Find $y^{\prime}$ from $x^{3}+y^{2}=43$ at $x=3$ , $y=4$ . Provide the exact answer (fractions, not decimals).

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Problem 4738

Bestimme die Punkte, wo $f(x)=2$ und berechne die Steigung. Vergleiche die mittlere Steigung für $1,5 \leq x \leq 2,5$ mit der Steigung bei $x=2$ .

See Solution

Problem 4739

Find the derivative of $f(t)=\frac{2}{4-t}$ at $t=-5$ using limits, then find the tangent line equation at that point.

See Solution

Problem 4740

Schreibe die Funktionen ohne Brüche und analysiere das Gelände-Modell $f(x)=\frac{1}{8} x \cdot \sin \left(\frac{\pi}{2} x\right)+2$ für $0 \leq x \leq 5$ . Bestimme die Punkte, die 2 km über NHN liegen, und berechne die Steigung dort. Vergleiche die mittlere Steigung für $1,5 \leq x \leq 2,5$ mit der Steigung bei $x=2$ .

See Solution

Problem 4741

Bestimmen Sie die ersten beiden Ableitungen für die Funktionen: a) $f(x)=x^{2} \cdot g(x)$ b) $f(x)=x \cdot g^{\prime}(x)$ c) $f(x)=x \cdot(g(x))^{2}$ d) $f(x)=g(x) \cdot g^{\prime}(x)$

See Solution

Problem 4742

Find the function $f(x)$ if $f'(x)=x e^{-4x}-4e e^{-4x}$ . Factor out $e^{-4x}$ .

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Problem 4743

Untersuchen Sie, ob der Graph von $f(x)=x \cdot (g(x))^{2}$ den Graphen von $g$ in $P(1,1)$ berührt, wo die Steigung -1 ist.

See Solution

Problem 4744

Bestimme die Ableitungsfunktion von $f$ und die Tangentensteigung an $x=-1$ . Funktionen: a) $f(x)=x^{5}$ , b) $f(x)=x^{4}$ , c) $f(x)=x^{3}$ , d) $f(x)=x^{6}$ , e) $f(x)=x^{9}$ , f) $f(x)=x^{11}$ , g) $f(x)=x^{12}$ , h) $f(x)=x^{25}$ , i) $f(x)=x^{2}$ .

See Solution

Problem 4745

Determine the horizontal asymptotes of the function $f(x)=\frac{2(3 x+8)(3 x-8)}{3(x+2)(x-2)}$ .

See Solution

Problem 4746

Determine the horizontal asymptotes of the function $f(x)=\frac{2 x(5 x+4)}{(x+3)(x-3)}$ .

See Solution

Problem 4747

Determine the horizontal asymptotes of the function $f(x)=\frac{3 x^{2}+51 x+210}{2 x+20}$ .

See Solution

Problem 4748

Find the 6th derivative of $f(x)=\cos(\ln(1+x^{2}))$ at $x=0$ using the Taylor series.

See Solution

Problem 4749

Determine the horizontal asymptotes of the function $f(x)=\frac{x^{2}-12 x+20}{2 x-4}$ .

See Solution

Problem 4750

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

See Solution

Problem 4751

Berechne den Flächeninhalt zwischen dem Graphen von $f$ und der $x$ -Achse für die Intervalle: a) $f(x)=\frac{1}{6} x^{3}-\frac{1}{2} x^{2}$ , $I=[-1 ; 2]$ und $I=[-3 ; 2]$ b) $f(x)=x^{3}-4 x$ c) $f(x)=\frac{1}{4}(x+3)(x-1)(x-2)$ , $I=[-3 ; 2]$ d) $f(x)=\frac{2}{x^{2}}$ , $I=[1 ; 3]$

See Solution

Problem 4752

Determine the horizontal asymptotes of the function $f(x)=\frac{2 x(x-8)}{3(x+9)(x-7)}$ .

See Solution

Problem 4753

Determine the horizontal asymptotes for the function $f(x)=\frac{x-8}{x^{2}-5 x-24}$ .

See Solution

Problem 4754

Berechne das Integral $\int \frac{1}{1-4 x} \cdot d x$ durch Substitution.

See Solution

Problem 4755

Berechne das unbestimmte Integral $\int 3 x \cdot \sin (4 x) \, dx$ .

See Solution

Problem 4756

Berechne das unbestimmte Integral $\int 3 x \cdot \ln (4 x) \, dx$ .

See Solution

Problem 4757

Find the slope of the tangent line to $f(x)=x^{3}-6x+3$ at the point $(0,3)$ .

See Solution

Problem 4758

Find the first and second derivatives of $f(x)=(x^{2}-x) \cdot e^{-0.5 x}$ and simplify by factoring common terms.

See Solution

Problem 4759

Find the derivative $\frac{d h}{d x}$ for the function $h(x)=c \pi$ , where $c$ is a constant.

See Solution

Problem 4760

Find the derivative $\frac{d y}{d x}$ for the function $y=\sin^{-1}\left(\frac{1}{2 x+5}\right)$ .

See Solution

Problem 4761

Find the derivative of $f(x)=x \cdot e^{-0.5 x}$ and simplify by factoring out common terms.

See Solution

Problem 4762

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $-x^{2}-8 y^{2}=3$ .

See Solution

Problem 4763

Find the derivative $\frac{d y}{d x}$ for the function $y=\sin ^{-1}\left(\frac{1}{2 x+5}\right)$ .

See Solution

Problem 4764

Find the first three derivatives of $f(x)=(1-x) \cdot e^{-x}$ and simplify by factoring out common terms.

See Solution

Problem 4765

How long (in hours) to metabolize half the caffeine? Use the equation $\frac{d C}{d t}=-0.14 C$ . Round to nearest integer.

See Solution

Problem 4766

Graph the function $\frac{1}{x^{5}-2 x^{3}-3 x^{2}+6}$ on given intervals and estimate vertical asymptotes.

See Solution

Problem 4767

Find the derivative of the function $s=8 t^{9/4}+3 e^{5}$ .

See Solution

Problem 4768

Find the derivative of $r$ with respect to $s$ for $r=\frac{1}{4 s^{3}}-\frac{5}{3 s^{2}}$ .

See Solution

Problem 4769

Find the first derivative of $r$ with respect to $s$ where $r=\frac{1}{2 s}-\frac{7}{3 s^{2}}$ .

See Solution

Problem 4770

Find the first and second derivatives of $f(x)=(x^{2}-1) \cdot e^{-x}$ and simplify by factoring common terms.

See Solution

Problem 4771

Find the derivative of $r$ with respect to $s$ for $r=\frac{1}{4 s^{3}}-\frac{5}{3 s^{2}}$ .

See Solution

Problem 4772

Find the derivative of $f(x)=(x+1) \cdot e^{-0.5 x}$ using the product rule.

See Solution

Problem 4773

Find the tangent line equation for $f(x)$ at $x=1$ where $f(x)=x^{4}-2 x^{3}-5 x-4$ .

See Solution

Problem 4774

Find the tangent line to $f(x)=\ln (x)$ at $x=e^{-1}$ : determine $m$ and $b$ for $y=m x+b$ .

See Solution

Problem 4775

Find the value of $c$ where the black line is tangent to $y=1 \cdot 5^{x}$ at the point $(0,1)$ .

See Solution

Problem 4776

Calculate the grams of carbon-14 left after 5944 years using the model $A=16 e^{-0.000121 t}$ . Round to the nearest whole number.

See Solution

Problem 4777

Find the tangent line equation for $y=2^{x}$ at the point $(2,4)$ . Tangent line: $y=$

See Solution

Problem 4778

Find the rate of increase of the rocking chair's value given by $V=65(1.65)^{t}$ at time $t$ (years since 1975).

See Solution

Problem 4779

Differentiate the function: $y=4x-6 \cdot 7^{x}$ .

See Solution

Problem 4780

Show that $\int f(x) d x=\frac{-1}{2} \cos ^{2}(x)+C$ for $f(x)=\sin (x) \cos (x)$ using differentiation.

See Solution

Problem 4781

Find the population change rate in Slim Chance on January 1, 1993, using $P=27000(0.94)^{t}$ .

See Solution

Problem 4782

Show that $\int f(x) dx = \frac{1}{2} \sin^2(x) + C$ for $f(x) = \sin(x) \cos(x)$ using differentiation.

See Solution

Problem 4783

Calculate the derivative $\frac{d}{d x} \int_{\sin (x)}^{\tan (x)} e^{t^{2}} d t$ for $x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$ .

See Solution

Problem 4784

Given $f(x)=\sin (x) \cos (x)$ , prove: a) $\int f(x) dx=\frac{1}{2} \sin ^{2}(x)+C$ ; b) $\int f(x) dx=\frac{-1}{2} \cos ^{2}(x)+C$ .

See Solution

Problem 4785

Find the tangent line equation to $y=f(x)$ at $x=1$ for $f(x)=-4 x^{3}-6 x+3$ .

See Solution

Problem 4786

Find the tangent line equation at $x=9$ for the function $y=6 \sqrt{x}$ .

See Solution

Problem 4787

Find the tangent line equation for $f(t)= \frac{2}{4-t}$ at $t=-5$ .

See Solution

Problem 4788

Find the normal line to $f(x)=a x^{2}-7 a x$ at $x=9$ , where $a$ is a positive constant.

See Solution

Problem 4789

Given $f(x)=\sin (x) \cos (x)$ , show both integrals yield the same result plus a constant.

See Solution

Problem 4790

Find the derivative of $f(t)=\frac{2}{4-t}$ at $t=-5$ using the limit definition.

See Solution

Problem 4791

Find the value(s) of $x$ where the tangent line to $f(x)=2 x^{2}-5 x$ is parallel to $-6 x+2 y=2$ .

See Solution

Problem 4792

Find points on the graph of $y=f(x)$ with horizontal tangents for $f(x)=-5 x^{4}+x^{3}$ .

See Solution

Problem 4793

Find the difference quotient $\frac{f(x)-f(3)}{x-3}$ for $f(x)=5+5x-5x^{2}$ and simplify it as $x \rightarrow 3$ .

See Solution

Problem 4794

Find the derivative of the function $k(z)=4z+13$ at $z=a$ using the limit definition. What is $k^{\prime}(a)=?$

See Solution

Problem 4795

Find the tangent line equation at $(3, 11)$ with slope $3$ .

See Solution

Problem 4796

Estimate the derivative of $f(x) = 2^x$ at points 0, 1, 2, and 3 using small increment $h=0.000001$ .
a. Find $f^{\prime}(0)$ to four decimal places. $f^{\prime}(0) \approx \text{?}$
b. Find $f^{\prime}(1)$ , $f^{\prime}(2)$ , and $f^{\prime}(3)$ to four decimal places. $f^{\prime}(1) \approx \text{?}$ $f^{\prime}(2) \approx \text{?}$ $f^{\prime}(3) \approx \text{?}$

See Solution

Problem 4797

Set up a definite integral for the volume of the solid formed by rotating the area bounded by $x=2 \sqrt{y}, x=0, y=9$ around $x=7$ .

See Solution

Problem 4798

Find the derivative $f^{\prime}(1)$ of the piecewise function $f(x) = 5 - 2x^{2}$ for $x > 1$ and $f(x) = 7 - 4x$ for $x \leq 1$ . If not differentiable at $c=1$ , enter DNE.

See Solution

Problem 4799

Find the tangent line equation for $f(x)=\frac{5}{9+x^{2}}$ at $x=3$ . Use $y=f(x)$ for the equation.

See Solution

Problem 4800

Find the instantaneous rate of change of the volume $v=\frac{15+5 \sqrt{5}}{12} x^{3}$ of an icosahedron at $x=a$ . $v^{\prime}(a)=$

See Solution

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Calculus

Problem 4701

Find f′(−1)f^{\prime}(-1)f′(−1) for the piecewise function: f(x)=−3x+5f(x) = -3x + 5f(x)=−3x+5 (if x<−1x < -1x<−1) and f(x)=−x2+3f(x) = -x^2 + 3f(x)=−x2+3 (if x≥−1x \geq -1x≥−1). Options: A. -3 B. -2 C. 2 D. 3 E. nonexistent.

Problem 4702

Find the limit as xxx approaches infinity for the expression 1x\frac{1}{x}x1​.

Problem 4703

Problem 4704

Problem 4705

Find h′(1)h'(1)h′(1) for h(x)=f(g(x))h(x)=f(g(x))h(x)=f(g(x)) given f(1)=2f(1)=2f(1)=2, f′(1)=3f'(1)=3f′(1)=3, f′(2)=−4f'(2)=-4f′(2)=−4, g(1)=2g(1)=2g(1)=2, g′(1)=−3g'(1)=-3g′(1)=−3, g′(2)=5g'(2)=5g′(2)=5.

Problem 4706

Find the tangent line equation for g(x)=x⋅f(x)g(x)=x \cdot f(x)g(x)=x⋅f(x) at x=2x=2x=2 given f(2)=6f(2)=6f(2)=6 and f′f'f′ has horizontal tangents at x=1,3x=1,3x=1,3.

Problem 4707

Find the limit: lim⁡x→−∞f(x)\lim _{x \rightarrow-\infty} f(x)limx→−∞​f(x) for the function f(x)=2xe2xf(x)=2 x e^{2 x}f(x)=2xe2x.

Problem 4708

Find lim⁡x→−∞f(x)\lim_{x \rightarrow -\infty} f(x)limx→−∞​f(x) for the function f(x)=2xe2xf(x) = 2x e^{2x}f(x)=2xe2x.

Problem 4709

Find the limit: lim⁡x→−∞f(x)\lim _{x \rightarrow-\infty} f(x)limx→−∞​f(x) where f(x)=2xe2xf(x)=2 x e^{2 x}f(x)=2xe2x.

Problem 4710

Find the curve equation through (0,2)(0,2)(0,2) with slope m=xym = \frac{x}{y}m=yx​.

Problem 4711

Solve the differential equation: dzdt+et+z=0\frac{d z}{d t}+e^{t+z}=0dtdz​+et+z=0.

Problem 4712

Solve 2y′=x+y2 y' = x + y2y′=x+y using the substitution u=x+yu = x + yu=x+y.

Problem 4713

Solve dydx=ytan⁡x\frac{d y}{d x}=y \tan xdxdy​=ytanx, with y=1y=1y=1 at x=0x=0x=0. Also, show dNdt=−kN\frac{d N}{d t}=-\frac{k}{\sqrt{N}}dtdN​=−N​k​ for cockroaches.

Problem 4714

Bestimmen Sie die Werte von t∈Rt \in \mathbb{R}t∈R, für die ft(x)=12x3+tx2+6x−2f_{t}(x)=\frac{1}{2} x^{3}+t x^{2}+6 x-2ft​(x)=21​x3+tx2+6x−2 a) keine und b) genau einen waagerechten Tangentenpunkt hat. Zeigen Sie, dass dieser Punkt kein Extrempunkt ist.

Problem 4715

Find the oscillation rate of the spring at t=5t=5t=5 s for s(t)=3sin⁡ts(t)=3 \sin ts(t)=3sint. Round to four decimal places.

Problem 4716

Find the first three derivatives of f(x)=2x−3+e−xf(x)=2x-3+e^{-x}f(x)=2x−3+e−x and identify its local extrema. Show your work.

Problem 4717

Find xxx in f(x)=5x−8cos⁡xf(x)=5x-8\cos xf(x)=5x−8cosx for 0<x<2π0<x<2\pi0<x<2π where the tangent line slope is 9. x=x=x=

Problem 4718

Find the 100th, 101st, and 102nd derivatives of the function f(x)=∑n=0∞(x−2)4nf(x) = \sum_{n=0}^{\infty}(x-2)^{4n}f(x)=∑n=0∞​(x−2)4n at x=2x=2x=2.

Problem 4719

A patient receives 50 mg of medicine that decreases by 10%10\%10% each hour. How much remains after 6 hours using A(t)=lertA(t)=l e^{r t}A(t)=lert?

Problem 4720

Find the first and second derivatives of f(x)=(x+1)⋅e−0.5xf(x)=(x+1) \cdot e^{-0.5 x}f(x)=(x+1)⋅e−0.5x.

Problem 4721

Determine the horizontal asymptotes of the function f(x)=(3x−7)(x+2)3(x−10)(x+2)f(x) = \frac{(3x-7)(x+2)}{3(x-10)(x+2)}f(x)=3(x−10)(x+2)(3x−7)(x+2)​.

Problem 4722

Determine the horizontal asymptotes for the function f(x)=2x+23x2−x−4f(x)=\frac{2 x+2}{3 x^{2}-x-4}f(x)=3x2−x−42x+2​.

Problem 4723

Determine the horizontal asymptotes of the function f(x)=(2x+1)(x−2)x(2x+1)f(x)=\frac{(2 x+1)(x-2)}{x(2 x+1)}f(x)=x(2x+1)(2x+1)(x−2)​.

Problem 4724

Determine the horizontal asymptotes of the function f(x)=2x−65x2−16x+3f(x)=\frac{2 x-6}{5 x^{2}-16 x+3}f(x)=5x2−16x+32x−6​.

Problem 4725

Determine the horizontal asymptotes of the function f(x)=2(x−8)(2x−3)(x−8)f(x)=\frac{2(x-8)}{(2 x-3)(x-8)}f(x)=(2x−3)(x−8)2(x−8)​.

Problem 4726

Determine the horizontal asymptotes of the function f(x)=2x2−82x2+x−45f(x)=\frac{2 x^{2}-8}{2 x^{2}+x-45}f(x)=2x2+x−452x2−8​.

Problem 4727

Evaluate the integral ∭Exy dV\iiint_{E} x y \, dV∭E​xydV for the tetrahedron with vertices (0,0,0),(4,0,0),(0,1,0),(0,0,7)(0,0,0),(4,0,0),(0,1,0),(0,0,7)(0,0,0),(4,0,0),(0,1,0),(0,0,7).

Problem 4728

Determine the horizontal asymptotes for the function f(x)=3(x+4)(x−4)x(2x−1)f(x)=\frac{3(x+4)(x-4)}{x(2 x-1)}f(x)=x(2x−1)3(x+4)(x−4)​.

Problem 4729

Determine the horizontal asymptotes of the function f(x)=3x−22x2+2x−112f(x)=\frac{3 x-2}{2 x^{2}+2 x-112}f(x)=2x2+2x−1123x−2​.

Problem 4730

Find the first and second derivatives of f(x)=x⋅e−0.5xf(x)=x \cdot e^{-0.5 x}f(x)=x⋅e−0.5x.

Problem 4731

Evaluate the integral ∫y2dy(y2+16)3/2\int \frac{y^{2} d y}{\left(y^{2}+16\right)^{3 / 2}}∫(y2+16)3/2y2dy​.

Problem 4732

Differentiate g(n)=an2+cna−(an−c)6g(n)=\frac{a n^{2}+c n}{a}-(a n-c)^{6}g(n)=aan2+cn​−(an−c)6 with respect to nnn. Result: 2an+c2 a n+c2an+c.

Problem 4733

Differentiate g(x)=xe+e2x+ex−1+e3x−4g(x)=\frac{x}{e}+e^{2} x+e x^{-1}+e \sqrt{3 x-4}g(x)=ex​+e2x+ex−1+e3x−4​.

Problem 4734

Find the xxx-coordinates where the graph of f(x)=8x6−3x4+17f(x)=8 x^{6}-3 x^{4}+17f(x)=8x6−3x4+17 has horizontal tangents. df(x)dx=48x5−12x3\frac{d f(x)}{d x}=48 x^{5}-12 x^{3}dxdf(x)​=48x5−12x3

Problem 4735

Differentiate g(n)=an2+cna−(an−c)6g(n)=\frac{a n^{2}+c n}{a}-(a n-c)^{6}g(n)=aan2+cn​−(an−c)6 with respect to nnn.

Problem 4736

Zeichne den Graphen von h(t)=18t2−2t+8h(t)=\frac{1}{8} t^{2}-2 t+8h(t)=81​t2−2t+8. Bestimme, wann der Behälter leer und halb voll ist und die Sinkgeschwindigkeit.

Problem 4737

Find y′y^{\prime}y′ from x3+y2=43x^{3}+y^{2}=43x3+y2=43 at x=3x=3x=3, y=4y=4y=4. Provide the exact answer (fractions, not decimals).

Problem 4738

Bestimme die Punkte, wo f(x)=2f(x)=2f(x)=2 und berechne die Steigung. Vergleiche die mittlere Steigung für 1,5≤x≤2,51,5 \leq x \leq 2,51,5≤x≤2,5 mit der Steigung bei x=2x=2x=2.

Problem 4739

Find the derivative of f(t)=24−tf(t)=\frac{2}{4-t}f(t)=4−t2​ at t=−5t=-5t=−5 using limits, then find the tangent line equation at that point.

Problem 4740

Problem 4741

Problem 4742

Find $f^{\prime}(-1)$ for the piecewise function: $f(x) = -3x + 5$ (if $x < -1$ ) and $f(x) = -x^2 + 3$ (if $x \geq -1$ ). Options: A. -3 B. -2 C. 2 D. 3 E. nonexistent.

Find the limit as $x$ approaches infinity for the expression $\frac{1}{x}$ .

Find $h'(1)$ for $h(x)=f(g(x))$ given $f(1)=2$ , $f'(1)=3$ , $f'(2)=-4$ , $g(1)=2$ , $g'(1)=-3$ , $g'(2)=5$ .

Find the tangent line equation for $g(x)=x \cdot f(x)$ at $x=2$ given $f(2)=6$ and $f'$ has horizontal tangents at $x=1,3$ .

Find the limit: $\lim _{x \rightarrow-\infty} f(x)$ for the function $f(x)=2 x e^{2 x}$ .

Find $\lim_{x \rightarrow -\infty} f(x)$ for the function $f(x) = 2x e^{2x}$ .

Find the limit: $\lim _{x \rightarrow-\infty} f(x)$ where $f(x)=2 x e^{2 x}$ .

Find the curve equation through $(0,2)$ with slope $m = \frac{x}{y}$ .

Solve the differential equation: $\frac{d z}{d t}+e^{t+z}=0$ .

Solve $2 y' = x + y$ using the substitution $u = x + y$ .

Solve $\frac{d y}{d x}=y \tan x$ , with $y=1$ at $x=0$ . Also, show $\frac{d N}{d t}=-\frac{k}{\sqrt{N}}$ for cockroaches.

Bestimmen Sie die Werte von $t \in \mathbb{R}$ , für die $f_{t}(x)=\frac{1}{2} x^{3}+t x^{2}+6 x-2$ a) keine und b) genau einen waagerechten Tangentenpunkt hat. Zeigen Sie, dass dieser Punkt kein Extrempunkt ist.

Find the oscillation rate of the spring at $t=5$ s for $s(t)=3 \sin t$ . Round to four decimal places.

Find the first three derivatives of $f(x)=2x-3+e^{-x}$ and identify its local extrema. Show your work.

Find $x$ in $f(x)=5x-8\cos x$ for $0<x<2\pi$ where the tangent line slope is 9. $x=$

Find the 100th, 101st, and 102nd derivatives of the function $f(x) = \sum_{n=0}^{\infty}(x-2)^{4n}$ at $x=2$ .

A patient receives 50 mg of medicine that decreases by $10\%$ each hour. How much remains after 6 hours using $A(t)=l e^{r t}$ ?

Find the first and second derivatives of $f(x)=(x+1) \cdot e^{-0.5 x}$ .

Determine the horizontal asymptotes of the function $f(x) = \frac{(3x-7)(x+2)}{3(x-10)(x+2)}$ .

Determine the horizontal asymptotes for the function $f(x)=\frac{2 x+2}{3 x^{2}-x-4}$ .

Determine the horizontal asymptotes of the function $f(x)=\frac{(2 x+1)(x-2)}{x(2 x+1)}$ .

Determine the horizontal asymptotes of the function $f(x)=\frac{2 x-6}{5 x^{2}-16 x+3}$ .

Determine the horizontal asymptotes of the function $f(x)=\frac{2(x-8)}{(2 x-3)(x-8)}$ .

Determine the horizontal asymptotes of the function $f(x)=\frac{2 x^{2}-8}{2 x^{2}+x-45}$ .

Evaluate the integral $\iiint_{E} x y \, dV$ for the tetrahedron with vertices $(0,0,0),(4,0,0),(0,1,0),(0,0,7)$ .

Determine the horizontal asymptotes for the function $f(x)=\frac{3(x+4)(x-4)}{x(2 x-1)}$ .

Determine the horizontal asymptotes of the function $f(x)=\frac{3 x-2}{2 x^{2}+2 x-112}$ .

Find the first and second derivatives of $f(x)=x \cdot e^{-0.5 x}$ .

Evaluate the integral $\int \frac{y^{2} d y}{\left(y^{2}+16\right)^{3 / 2}}$ .

Differentiate $g(n)=\frac{a n^{2}+c n}{a}-(a n-c)^{6}$ with respect to $n$ . Result: $2 a n+c$ .

Differentiate $g(x)=\frac{x}{e}+e^{2} x+e x^{-1}+e \sqrt{3 x-4}$ .

Find the $x$ -coordinates where the graph of $f(x)=8 x^{6}-3 x^{4}+17$ has horizontal tangents. $\frac{d f(x)}{d x}=48 x^{5}-12 x^{3}$

Differentiate $g(n)=\frac{a n^{2}+c n}{a}-(a n-c)^{6}$ with respect to $n$ .

Zeichne den Graphen von $h(t)=\frac{1}{8} t^{2}-2 t+8$ . Bestimme, wann der Behälter leer und halb voll ist und die Sinkgeschwindigkeit.

Find $y^{\prime}$ from $x^{3}+y^{2}=43$ at $x=3$ , $y=4$ . Provide the exact answer (fractions, not decimals).

Bestimme die Punkte, wo $f(x)=2$ und berechne die Steigung. Vergleiche die mittlere Steigung für $1,5 \leq x \leq 2,5$ mit der Steigung bei $x=2$ .

Find the derivative of $f(t)=\frac{2}{4-t}$ at $t=-5$ using limits, then find the tangent line equation at that point.

Find the function $f(x)$ if $f'(x)=x e^{-4x}-4e e^{-4x}$ . Factor out $e^{-4x}$ .

Untersuchen Sie, ob der Graph von $f(x)=x \cdot (g(x))^{2}$ den Graphen von $g$ in $P(1,1)$ berührt, wo die Steigung -1 ist.

Determine the horizontal asymptotes of the function $f(x)=\frac{2(3 x+8)(3 x-8)}{3(x+2)(x-2)}$ .

Determine the horizontal asymptotes of the function $f(x)=\frac{2 x(5 x+4)}{(x+3)(x-3)}$ .

Determine the horizontal asymptotes of the function $f(x)=\frac{3 x^{2}+51 x+210}{2 x+20}$ .

Find the 6th derivative of $f(x)=\cos(\ln(1+x^{2}))$ at $x=0$ using the Taylor series.

Determine the horizontal asymptotes of the function $f(x)=\frac{x^{2}-12 x+20}{2 x-4}$ .

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

Determine the horizontal asymptotes of the function $f(x)=\frac{2 x(x-8)}{3(x+9)(x-7)}$ .

Determine the horizontal asymptotes for the function $f(x)=\frac{x-8}{x^{2}-5 x-24}$ .

Berechne das Integral $\int \frac{1}{1-4 x} \cdot d x$ durch Substitution.

Berechne das unbestimmte Integral $\int 3 x \cdot \sin (4 x) \, dx$ .

Berechne das unbestimmte Integral $\int 3 x \cdot \ln (4 x) \, dx$ .

Find the slope of the tangent line to $f(x)=x^{3}-6x+3$ at the point $(0,3)$ .

Find the first and second derivatives of $f(x)=(x^{2}-x) \cdot e^{-0.5 x}$ and simplify by factoring common terms.

Find the derivative $\frac{d h}{d x}$ for the function $h(x)=c \pi$ , where $c$ is a constant.

Find the derivative $\frac{d y}{d x}$ for the function $y=\sin^{-1}\left(\frac{1}{2 x+5}\right)$ .

Find the derivative of $f(x)=x \cdot e^{-0.5 x}$ and simplify by factoring out common terms.

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $-x^{2}-8 y^{2}=3$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\sin ^{-1}\left(\frac{1}{2 x+5}\right)$ .

Find the first three derivatives of $f(x)=(1-x) \cdot e^{-x}$ and simplify by factoring out common terms.

How long (in hours) to metabolize half the caffeine? Use the equation $\frac{d C}{d t}=-0.14 C$ . Round to nearest integer.

Graph the function $\frac{1}{x^{5}-2 x^{3}-3 x^{2}+6}$ on given intervals and estimate vertical asymptotes.

Find the derivative of the function $s=8 t^{9/4}+3 e^{5}$ .

Find the derivative of $r$ with respect to $s$ for $r=\frac{1}{4 s^{3}}-\frac{5}{3 s^{2}}$ .

Find the first derivative of $r$ with respect to $s$ where $r=\frac{1}{2 s}-\frac{7}{3 s^{2}}$ .

Find the first and second derivatives of $f(x)=(x^{2}-1) \cdot e^{-x}$ and simplify by factoring common terms.

Find the derivative of $r$ with respect to $s$ for $r=\frac{1}{4 s^{3}}-\frac{5}{3 s^{2}}$ .

Find the derivative of $f(x)=(x+1) \cdot e^{-0.5 x}$ using the product rule.

Find the tangent line equation for $f(x)$ at $x=1$ where $f(x)=x^{4}-2 x^{3}-5 x-4$ .

Find the tangent line to $f(x)=\ln (x)$ at $x=e^{-1}$ : determine $m$ and $b$ for $y=m x+b$ .