Calculus

Problem 8701

Find the sum of the series $1 + \frac{2^1}{1!} + \frac{2^2}{2!} + \frac{2^3}{3!} + \ldots$ . Choices: (A) $\ln 2$ (B) $e^2$ (C) $\cos 2$ (D) $\sin 2$ (E) nonexistent.

See Solution

Problem 8702

Find the hole in the graph of $g(x)=\frac{3 x^{2}-8 x-3}{x^{2}-9}$ and its limits as $x$ approaches -3 and 3.

See Solution

Problem 8703

Find the integral of $t^6 \, dt$ . What is the result? A. $6 t^5$ B. $6 t^5 + C$ C. $\frac{1}{7} t^7 + C$ D. $t^7 + C$

See Solution

Problem 8704

Find the derivative of $f(x)=x^{\wedge} 3+x^{\wedge} 2+3$ . Choose from A. $3 x^{\wedge} 2+2 x$ , B. $3 x+2 x$ , C. $3 x+2 x+3$ , D. $x^{\wedge} 3+x^{\wedge} 2$ .

See Solution

Problem 8705

How many gallons of oil are pumped out in $0 \leq t \leq 6$ minutes if the rate is $20 e^{-0.1t}/(1+e^{-t})$ ? (A) 62 (B) 78 (C) 85 (D) 93

See Solution

Problem 8706

Find the limits of the function $k(x)=\frac{2 x^{2}-8 x+6}{x^{2}+x-2}$ as $x \to -\infty$ , $x \to \infty$ , $x \to -2^{-}$ , $x \to -2^{+}$ , $x \to 1^{-}$ , $x \to 1^{+}$ . Also, determine the domain of $k(x)$ .

See Solution

Problem 8707

Find the volume of the solid formed by revolving the region between $y=1+\sin(\pi x)$ and $y=x^3$ from $x=0$ to $x=1$ .

See Solution

Problem 8708

Find the second derivative of $f(x)=2x-5x^{\wedge}6$ . What is $f^{\prime \prime}(x)$ ? A. $2-30x$ B. $2-30x^{\wedge}5$ C. $-30x^{\wedge}5$ D. $-150x^{\wedge}4$

See Solution

Problem 8709

Find the derivative using implicit differentiation for $x^{2} y - y^{2} = 6$ : $\frac{d y}{d x} = \square$ .

See Solution

Problem 8710

Integrate $\int \cos^{2}(4x) \sin^{3}(4x) \, dx$ .

See Solution

Problem 8711

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $x y^{3}-y=x$ .

See Solution

Problem 8712

Find the derivative of $y=x^{2}(\sin^{-1} x)^{3}$ with respect to $x$ .

See Solution

Problem 8713

Calculate $\lim _{h \rightarrow 0} \frac{10^{h}-1}{h}$ .

See Solution

Problem 8714

Find $h^{\prime}(9)$ if $h(x)=5 f(x)+4 g(x)+4$ with $f^{\prime}(9)=9$ and $g^{\prime}(9)=6$ .

See Solution

Problem 8715

Find the limit: $\lim _{h \rightarrow 0} \frac{\tan ^{-1}(1+h)-\pi / 4}{h}$ .

See Solution

Problem 8716

Find the rate of change of surface area $A$ with respect to weight $w$ when $A(w)=5(10)^{1-\log(w)}+1$ and $A=6 \mathrm{in}^2$ .

See Solution

Problem 8717

Given the function $f(x)=\frac{x^{2}}{x^{2}+3}$ , find:
(a) the intervals where $f$ is increasing and decreasing. (b) the local minimum value of $f$ . (c) the inflection points and intervals of concavity (up and down).

See Solution

Problem 8718

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{(2+\Delta x)^{(2+\Delta x)}-4}{\Delta x}$ .

See Solution

Problem 8719

Bestimmen Sie die Tangentengleichung an $f(x)=\frac{1}{2} x^{2}-3 x+1$ im Punkt $P$ für die gegebenen Koordinaten.

See Solution

Problem 8720

Given $f(x)=3 x^{4}-24 x^{2}+6$ , find: (A) $f^{\prime}(x)$ , (B) slope at $x=-1$ , (C) tangent line equation at $x=-1$ , (D) where tangent is horizontal.

See Solution

Problem 8721

Find the limit: $\lim _{w \rightarrow 2} \frac{3 \sec ^{-1} w-\pi}{w-2}$ .

See Solution

Problem 8722

A bacteria colony shrinks by $25\%$ daily. Find the decrease rate when it is half its original size. Also, find $\frac{d F}{-}$ and $\frac{d r}{-}$ for $F(r)=\frac{G m_{1} m_{2}}{2}$ .

See Solution

Problem 8723

Find the derivatives $\frac{d F}{d r}$ and $\frac{d r}{d F}$ for the function $F(r)=\frac{G m_{1} m_{2}}{r^{2}}$ .

See Solution

Problem 8724

Analyze the function $f(x)=\frac{x-x^{2}}{2-3 x+x^{2}}$ for intercepts, asymptotes, extrema, concavity, and inflection points, then sketch the curve.

See Solution

Problem 8725

Find the concentration $C(t)=\frac{0.2 t}{1+t^{2}}$ when it decreases by $0.016 \mathrm{mg} / \mathrm{cm}^{3}$ per hour.

See Solution

Problem 8726

A marine manufacturer sells $\mathrm{N}(x)$ power boats after spending $\$x$ thousand on ads: $N(x)=1,010-\frac{3,810}{x}, 5 \leq x \leq 30$ .
(A) Find $N'(x)$ .
(B) Calculate $N'(10)$ and $N'(20)$ , then interpret these results.

See Solution

Problem 8727

Differentiate implicitly to find the slope and equation of the tangent line at $(-3,-3)$ for $2 x^{2}+x y=3 y^{2}$ .
(a) Find $\frac{d y}{d x}=$
(b) Find $y=$

See Solution

Problem 8728

Find the tangent line equations for these functions at the specified $x$ values: a. $x^{2} y - x y^{3} = y - 8$ at $x=1$ b. $x y + 1 = y + 2 y^{-1}$ at $x=2$ c. $\frac{3 x}{y} = e^{x-1}$ at $x=1$ d. $x = \sqrt{\frac{1-y}{1+y}}$ at $x=1$ e. $e^{x y} = \frac{x^{2}-y^{2}}{x+y}$ at $x=0$

See Solution

Problem 8729

Find the limit: $\lim _{x \rightarrow \infty} \frac{6}{\sqrt{x}}$ .

See Solution

Problem 8730

Find the instantaneous acceleration of a particle at $t=3.5$ given its displacement function $f(t)=4 t^{2} \ln (t)$ .

See Solution

Problem 8731

Find the slope of the tangent to the curve defined by $x=t^2+3t-8$ , $y=2t^2-2t-5$ at the point $(2,-1)$ .

See Solution

Problem 8732

Simplify the expression: $d = \frac{4(x-1)}{2}$

See Solution

Problem 8733

Find $d y / d x$ using logarithmic differentiation for $y=\frac{(4 x+1)^{2}}{6(2 x-1)^{3}}$ .

See Solution

Problem 8734

Find the horizontal asymptote of $f(x)=\frac{5x+6}{x^{2}+9x-10}$ .

See Solution

Problem 8735

Determine the horizontal asymptote of $f(x)=\frac{9x+2}{4x+9}$ .

See Solution

Problem 8736

Find the max and min values of $f(x)=6x-2$ on the interval [-4,3]. State the $\mathrm{x}$ -values where they occur.

See Solution

Problem 8737

Find the profit rate change if production increases by 500 games/week at $x=1500$ , given $C(x)=72000+60x$ and $R(x)=200x-\frac{x^{2}}{30}$ .

See Solution

Problem 8738

Find the rate of change of total resistance when $R_{1}=80$ ohms and $R_{2}=100$ ohms, given their rates of change.

See Solution

Problem 8739

Bestimmen Sie die Extrema und Sattelpunkte von $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-2 x$ für $-3,5 \leq x \leq 2,5$ .

See Solution

Problem 8740

Sketch a graph of $g(x)$ with these limits: $\lim_{x \to -\infty} g(x) = 2$ , $\lim_{x \to \infty} g(x) = 2$ , $\lim_{x \to -3} g(x) = \infty$ , $\lim_{x \to 3^-} g(x) = -\infty$ , $\lim_{x \to 3^+} g(x) = \infty$ .

See Solution

Problem 8741

Two vehicles are moving towards each other: one at $50 \mathrm{mph}$ west and the other at $60 \mathrm{mph}$ north. Find their approach speed when the west vehicle is 0.3 miles and the north vehicle is 0.4 miles from collision.

See Solution

Problem 8742

For the function $f(x)=\frac{x+1}{x-2}$ , find the average rate of change over the intervals $[0,1]$ , $[0,0.5]$ , $[0,0.1]$ , $[0,0.01]$ , estimate the instantaneous rate at $x=0$ , and state the tangent line equation at $x=0$ .

See Solution

Problem 8743

Untersuchen Sie $f$ auf Extrema und Sattelpunkte für die Funktionen a) $f(x)=\frac{1}{2} x^{2}-2 x+3$ , b) $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-2 x$ , c) $f(x)=\frac{1}{3} x^{3}+x^{2}+x$ im Bereich $-1 \leq x \leq 5$ , $-3,5 \leq x \leq 2,5$ , $-3 \leq x \leq 2$ .

See Solution

Problem 8744

Find the instantaneous rate of change of blood sugar level $G(x)=450-0.2 x^{2}$ after 10 insulin units.

See Solution

Problem 8745

Given $f(x)=\frac{x+1}{x-2}$ , find the average rate of change over the intervals $[0,1]$ , $[0,0.5]$ , $[0,0.1]$ , $[0,0.01]$ , estimate the instantaneous rate at $x=0$ , and state the tangent line equation at that point.

See Solution

Problem 8746

Erklären Sie, warum die Potenzregel $g(x)=x^{0}=1$ für $x=0$ nicht gilt, obwohl $f^{\prime}(x)=1$ und $g^{\prime}(x)=0$ sind.

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Problem 8747

Bestimme die Ableitung von $f(x)$ an den Punkten $x_{0}$ : a) $x^{5}$ bei 2, b) $x^{2}$ bei -3, c) $x^{3}$ bei 4, d) $x^{4}$ bei -2, e) $x^{7}$ bei -1, f) $x$ bei 9.

See Solution

Problem 8748

Solve the differential equation: $y^{\prime \prime}-y^{\prime}-30 y=0$ using constants $c_{1}$ and $c_{2}$ .

See Solution

Problem 8749

Find the instantaneous rate of change for $f(x)=\frac{1}{x+1}$ and $g(x)=2x^2-5x+40$ . Then, find tangent lines at $x=2$ and $x=-1$ .

See Solution

Problem 8750

Find the instantaneous rate of change for: a. $f(x)=\frac{1}{x+1}$ , b. $g(x)=2 x^{2}-5 x+40$ . Then, find the tangent line at $x=2$ for $f(x)$ and at $x=-1$ for $g(x)$ .

See Solution

Problem 8751

Estimate the additional revenue from producing 5000 to 5001 units using the limit definition of the derivative for $R(x)=50 x-0.5 x^{2}$ .

See Solution

Problem 8752

Find the marginal price function for $p(x)=\frac{10 x}{1+x^{2}}$ and the tangent line at $x=2$ for given cases.

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Problem 8753

Bestimme die Ableitungen der Funktionen: a) $f(x)=x^{7}$ , b) $f(x)=x^{9}$ , c) $h(x)=x^{100}$ , d) $f(x)=x^{m+1}$ , e) $g(x)=x^{2 n}$ , f) $v(t)=t^{2}$ .

See Solution

Problem 8754

Solve the initial-value problem: $y^{\prime \prime}+3 y=0$ , with $y(0)=1$ and $y^{\prime}(0)=3$ .

See Solution

Problem 8755

Gegeben ist die Funktion $f(x)=-x^{2}+9$ . Bestimme für $0 \leq u \leq 3$ :
a) den Wert von $u$ , der den maximalen Flächeninhalt des Rechtecks gibt. b) den Wert von $u$ , der den maximalen Umfang des Rechtecks gibt.

See Solution

Problem 8756

Find the unemployment increase in 2016 using the derivative limit for $U(t)=\frac{\sqrt{t}+t}{10}$ , $t$ since 2000.

See Solution

Problem 8757

Find the unemployment increase in 2016 using the limit definition of the derivative for $U(t)=\frac{\sqrt{t}+t}{10}$ .

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Problem 8758

Find the derivatives of these functions: a. $y=\frac{1}{\sqrt[3]{s^{2}}}-\frac{12}{s^{5}}$ b. $f(x)=(x+x^{-1})^{2}$ c. $G(t)=\frac{t^{2}+4t+3}{\sqrt{t}}$ d. $y=(1-e^{x})(1+e^{x})$ e. $h(z)=\frac{z}{z+\frac{c}{z}}+1$ f. $f(x)=A\left(\frac{1}{B}+\frac{x}{C}\right)$ g. $y=x \ln (x)-x$ h. $y=\ln (\ln (x))$ i. $y=\ln \left(e^{-x}+x e^{-x}\right)$

See Solution

Problem 8759

Berechne den Differenzenquotienten für die Funktionen in den angegebenen Intervallen: a) $f(x)=x^{2}$ , I=[0; 2]; b) $f(x)=3x^{2}$ , I=[-1; 3]; c) $f(x)=2x^{3}$ , I=[-1; 1]; d) $f(x)=3x^{2}-4$ , I=[1; 3]; e) $f(x)=2x^{2}-3$ , I=[2; 4]; f) $f(x)=\frac{1}{2}x^{2}-3$ , I=[-2; 1]; g) $f(x)=\sqrt{x}$ , I=[4; 9]; h) $f(x)=\frac{1}{x}$ , I=[-1; -0.5].

See Solution

Problem 8760

Find where the horizontal tangent line occurs for: a. $y=2 x^{3}+3 x^{2}-12 x+1$ b. $y=2 e^{x}-4 x$

See Solution

Problem 8761

Find tangent line equations for these functions at specified points: a. $p(x)=x^{3}-x+1$ at $x=-1$ b. $q(x)=\sqrt{x+5}$ at $x=4$ c. $f(x)=x-\sqrt{x}$ at $x=4$ d. $g(x)=3 x^{2}+4 x-6$ at $x=-1$ e. $h(x)=e^{x}+e^{-x}$ at $x=0$ f. $H(x)$ at $x=-5$ , $H(5)=3$ , $H^{\prime}(5)=-1$ g. $G(x)=e^{x} K(x)$ at $x=0$ , $K(0)=2$ , $K^{\prime}(0)=-3$

See Solution

Problem 8762

Find the rate of change of community awareness $A(t)=10 \ln (2) t^{2} \cdot 2^{-t}$ after 4 weeks.

See Solution

Problem 8763

Find the marginal profit for $P(x)=30 \ln (10) \cdot \log (3 x+1)$ at $x=33$ . Also, estimate profit change from 4500 to 4510 games with $C(x)=72000+60 x$ and $R(x)=200 x-\frac{x^{2}}{30}$ .

See Solution

Problem 8764

Evaluate the integral $\int_{-1}^{3} e^{x}+2 x^{2} \, dx$ .

See Solution

Problem 8765

Find the marginal profit for $P(x)=30 \ln (10) \cdot \log (3 x+1)$ at $x=33$ . Also, estimate profit change from 4500 to 4510 games.

See Solution

Problem 8766

Approximate the extra steel needed if a bearing's radius increases from $5 \mathrm{~mm}$ to $5.3 \mathrm{~mm}$ . Also, find the instantaneous rate of change for a \ $25,000 investment compounding at$ 8.4\%$ after 2 years.

See Solution

Problem 8767

Berechne die Kathetenlängen eines rechtwinkligen Dreiecks mit Hypotenuse an einer Wand, um die Fläche zu maximieren. $A(x)=-0,5 x^2 + 25 x$

See Solution

Problem 8768

Find the limit as $n$ approaches infinity: $\lim _{n \rightarrow \infty}\left(\frac{n+1}{n-m}\right)^{n}=$

See Solution

Problem 8769

Find the limit as $u$ approaches infinity: $\lim _{u \rightarrow \infty}\left(\frac{2+u}{3+u}\right)^{n}=$

See Solution

Problem 8770

Find the limit: $\lim _{u \rightarrow a}\left(\frac{u+2}{u+3}\right)^{u}$ .

See Solution

Problem 8771

Find the value of $x$ for $-4 \leq x \leq 4$ where $f$ has an absolute maximum, given $f'(x)$ and $f(0)=5$ .

See Solution

Problem 8772

Find where the function $f(x)=\sin^{2}(x)-\sin(x)$ is increasing for $0 \leq x \leq \frac{3\pi}{2}$ . Justify your answer.

See Solution

Problem 8773

Find the $I^{\text{th}}$ derivative of $f$ at the origin, where $I=(2,1)$ . Also, find the Taylor expansion of $\partial f / \partial x$ .

See Solution

Problem 8774

Find the limit: $\lim _{n \rightarrow \infty}\left(\frac{n+2}{n+3}\right)^{n}$ .

See Solution

Problem 8775

Find the limit of $a_{n}=\left(\frac{n+1}{n-3}\right)^{n}$ as $n \to \infty$ .

See Solution

Problem 8776

Show that there exists a value $c$ in $(2, 5)$ where $f'(c) = -1$ given $f(2) = 5$ and $f(5) = 2$ .

See Solution

Problem 8777

Find the interval where the function $f(x)=\frac{1}{3} x^{3}+2 x^{2}+4 x$ is both concave up and increasing. Justify.

See Solution

Problem 8778

A function $f(x, y)$ has a Taylor expansion at the origin. Find the $I^{\text{th}}$ derivative at $(2,1)$ and the expansion of $\partial f / \partial x$ .

See Solution

Problem 8779

How long does it take for a ball to fall 84 meters due to gravity? Use $d = \frac{1}{2}gt^2$ to find $t$ .

See Solution

Problem 8780

Find the derivative of $f(x) = \frac{1}{x}$ using the limit definition of derivatives.

See Solution

Problem 8781

Find the first and second derivatives of $f(x, y)=y^{2}-2y+x^{2}-xy-4x-2$ , and determine its global max/min for $x \geq 0$ .

See Solution

Problem 8782

An object is dropped from a 50m building. Height is $h(t)=50-4.9 t^{\wedge} 2$ . Find its velocity when it hits the ground.

See Solution

Problem 8783

Find the general solution of the differential equation: $\frac{d x}{d x} + \frac{5}{x} = \sinh(x^{2})$ , for $x > 0$ .

See Solution

Problem 8784

Calculate the percent rate of change for the function $f(x)=(0.25)^{xy}$ with a constant $y$ .

See Solution

Problem 8785

Find the average rate of change of $f(x)=x^{2}-5x+14$ from $x=3$ to $x=5$ .

See Solution

Problem 8786

Find the average rate of change of $h(x)=6-3 x^{2}$ from $x=-2$ to $x=1$ .

See Solution

Problem 8787

Does the function $p(x)=\frac{(3 x-1)^{2}}{2 x^{2}+3 x+5}$ have a horizontal asymptote? If yes, what is the equation?

See Solution

Problem 8788

Does the function $y=\frac{(x-2)(4-x)}{(x+3)^{2}}$ have a horizontal asymptote? Answer Y or N and provide the equation if Y.

See Solution

Problem 8789

Find if the function $s(x)=\frac{(2x^{2}+3)^{2}(x-4)}{(x^{2}+5)(x-2)}$ has a horizontal asymptote. Y or N? If yes, what is the equation?

See Solution

Problem 8790

Find the derivative of $f(x)=x^{5} e^{8 x}$ .

See Solution

Problem 8791

Calculate the average rate of change of $k(x)=-3 x^{3}-8 x^{2}+19 x$ from $x=-4$ to $x=3$ . Round to the nearest tenth.

See Solution

Problem 8792

Find the derivative of the function $e^{-5 x^{2}+9 x}$ .

See Solution

Problem 8793

Differentiate the function: $f(x)=4 e^{5\left(x^{2}+2\right)^{2}}$ .

See Solution

Problem 8794

Find the derivative of $f_{t}(x)=-t \cdot(x+2) \cdot(x-1)^{2}$ .

See Solution

Problem 8795

Find the derivative of $h(x)=\log _{3}(2 e^{x}-7)+\arctan(5 x)$ .

See Solution

Problem 8796

Differentiate $g(n)=\ln \left(\frac{n^{3}-n}{n^{2}+1}\right)$ .

See Solution

Problem 8797

Find the first derivative of $f(x)=\left(\frac{4}{5} e^{x} \cos (\pi x+3)\right)^{x}$ using logarithmic differentiation.

See Solution

Problem 8798

Calculate the average rate of change of $k(x)=-2 \sqrt{x-5}$ from $x=5$ to $x=14$ . Round to the nearest tenth.

See Solution

Problem 8799

Find the rate of change of a $120 \mathrm{mg}$ mercury-195 sample after $t$ hours. Does it decay faster at 15 or 45 hours?

See Solution

Problem 8800

Find the derivative of $f(x)=\left(\frac{4}{5} e^{x} \cos (\pi x+3)\right)^{x}$ using logarithmic differentiation.

See Solution

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Calculus

Problem 8701

Find the sum of the series 1+211!+222!+233!+…1 + \frac{2^1}{1!} + \frac{2^2}{2!} + \frac{2^3}{3!} + \ldots1+1!21​+2!22​+3!23​+…. Choices: (A) ln⁡2\ln 2ln2 (B) e2e^2e2 (C) cos⁡2\cos 2cos2 (D) sin⁡2\sin 2sin2 (E) nonexistent.

Problem 8702

Find the hole in the graph of g(x)=3x2−8x−3x2−9g(x)=\frac{3 x^{2}-8 x-3}{x^{2}-9}g(x)=x2−93x2−8x−3​ and its limits as xxx approaches -3 and 3.

Problem 8703

Find the integral of t6 dtt^6 \, dtt6dt. What is the result? A. 6t56 t^56t5 B. 6t5+C6 t^5 + C6t5+C C. 17t7+C\frac{1}{7} t^7 + C71​t7+C D. t7+Ct^7 + Ct7+C

Problem 8704

Find the derivative of f(x)=x∧3+x∧2+3f(x)=x^{\wedge} 3+x^{\wedge} 2+3f(x)=x∧3+x∧2+3. Choose from A. 3x∧2+2x3 x^{\wedge} 2+2 x3x∧2+2x, B. 3x+2x3 x+2 x3x+2x, C. 3x+2x+33 x+2 x+33x+2x+3, D. x∧3+x∧2x^{\wedge} 3+x^{\wedge} 2x∧3+x∧2.

Problem 8705

How many gallons of oil are pumped out in 0≤t≤60 \leq t \leq 60≤t≤6 minutes if the rate is 20e−0.1t/(1+e−t)20 e^{-0.1t}/(1+e^{-t})20e−0.1t/(1+e−t)? (A) 62 (B) 78 (C) 85 (D) 93

Problem 8706

Problem 8707

Find the volume of the solid formed by revolving the region between y=1+sin⁡(πx)y=1+\sin(\pi x)y=1+sin(πx) and y=x3y=x^3y=x3 from x=0x=0x=0 to x=1x=1x=1.

Problem 8708

Find the second derivative of f(x)=2x−5x∧6f(x)=2x-5x^{\wedge}6f(x)=2x−5x∧6. What is f′′(x)f^{\prime \prime}(x)f′′(x)? A. 2−30x2-30x2−30x B. 2−30x∧52-30x^{\wedge}52−30x∧5 C. −30x∧5-30x^{\wedge}5−30x∧5 D. −150x∧4-150x^{\wedge}4−150x∧4

Problem 8709

Find the derivative using implicit differentiation for x2y−y2=6x^{2} y - y^{2} = 6x2y−y2=6: dydx=□\frac{d y}{d x} = \squaredxdy​=□.

Problem 8710

Integrate ∫cos⁡2(4x)sin⁡3(4x) dx\int \cos^{2}(4x) \sin^{3}(4x) \, dx∫cos2(4x)sin3(4x)dx.

Problem 8711

Find the derivative dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the equation xy3−y=xx y^{3}-y=xxy3−y=x.

Problem 8712

Find the derivative of y=x2(sin⁡−1x)3y=x^{2}(\sin^{-1} x)^{3}y=x2(sin−1x)3 with respect to xxx.

Problem 8713

Calculate lim⁡h→010h−1h\lim _{h \rightarrow 0} \frac{10^{h}-1}{h}limh→0​h10h−1​.

Problem 8714

Find h′(9)h^{\prime}(9)h′(9) if h(x)=5f(x)+4g(x)+4h(x)=5 f(x)+4 g(x)+4h(x)=5f(x)+4g(x)+4 with f′(9)=9f^{\prime}(9)=9f′(9)=9 and g′(9)=6g^{\prime}(9)=6g′(9)=6.

Problem 8715

Find the limit: lim⁡h→0tan⁡−1(1+h)−π/4h\lim _{h \rightarrow 0} \frac{\tan ^{-1}(1+h)-\pi / 4}{h}limh→0​htan−1(1+h)−π/4​.

Problem 8716

Find the rate of change of surface area AAA with respect to weight www when A(w)=5(10)1−log⁡(w)+1A(w)=5(10)^{1-\log(w)}+1A(w)=5(10)1−log(w)+1 and A=6in2A=6 \mathrm{in}^2A=6in2.

Problem 8717

Given the function f(x)=x2x2+3f(x)=\frac{x^{2}}{x^{2}+3}f(x)=x2+3x2​, find:(a) the intervals where fff is increasing and decreasing. (b) the local minimum value of fff. (c) the inflection points and intervals of concavity (up and down).

Problem 8718

Find the limit: lim⁡Δx→0(2+Δx)(2+Δx)−4Δx\lim _{\Delta x \rightarrow 0} \frac{(2+\Delta x)^{(2+\Delta x)}-4}{\Delta x}limΔx→0​Δx(2+Δx)(2+Δx)−4​.

Problem 8719

Bestimmen Sie die Tangentengleichung an f(x)=12x2−3x+1f(x)=\frac{1}{2} x^{2}-3 x+1f(x)=21​x2−3x+1 im Punkt PPP für die gegebenen Koordinaten.

Problem 8720

Given f(x)=3x4−24x2+6f(x)=3 x^{4}-24 x^{2}+6f(x)=3x4−24x2+6, find: (A) f′(x)f^{\prime}(x)f′(x), (B) slope at x=−1x=-1x=−1, (C) tangent line equation at x=−1x=-1x=−1, (D) where tangent is horizontal.

Problem 8721

Find the limit: lim⁡w→23sec⁡−1w−πw−2\lim _{w \rightarrow 2} \frac{3 \sec ^{-1} w-\pi}{w-2}limw→2​w−23sec−1w−π​.

Problem 8722

A bacteria colony shrinks by 25%25\%25% daily. Find the decrease rate when it is half its original size. Also, find dF−\frac{d F}{-}−dF​ and dr−\frac{d r}{-}−dr​ for F(r)=Gm1m22F(r)=\frac{G m_{1} m_{2}}{2}F(r)=2Gm1​m2​​.

Problem 8723

Find the derivatives dFdr\frac{d F}{d r}drdF​ and drdF\frac{d r}{d F}dFdr​ for the function F(r)=Gm1m2r2F(r)=\frac{G m_{1} m_{2}}{r^{2}}F(r)=r2Gm1​m2​​.

Problem 8724

Analyze the function f(x)=x−x22−3x+x2f(x)=\frac{x-x^{2}}{2-3 x+x^{2}}f(x)=2−3x+x2x−x2​ for intercepts, asymptotes, extrema, concavity, and inflection points, then sketch the curve.

Problem 8725

Find the concentration C(t)=0.2t1+t2C(t)=\frac{0.2 t}{1+t^{2}}C(t)=1+t20.2t​ when it decreases by 0.016mg/cm30.016 \mathrm{mg} / \mathrm{cm}^{3}0.016mg/cm3 per hour.

Problem 8726

Problem 8727

Differentiate implicitly to find the slope and equation of the tangent line at (−3,−3)(-3,-3)(−3,−3) for 2x2+xy=3y22 x^{2}+x y=3 y^{2}2x2+xy=3y2. (a) Find dydx=\frac{d y}{d x}=dxdy​=(b) Find y=y=y=

Problem 8728

Problem 8729

Find the limit: lim⁡x→∞6x\lim _{x \rightarrow \infty} \frac{6}{\sqrt{x}}limx→∞​x​6​.

Problem 8730

Find the instantaneous acceleration of a particle at t=3.5t=3.5t=3.5 given its displacement function f(t)=4t2ln⁡(t)f(t)=4 t^{2} \ln (t)f(t)=4t2ln(t).

Problem 8731

Find the slope of the tangent to the curve defined by x=t2+3t−8x=t^2+3t-8x=t2+3t−8, y=2t2−2t−5y=2t^2-2t-5y=2t2−2t−5 at the point (2,−1)(2,-1)(2,−1).

Problem 8732

Simplify the expression: d=4(x−1)2d = \frac{4(x-1)}{2}d=24(x−1)​

Problem 8733

Find dy/dxd y / d xdy/dx using logarithmic differentiation for y=(4x+1)26(2x−1)3y=\frac{(4 x+1)^{2}}{6(2 x-1)^{3}}y=6(2x−1)3(4x+1)2​.

Problem 8734

Find the horizontal asymptote of f(x)=5x+6x2+9x−10f(x)=\frac{5x+6}{x^{2}+9x-10}f(x)=x2+9x−105x+6​.

Problem 8735

Determine the horizontal asymptote of f(x)=9x+24x+9f(x)=\frac{9x+2}{4x+9}f(x)=4x+99x+2​.

Problem 8736

Find the max and min values of f(x)=6x−2f(x)=6x-2f(x)=6x−2 on the interval [-4,3]. State the x\mathrm{x}x-values where they occur.

Problem 8737

Find the profit rate change if production increases by 500 games/week at x=1500x=1500x=1500, given C(x)=72000+60xC(x)=72000+60xC(x)=72000+60x and R(x)=200x−x230R(x)=200x-\frac{x^{2}}{30}R(x)=200x−30x2​.

Problem 8738

Find the rate of change of total resistance when R1=80R_{1}=80R1​=80 ohms and R2=100R_{2}=100R2​=100 ohms, given their rates of change.

Problem 8739

Bestimmen Sie die Extrema und Sattelpunkte von f(x)=13x3+12x2−2xf(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-2 xf(x)=31​x3+21​x2−2x für −3,5≤x≤2,5-3,5 \leq x \leq 2,5−3,5≤x≤2,5.

Problem 8740

Problem 8741

Two vehicles are moving towards each other: one at 50mph50 \mathrm{mph}50mph west and the other at 60mph60 \mathrm{mph}60mph north. Find their approach speed when the west vehicle is 0.3 miles and the north vehicle is 0.4 miles from collision.

Problem 8742

Find the sum of the series $1 + \frac{2^1}{1!} + \frac{2^2}{2!} + \frac{2^3}{3!} + \ldots$ . Choices: (A) $\ln 2$ (B) $e^2$ (C) $\cos 2$ (D) $\sin 2$ (E) nonexistent.

Find the hole in the graph of $g(x)=\frac{3 x^{2}-8 x-3}{x^{2}-9}$ and its limits as $x$ approaches -3 and 3.

Find the integral of $t^6 \, dt$ . What is the result? A. $6 t^5$ B. $6 t^5 + C$ C. $\frac{1}{7} t^7 + C$ D. $t^7 + C$

Find the derivative of $f(x)=x^{\wedge} 3+x^{\wedge} 2+3$ . Choose from A. $3 x^{\wedge} 2+2 x$ , B. $3 x+2 x$ , C. $3 x+2 x+3$ , D. $x^{\wedge} 3+x^{\wedge} 2$ .

How many gallons of oil are pumped out in $0 \leq t \leq 6$ minutes if the rate is $20 e^{-0.1t}/(1+e^{-t})$ ? (A) 62 (B) 78 (C) 85 (D) 93

Find the volume of the solid formed by revolving the region between $y=1+\sin(\pi x)$ and $y=x^3$ from $x=0$ to $x=1$ .

Find the second derivative of $f(x)=2x-5x^{\wedge}6$ . What is $f^{\prime \prime}(x)$ ? A. $2-30x$ B. $2-30x^{\wedge}5$ C. $-30x^{\wedge}5$ D. $-150x^{\wedge}4$

Find the derivative using implicit differentiation for $x^{2} y - y^{2} = 6$ : $\frac{d y}{d x} = \square$ .

Integrate $\int \cos^{2}(4x) \sin^{3}(4x) \, dx$ .

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for the equation $x y^{3}-y=x$ .

Find the derivative of $y=x^{2}(\sin^{-1} x)^{3}$ with respect to $x$ .

Calculate $\lim _{h \rightarrow 0} \frac{10^{h}-1}{h}$ .

Find $h^{\prime}(9)$ if $h(x)=5 f(x)+4 g(x)+4$ with $f^{\prime}(9)=9$ and $g^{\prime}(9)=6$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\tan ^{-1}(1+h)-\pi / 4}{h}$ .

Find the rate of change of surface area $A$ with respect to weight $w$ when $A(w)=5(10)^{1-\log(w)}+1$ and $A=6 \mathrm{in}^2$ .

Given the function $f(x)=\frac{x^{2}}{x^{2}+3}$ , find:
(a) the intervals where $f$ is increasing and decreasing. (b) the local minimum value of $f$ . (c) the inflection points and intervals of concavity (up and down).

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{(2+\Delta x)^{(2+\Delta x)}-4}{\Delta x}$ .

Bestimmen Sie die Tangentengleichung an $f(x)=\frac{1}{2} x^{2}-3 x+1$ im Punkt $P$ für die gegebenen Koordinaten.

Given $f(x)=3 x^{4}-24 x^{2}+6$ , find: (A) $f^{\prime}(x)$ , (B) slope at $x=-1$ , (C) tangent line equation at $x=-1$ , (D) where tangent is horizontal.

Find the limit: $\lim _{w \rightarrow 2} \frac{3 \sec ^{-1} w-\pi}{w-2}$ .

A bacteria colony shrinks by $25\%$ daily. Find the decrease rate when it is half its original size. Also, find $\frac{d F}{-}$ and $\frac{d r}{-}$ for $F(r)=\frac{G m_{1} m_{2}}{2}$ .

Find the derivatives $\frac{d F}{d r}$ and $\frac{d r}{d F}$ for the function $F(r)=\frac{G m_{1} m_{2}}{r^{2}}$ .

Analyze the function $f(x)=\frac{x-x^{2}}{2-3 x+x^{2}}$ for intercepts, asymptotes, extrema, concavity, and inflection points, then sketch the curve.

Find the concentration $C(t)=\frac{0.2 t}{1+t^{2}}$ when it decreases by $0.016 \mathrm{mg} / \mathrm{cm}^{3}$ per hour.

Differentiate implicitly to find the slope and equation of the tangent line at $(-3,-3)$ for $2 x^{2}+x y=3 y^{2}$ .
(a) Find $\frac{d y}{d x}=$
(b) Find $y=$

Find the limit: $\lim _{x \rightarrow \infty} \frac{6}{\sqrt{x}}$ .

Find the instantaneous acceleration of a particle at $t=3.5$ given its displacement function $f(t)=4 t^{2} \ln (t)$ .

Find the slope of the tangent to the curve defined by $x=t^2+3t-8$ , $y=2t^2-2t-5$ at the point $(2,-1)$ .

Simplify the expression: $d = \frac{4(x-1)}{2}$

Find $d y / d x$ using logarithmic differentiation for $y=\frac{(4 x+1)^{2}}{6(2 x-1)^{3}}$ .

Find the horizontal asymptote of $f(x)=\frac{5x+6}{x^{2}+9x-10}$ .

Determine the horizontal asymptote of $f(x)=\frac{9x+2}{4x+9}$ .

Find the max and min values of $f(x)=6x-2$ on the interval [-4,3]. State the $\mathrm{x}$ -values where they occur.

Find the profit rate change if production increases by 500 games/week at $x=1500$ , given $C(x)=72000+60x$ and $R(x)=200x-\frac{x^{2}}{30}$ .

Find the rate of change of total resistance when $R_{1}=80$ ohms and $R_{2}=100$ ohms, given their rates of change.

Bestimmen Sie die Extrema und Sattelpunkte von $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-2 x$ für $-3,5 \leq x \leq 2,5$ .

Two vehicles are moving towards each other: one at $50 \mathrm{mph}$ west and the other at $60 \mathrm{mph}$ north. Find their approach speed when the west vehicle is 0.3 miles and the north vehicle is 0.4 miles from collision.

Find the instantaneous rate of change of blood sugar level $G(x)=450-0.2 x^{2}$ after 10 insulin units.

Given $f(x)=\frac{x+1}{x-2}$ , find the average rate of change over the intervals $[0,1]$ , $[0,0.5]$ , $[0,0.1]$ , $[0,0.01]$ , estimate the instantaneous rate at $x=0$ , and state the tangent line equation at that point.

Erklären Sie, warum die Potenzregel $g(x)=x^{0}=1$ für $x=0$ nicht gilt, obwohl $f^{\prime}(x)=1$ und $g^{\prime}(x)=0$ sind.

Bestimme die Ableitung von $f(x)$ an den Punkten $x_{0}$ : a) $x^{5}$ bei 2, b) $x^{2}$ bei -3, c) $x^{3}$ bei 4, d) $x^{4}$ bei -2, e) $x^{7}$ bei -1, f) $x$ bei 9.

Solve the differential equation: $y^{\prime \prime}-y^{\prime}-30 y=0$ using constants $c_{1}$ and $c_{2}$ .

Find the instantaneous rate of change for $f(x)=\frac{1}{x+1}$ and $g(x)=2x^2-5x+40$ . Then, find tangent lines at $x=2$ and $x=-1$ .

Find the instantaneous rate of change for: a. $f(x)=\frac{1}{x+1}$ , b. $g(x)=2 x^{2}-5 x+40$ . Then, find the tangent line at $x=2$ for $f(x)$ and at $x=-1$ for $g(x)$ .

Estimate the additional revenue from producing 5000 to 5001 units using the limit definition of the derivative for $R(x)=50 x-0.5 x^{2}$ .

Find the marginal price function for $p(x)=\frac{10 x}{1+x^{2}}$ and the tangent line at $x=2$ for given cases.

Bestimme die Ableitungen der Funktionen: a) $f(x)=x^{7}$ , b) $f(x)=x^{9}$ , c) $h(x)=x^{100}$ , d) $f(x)=x^{m+1}$ , e) $g(x)=x^{2 n}$ , f) $v(t)=t^{2}$ .

Solve the initial-value problem: $y^{\prime \prime}+3 y=0$ , with $y(0)=1$ and $y^{\prime}(0)=3$ .

Gegeben ist die Funktion $f(x)=-x^{2}+9$ . Bestimme für $0 \leq u \leq 3$ :
a) den Wert von $u$ , der den maximalen Flächeninhalt des Rechtecks gibt. b) den Wert von $u$ , der den maximalen Umfang des Rechtecks gibt.

Find the unemployment increase in 2016 using the derivative limit for $U(t)=\frac{\sqrt{t}+t}{10}$ , $t$ since 2000.

Find the unemployment increase in 2016 using the limit definition of the derivative for $U(t)=\frac{\sqrt{t}+t}{10}$ .

Find where the horizontal tangent line occurs for: a. $y=2 x^{3}+3 x^{2}-12 x+1$ b. $y=2 e^{x}-4 x$

Find the rate of change of community awareness $A(t)=10 \ln (2) t^{2} \cdot 2^{-t}$ after 4 weeks.

Evaluate the integral $\int_{-1}^{3} e^{x}+2 x^{2} \, dx$ .

Find the marginal profit for $P(x)=30 \ln (10) \cdot \log (3 x+1)$ at $x=33$ . Also, estimate profit change from 4500 to 4510 games.

Approximate the extra steel needed if a bearing's radius increases from $5 \mathrm{~mm}$ to $5.3 \mathrm{~mm}$ . Also, find the instantaneous rate of change for a \ $25,000 investment compounding at$ 8.4\%$ after 2 years.

Berechne die Kathetenlängen eines rechtwinkligen Dreiecks mit Hypotenuse an einer Wand, um die Fläche zu maximieren. $A(x)=-0,5 x^2 + 25 x$

Find the limit as $n$ approaches infinity: $\lim _{n \rightarrow \infty}\left(\frac{n+1}{n-m}\right)^{n}=$

Find the limit as $u$ approaches infinity: $\lim _{u \rightarrow \infty}\left(\frac{2+u}{3+u}\right)^{n}=$

Find the limit: $\lim _{u \rightarrow a}\left(\frac{u+2}{u+3}\right)^{u}$ .

Find the value of $x$ for $-4 \leq x \leq 4$ where $f$ has an absolute maximum, given $f'(x)$ and $f(0)=5$ .

Find where the function $f(x)=\sin^{2}(x)-\sin(x)$ is increasing for $0 \leq x \leq \frac{3\pi}{2}$ . Justify your answer.

Find the $I^{\text{th}}$ derivative of $f$ at the origin, where $I=(2,1)$ . Also, find the Taylor expansion of $\partial f / \partial x$ .

Find the limit: $\lim _{n \rightarrow \infty}\left(\frac{n+2}{n+3}\right)^{n}$ .