Calculus

Problem 4901

Differentiate the equation $x^{3}+y^{2}=24$ to find $\frac{d y}{d x}$ .

See Solution

Problem 4902

Find the derivative $\frac{d y}{d t}$ for the equation: $7 x^{3}+7 y^{3}=9$ .

See Solution

Problem 4903

Find the derivative $\frac{d y}{d t}$ for the equation: $5 x^{3}+y^{4}=3$ .

See Solution

Problem 4904

Find the derivative $\frac{d y}{d x}$ for the equation: $8 \sqrt[4]{x}+2 \sqrt{y}=5$ .

See Solution

Problem 4905

Find the slope of the tangent line to the curve $x^{3}+y^{2}=24$ at the point $(2,-4)$ .

See Solution

Problem 4906

Find the derivative $\frac{d y}{d x}$ for the equation: $6 x^{2}+4 x y+4 y^{3}=5$ .

See Solution

Problem 4907

Find $\frac{d y}{d t}$ for the equation $2 x^{4}-8 y^{4}=2$ .

See Solution

Problem 4908

Find $\frac{d y}{d t}$ for the equation $2 x^{2}+3 x y-6 y^{3}=8$ .

See Solution

Problem 4909

Gegeben ist $f(x)=e^{x}+\frac{1}{2} x$ und $g_{c}(x)=\frac{1}{2} x-c$ . Bestimmen Sie $c$ , sodass die Fläche zwischen $f$ und $g_{c}$ von $x=0$ bis $x=1$ gleich 3 ist.

See Solution

Problem 4910

A 1962 ft pole leans against a wall. If its base moves away at $3 \mathrm{ft/sec}$ , how fast is the top descending when it's 1638 ft high?

See Solution

Problem 4911

Untersuchen Sie das Flussbettprofil $f(x)=-2 x^{2} e^{x}$ für $x \in[-7; 0]$ und beantworten Sie Fragen zu Graphen, Tiefpunkt und Fließgeschwindigkeit.

See Solution

Problem 4912

Find the tangent line in slope-intercept form at $x=5$ for $y=f(x)$ where $f(x)=(4x^{2}-7)(x-1)$ .

See Solution

Problem 4913

Find the tangent line equation to the curve $y=\frac{6 x}{x+5}$ at $x=-3$ .

See Solution

Problem 4914

Find $\lim _{x \rightarrow 7} \sqrt[3]{f(x) g(x)+9}$ given $\lim _{x \rightarrow 7} f(x)=9$ and $\lim _{x \rightarrow 7} g(x)=2$ .

See Solution

Problem 4915

Find the limit of $3f(x)$ as $x$ approaches 1, given that $f(x) = 8$ .

See Solution

Problem 4916

Find $p(0)$ if $\lim _{x \rightarrow 0} \frac{p(x)}{q(x)}=3$ and $q(0)=5$ .

See Solution

Problem 4917

Find $\lim _{x \rightarrow 7} \frac{f(x)}{g(x)-h(x)}$ given $\lim _{x \rightarrow 7} f(x)=22$ , $\lim _{x \rightarrow 7} g(x)=7$ , $\lim _{x \rightarrow 7} h(x)=5$ .

See Solution

Problem 4918

Find $h^{\prime}(6)$ for $h(x)=\frac{f(x)}{g(x)}$ given $f(6)=2, f^{\prime}(6)=9, g(6)=5, g^{\prime}(6)=4$ .

See Solution

Problem 4919

Find the limit: $\lim _{x \rightarrow 3} \frac{-6 x}{\sqrt{7 x-12}}$

See Solution

Problem 4920

Find the slopes of the tangents for $g(x)=x^{2}f(x)$ and $h(x)=\frac{f(x)}{x-4}$ at $x=3$ .

See Solution

Problem 4921

Find the limit: $\lim _{x \rightarrow b} \frac{(x-b)^{60}-3 x+3 b}{x-b}$ .

See Solution

Problem 4922

Find the limit: $\lim _{x \rightarrow-4} \frac{4(2 x+1)^{2}-196}{x+4}$ .

See Solution

Problem 4923

Find the limit as $x$ approaches $b$ of $\frac{(x-b)^{60}-3x+3b}{x-b}$ or state if it doesn't exist.

See Solution

Problem 4924

Find the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ .

See Solution

Problem 4925

Simplify the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ .

See Solution

Problem 4926

Find the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ or state if it doesn't exist.

See Solution

Problem 4927

Gegeben die Funktion $f_{a}: x \mapsto e^{x}(x-a)$ , finde die Schnittpunkte mit den Achsen und das Verhalten für $x \rightarrow+\infty$ .

See Solution

Problem 4928

Find the limit: $\lim _{h \rightarrow 0} \frac{(9+h)^{2}-81}{h}$ .

See Solution

Problem 4929

Find the limit: $\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{2 x+23}-5}$ .

See Solution

Problem 4930

Find the general solution for the equation $\frac{d x}{d t}=\frac{\sqrt{1-x^{2}}}{\sqrt{1-t^{2}}}$ .

See Solution

Problem 4931

Evaluate the limit: $\lim _{x \rightarrow 9^{+}} \sqrt{x-9}$ .

See Solution

Problem 4932

Find $\lim _{x \rightarrow 14} E(x)$ where $E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}$ .

See Solution

Problem 4933

Find the limits of the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}+11, & x<-11 \\ \sqrt{x+11}, & x \geq-11\end{array}\right.$ at $x \rightarrow -11$ .

See Solution

Problem 4934

Find the integrating factor for the equation $t^{2} \frac{d x}{d t}=4 t-t^{5} x$ . Options include $I(t)=e^{1 / 2 t^{2}}$ , $I(t)=e^{-t^{6} / 6}$ , $I(t)=e^{t^{6} / 6}$ , $I(t)=e^{t^{4} / 4}$ , $I(t)=e^{-1 / 2 t^{2}}$ , $I(t)=e^{-t^{4} / 4}$ .

See Solution

Problem 4935

Find $\lim _{x \rightarrow 15} E(x)$ where $E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}$ .

See Solution

Problem 4936

Find the limit using the squeeze theorem: $\lim _{x \rightarrow 0} x \sin \left(\frac{6}{x}\right)$ .

See Solution

Problem 4937

Calculate $f^{\prime}(4)$ for the function $f(x)=\frac{2 x^{2}-2}{\sqrt{x}}$ .

See Solution

Problem 4938

Find the stationary points and their nature for the curve $y=x^{3}+5 x^{2}-8 x+1$ .

See Solution

Problem 4939

Find the limit $\lim _{x \rightarrow a} \frac{x^{5}-a^{5}}{x-a}$ using the factorization $x^{n}-a^{n}=(x-a)(\text{sum})$ .

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Problem 4940

Differentiate the function: $f(t)=\sqrt[4]{t}-\frac{1}{\sqrt[4]{t}}$

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Problem 4941

Calculate the limit: $\lim _{x \rightarrow 9^{-}} \frac{81-x^{2}}{|9-x|}$ .

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Problem 4942

Find the derivative of $y=\frac{f(x)+1}{f(x)+x}$ at $x=2$ given $f(2)=-3$ and $f'(2)=3$ .

See Solution

Problem 4943

Find the derivative of $y=\frac{f(x)+1}{f(x)+x}$ at $x=2$ , given $f(2)=-3$ and $f^{\prime}(2)=3$ .

See Solution

Problem 4944

Find the limit: $\lim _{x \rightarrow 9^{+}} \frac{x^{2}-81}{|9-x|}$ .

See Solution

Problem 4945

Find relative maxima and minima for the function $f(x)=x^{3}-3 x^{2}+1$ . Options: A, B, C, D.

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Problem 4946

A 25.0 kg trunk falls at 12.5 m/s. Find the height it fell from, assuming no air resistance. (7.96 m)

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Problem 4947

Berechnen Sie $U_4$ , $O_4$ , $U_8$ und $O_8$ für $f(x)=2-x$ über das Intervall $I=[0; 2]$ .

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Problem 4948

Find the derivative of $y$ at $x=5$ given $f(5)=-3$ and $f^{\prime}(5)=3$ where $y=\frac{f(x)+1}{f(x)+x}$ .

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Problem 4949

Can Rolle's Theorem be applied to $f(x)=x^{2/3}-1$ on $[-8,8]$ ? If yes, find $c$ where $f'(c)=0$ . Enter NA if not applicable.

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Problem 4950

Find the limit as $x$ approaches 5 from the right: $\lim _{x \rightarrow 5^{+}} \frac{(x-6)^{3}(x+2)}{(x^{2}+8)(x^{2}-3 x-10)}$

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Problem 4951

A car (1850 kg) starts at 28 m, moving at 32 m/s. With a frictional force of 35,000 N over 25 m, find its height at 18.5 m/s. (14.5 m)

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Problem 4952

Can Rolle's Theorem be applied to $f(x)=\frac{x^{2}-3 x-10}{x+4}$ on $[-2,5]$ ? If yes, find $c$ in $(a,b)$ .

See Solution

Problem 4953

Berechnen Sie $U_{4}$ , $O_{4}$ , $U_{8}$ und $O_{8}$ für $f(x)=\frac{1}{2} x^{2}$ im Intervall $I=[0; 1]$ .

See Solution

Problem 4954

A girl on a swing is 3.2 m high at the top and has a speed of $4.2 \mathrm{~m/s}$ at the bottom. What is her height at the bottom? $(2.3 \mathrm{~m})$

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Problem 4955

Find the derivative of $f(x)=\frac{2}{3}x^{3}+3x^{2}+4x$ .

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Problem 4956

Find the average velocity of an object dropped from 200m in the first 2 seconds. Then use the Mean Value Theorem to find instantaneous velocity.

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Problem 4957

Can the Mean Value Theorem apply to $f(x)=\frac{x+8}{x}$ on $[-1,2]$ ? If yes, find $c$ where $f^{\prime}(c)=\frac{f(2)-f(-1)}{3}$ .

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Problem 4958

Find the negative and positive values of $x$ where the horizontal tangents of $f(x)=2 x^{3}+12 x^{2}-126 x+22$ occur.

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Problem 4959

Berechne den Flächeninhalt zwischen dem Graphen von $\mathrm{f}$ und der X-Achse für die Intervalle: a) $I=[-1 ; 2]$ , b) $I=[-3 ; 2]$ , c) $I=[-3 ; 2]$ , d) $I=[1 ; 3]$ .

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Problem 4960

Find $f^{\prime}(-1)$ given $f(x)=x^{9} h(x)$ , $h(-1)=4$ , and $h^{\prime}(-1)=7$ .

See Solution

Problem 4961

Find the marginal revenue for the revenue function $R(q)=-6 q^{2}+200 q$ . Simplify your result. Answer: $M R(q)=$

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Problem 4962

Given the function $f(x)=2 x^{3}+3 x^{2}-36 x$ :
(a) Find the critical numbers of $f$ .
(b) Determine the intervals where $f$ is increasing or decreasing.
(c) Use the First Derivative Test to find relative extrema.

See Solution

Problem 4963

Find the derivative of $8 \sqrt{8 x^{7}+4 x^{10}}$ using the chain rule. Avoid fractional or negative exponents.

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Problem 4964

Analyze the limits of $f(x)=x^{9}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What are the results?

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Problem 4965

Find the derivative of $r=(\sec \theta+\tan \theta)^{-5}$ .

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Problem 4966

Calculate the total amount and interest from a \ $3500 investment at a continuous rate of$ 7.5\%$ for 7 years.

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Problem 4967

How much should the corporation invest now to have \$1,000,000 in 6 years at a 4% continuous compounding rate?

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Problem 4968

Find the derivative of $f(x) = e^{2x} - 4x$ .

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Problem 4969

Find the general solution to $\frac{d x}{d t}=\frac{\sqrt{1-x^{2}}}{\sqrt{1-t^{2}}}$ . Options include $x(t)=\arcsin (\sin t+C)$ , $x(t)=\sin (\arcsin t+C)$ , $x(t)=C t$ , $x(t)=\sin \arcsin (t+C)$ , $x(t)=t+C$ .

See Solution

Problem 4970

Finde den Punkt, an dem die Funktion $h(x)=e^{2 \cdot x}-4 \cdot x$ eine waagerechte Tangente hat.

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Problem 4971

Find the correct integrating factor for the equation $t^{2} \frac{d x}{d t}=4 t-t^{5} x$ . Options include $I(t)=e^{-t^{6} / 6}$ , $I(t)=e^{1 / 2 t^{2}}$ , $I(t)=e^{-1 / 2 t^{2}}$ , $I(t)=e^{-t^{4} / 4}$ , $I(t)=e^{t^{6} / 6}$ , $I(t)=e^{t^{4} / 4}$ .

See Solution

Problem 4972

Find the integral of $x^{2} e^{x / 2}$ with respect to $x$ .

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Problem 4973

Evaluate the integral: $\int \frac{\ln \left(15 x^{5}\right)}{x} d x$

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Problem 4974

Find the limit: $\lim _{t \rightarrow 0} \frac{\sqrt{2+t}-\sqrt{2-t}}{t}$ .

See Solution

Problem 4975

Find the limit as $t$ approaches 0 for the expression $\frac{2}{t}-\frac{2}{t^{2}+t}$ .

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Problem 4976

Let $y=(f(u)+6 x)^{2}$ with $u=x^{3}-2 x$ . Given $f(4)=7$ and $\frac{d y}{d x}=19$ at $x=2$ , find $f^{\prime}(4)$ .

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Problem 4977

Find the derivative of $y=x^{2}-2$ at $(2,2)$ and the equation of the tangent line there.

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Problem 4978

Find the derivative of the function $\frac{2 x-1}{x+\sqrt{x}+1}$ with respect to $x$ .

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Problem 4979

Find the value of $x$ where the function $f(x)=\frac{1}{(x^{2}+1)(x-2)}$ is discontinuous.

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Problem 4980

Find $\lim_{x \rightarrow 0} f(x)$ for the function $f(x)=1$ if $x<0$ and $f(x)=x+1$ if $x>0$ .

See Solution

Problem 4981

Find the derivative of the function: $\frac{d}{d x} \sqrt{4 x^{2}+16}$ .

See Solution

Problem 4982

Find $\frac{d y}{d x}$ for the equation $y^{3}-2 x y+x^{2}=5$ .

See Solution

Problem 4983

Find the derivative of the function $(\sqrt{x}-2)(x^{3}-2x+1)$ .

See Solution

Problem 4984

Find the derivative of $(\sqrt{x}-2)(x^{3}-2x+1)$ . Choices include various expressions involving derivatives.

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Problem 4985

Find the derivative of the function $\frac{2 x-1}{x+\sqrt{x}+1}$ . Choose the correct expression for the derivative.

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Problem 4986

Find the time for a spherical object to drop from $h=6r$ to $h=2r$ using the equation $\frac{d h}{d t}=-\frac{\alpha}{r} h$ .

See Solution

Problem 4987

Find the general solution to $\frac{d x}{d t}=\frac{\sqrt{1-x^{2}}}{\sqrt{1-t^{2}}}$ . Options include $x(t)=C t$ , etc.

See Solution

Problem 4988

Find the integrating factor for the linear differential equation: $t^{2} \frac{d x}{d t}=4 t-t^{5} x$ .

See Solution

Problem 4989

Let $h(x) = x^{2} \operatorname{sgn}(x)$ .
(a) Sketch the graph of $h$ . (b) Is $h$ continuous at 0? Explain briefly. (c) Is $h$ differentiable at 0? Explain briefly. (d) Does $h$ have a second derivative at 0? Explain briefly.

See Solution

Problem 4990

Find the derivative $\frac{d y}{d t}$ for the function $y=2 t(3 t^{2}-5)^{4}$ .

See Solution

Problem 4991

Determine which statements about the derivative $C'(t)$ of COVID-19 cases in Ontario are true.

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Problem 4992

Find the derivative of $y=(5x+2)^{4}(3x+2)^{-2}$ .

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Problem 4993

Find the derivative of the function $y=\sqrt{x+\sqrt{x}}$ .

See Solution

Problem 4994

Find $h^{\prime}(5)$ given that $h(x)=f(x) \cdot g(x)$ and use the values from the chart.

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Problem 4995

Find $h^{\prime}(5)$ if $h(x)=f(x)-g(x)$ and $f^{\prime}(5)=-1$ , $g^{\prime}(5)=8$ .

See Solution

Problem 4996

Differentiate $h(s)=\left(\frac{4 s^{2}}{s+1}\right)^{4}$ with respect to $s$ .

See Solution

Problem 4997

Find the amount of carbon-14 left after 7246 years using the model $A=16 e^{-0.000121 t}$ .

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Problem 4998

Find the tangent line to the curve $y=\sqrt{x^{2}-x+2}$ at the point where $x=2$ .

See Solution

Problem 4999

Find the tangent line to the curve $y=\sqrt{x^{2}-x+2}$ at the point where $x=2$ .

See Solution

Problem 5000

Find the derivative of the function $y=2 x^{2} \cos x$ , i.e., compute $\frac{d y}{d x}$ .

See Solution

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Calculus

Problem 4901

Differentiate the equation x3+y2=24x^{3}+y^{2}=24x3+y2=24 to find dydx\frac{d y}{d x}dxdy​.

Problem 4902

Find the derivative dydt\frac{d y}{d t}dtdy​ for the equation: 7x3+7y3=97 x^{3}+7 y^{3}=97x3+7y3=9.

Problem 4903

Find the derivative dydt\frac{d y}{d t}dtdy​ for the equation: 5x3+y4=35 x^{3}+y^{4}=35x3+y4=3.

Problem 4904

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation: 8x4+2y=58 \sqrt[4]{x}+2 \sqrt{y}=584x​+2y​=5.

Problem 4905

Find the slope of the tangent line to the curve x3+y2=24x^{3}+y^{2}=24x3+y2=24 at the point (2,−4)(2,-4)(2,−4).

Problem 4906

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation: 6x2+4xy+4y3=56 x^{2}+4 x y+4 y^{3}=56x2+4xy+4y3=5.

Problem 4907

Find dydt\frac{d y}{d t}dtdy​ for the equation 2x4−8y4=22 x^{4}-8 y^{4}=22x4−8y4=2.

Problem 4908

Find dydt\frac{d y}{d t}dtdy​ for the equation 2x2+3xy−6y3=82 x^{2}+3 x y-6 y^{3}=82x2+3xy−6y3=8.

Problem 4909

Gegeben ist f(x)=ex+12xf(x)=e^{x}+\frac{1}{2} xf(x)=ex+21​x und gc(x)=12x−cg_{c}(x)=\frac{1}{2} x-cgc​(x)=21​x−c. Bestimmen Sie ccc, sodass die Fläche zwischen fff und gcg_{c}gc​ von x=0x=0x=0 bis x=1x=1x=1 gleich 3 ist.

Problem 4910

A 1962 ft pole leans against a wall. If its base moves away at 3ft/sec3 \mathrm{ft/sec}3ft/sec, how fast is the top descending when it's 1638 ft high?

Problem 4911

Untersuchen Sie das Flussbettprofil f(x)=−2x2exf(x)=-2 x^{2} e^{x}f(x)=−2x2ex für x∈[−7;0]x \in[-7; 0]x∈[−7;0] und beantworten Sie Fragen zu Graphen, Tiefpunkt und Fließgeschwindigkeit.

Problem 4912

Find the tangent line in slope-intercept form at x=5x=5x=5 for y=f(x)y=f(x)y=f(x) where f(x)=(4x2−7)(x−1)f(x)=(4x^{2}-7)(x-1)f(x)=(4x2−7)(x−1).

Problem 4913

Find the tangent line equation to the curve y=6xx+5y=\frac{6 x}{x+5}y=x+56x​ at x=−3x=-3x=−3.

Problem 4914

Find lim⁡x→7f(x)g(x)+93\lim _{x \rightarrow 7} \sqrt[3]{f(x) g(x)+9}limx→7​3f(x)g(x)+9​ given lim⁡x→7f(x)=9\lim _{x \rightarrow 7} f(x)=9limx→7​f(x)=9 and lim⁡x→7g(x)=2\lim _{x \rightarrow 7} g(x)=2limx→7​g(x)=2.

Problem 4915

Find the limit of 3f(x)3f(x)3f(x) as xxx approaches 1, given that f(x)=8f(x) = 8f(x)=8.

Problem 4916

Find p(0)p(0)p(0) if lim⁡x→0p(x)q(x)=3\lim _{x \rightarrow 0} \frac{p(x)}{q(x)}=3limx→0​q(x)p(x)​=3 and q(0)=5q(0)=5q(0)=5.

Problem 4917

Problem 4918

Find h′(6)h^{\prime}(6)h′(6) for h(x)=f(x)g(x)h(x)=\frac{f(x)}{g(x)}h(x)=g(x)f(x)​ given f(6)=2,f′(6)=9,g(6)=5,g′(6)=4f(6)=2, f^{\prime}(6)=9, g(6)=5, g^{\prime}(6)=4f(6)=2,f′(6)=9,g(6)=5,g′(6)=4.

Problem 4919

Find the limit: lim⁡x→3−6x7x−12\lim _{x \rightarrow 3} \frac{-6 x}{\sqrt{7 x-12}}limx→3​7x−12​−6x​

Problem 4920

Find the slopes of the tangents for g(x)=x2f(x)g(x)=x^{2}f(x)g(x)=x2f(x) and h(x)=f(x)x−4h(x)=\frac{f(x)}{x-4}h(x)=x−4f(x)​ at x=3x=3x=3.

Problem 4921

Find the limit: lim⁡x→b(x−b)60−3x+3bx−b\lim _{x \rightarrow b} \frac{(x-b)^{60}-3 x+3 b}{x-b}limx→b​x−b(x−b)60−3x+3b​.

Problem 4922

Find the limit: lim⁡x→−44(2x+1)2−196x+4\lim _{x \rightarrow-4} \frac{4(2 x+1)^{2}-196}{x+4}limx→−4​x+44(2x+1)2−196​.

Problem 4923

Find the limit as xxx approaches bbb of (x−b)60−3x+3bx−b\frac{(x-b)^{60}-3x+3b}{x-b}x−b(x−b)60−3x+3b​ or state if it doesn't exist.

Problem 4924

Find the limit: lim⁡x→64x−8x−64\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}limx→64​x−64x​−8​.

Problem 4925

Simplify the limit: lim⁡x→64x−8x−64\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}limx→64​x−64x​−8​.

Problem 4926

Find the limit: lim⁡x→64x−8x−64\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}x→64lim​x−64x​−8​ or state if it doesn't exist.

Problem 4927

Gegeben die Funktion fa:x↦ex(x−a)f_{a}: x \mapsto e^{x}(x-a)fa​:x↦ex(x−a), finde die Schnittpunkte mit den Achsen und das Verhalten für x→+∞x \rightarrow+\inftyx→+∞.

Problem 4928

Find the limit: lim⁡h→0(9+h)2−81h\lim _{h \rightarrow 0} \frac{(9+h)^{2}-81}{h}limh→0​h(9+h)2−81​.

Problem 4929

Find the limit: lim⁡x→1x−12x+23−5\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{2 x+23}-5}limx→1​2x+23​−5x−1​.

Problem 4930

Find the general solution for the equation dxdt=1−x21−t2 \frac{d x}{d t}=\frac{\sqrt{1-x^{2}}}{\sqrt{1-t^{2}}} dtdx​=1−t2​1−x2​​.

Problem 4931

Evaluate the limit: lim⁡x→9+x−9\lim _{x \rightarrow 9^{+}} \sqrt{x-9}limx→9+​x−9​.

Problem 4932

Find lim⁡x→14E(x)\lim _{x \rightarrow 14} E(x)limx→14​E(x) where E(x)=4.35xx2+0.01E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}E(x)=xx2+0.01​4.35​.

Problem 4933

Find the limits of the piecewise function f(x)={x2+11,x<−11x+11,x≥−11f(x)=\left\{\begin{array}{ll}x^{2}+11, & x<-11 \\ \sqrt{x+11}, & x \geq-11\end{array}\right.f(x)={x2+11,x+11​,​x<−11x≥−11​ at x→−11x \rightarrow -11x→−11.

Problem 4934

Problem 4935

Find lim⁡x→15E(x)\lim _{x \rightarrow 15} E(x)limx→15​E(x) where E(x)=4.35xx2+0.01E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}E(x)=xx2+0.01​4.35​.

Problem 4936

Find the limit using the squeeze theorem: lim⁡x→0xsin⁡(6x)\lim _{x \rightarrow 0} x \sin \left(\frac{6}{x}\right)limx→0​xsin(x6​).

Problem 4937

Calculate f′(4)f^{\prime}(4)f′(4) for the function f(x)=2x2−2xf(x)=\frac{2 x^{2}-2}{\sqrt{x}}f(x)=x​2x2−2​.

Problem 4938

Find the stationary points and their nature for the curve y=x3+5x2−8x+1y=x^{3}+5 x^{2}-8 x+1y=x3+5x2−8x+1.

Problem 4939

Find the limit lim⁡x→ax5−a5x−a\lim _{x \rightarrow a} \frac{x^{5}-a^{5}}{x-a}limx→a​x−ax5−a5​ using the factorization xn−an=(x−a)(sum)x^{n}-a^{n}=(x-a)(\text{sum})xn−an=(x−a)(sum).

Problem 4940

Differentiate the function: f(t)=t4−1t4f(t)=\sqrt[4]{t}-\frac{1}{\sqrt[4]{t}}f(t)=4t​−4t​1​

Problem 4941

Differentiate the equation $x^{3}+y^{2}=24$ to find $\frac{d y}{d x}$ .

Find the derivative $\frac{d y}{d t}$ for the equation: $7 x^{3}+7 y^{3}=9$ .

Find the derivative $\frac{d y}{d t}$ for the equation: $5 x^{3}+y^{4}=3$ .

Find the derivative $\frac{d y}{d x}$ for the equation: $8 \sqrt[4]{x}+2 \sqrt{y}=5$ .

Find the slope of the tangent line to the curve $x^{3}+y^{2}=24$ at the point $(2,-4)$ .

Find the derivative $\frac{d y}{d x}$ for the equation: $6 x^{2}+4 x y+4 y^{3}=5$ .

Find $\frac{d y}{d t}$ for the equation $2 x^{4}-8 y^{4}=2$ .

Find $\frac{d y}{d t}$ for the equation $2 x^{2}+3 x y-6 y^{3}=8$ .

Gegeben ist $f(x)=e^{x}+\frac{1}{2} x$ und $g_{c}(x)=\frac{1}{2} x-c$ . Bestimmen Sie $c$ , sodass die Fläche zwischen $f$ und $g_{c}$ von $x=0$ bis $x=1$ gleich 3 ist.

A 1962 ft pole leans against a wall. If its base moves away at $3 \mathrm{ft/sec}$ , how fast is the top descending when it's 1638 ft high?

Untersuchen Sie das Flussbettprofil $f(x)=-2 x^{2} e^{x}$ für $x \in[-7; 0]$ und beantworten Sie Fragen zu Graphen, Tiefpunkt und Fließgeschwindigkeit.

Find the tangent line in slope-intercept form at $x=5$ for $y=f(x)$ where $f(x)=(4x^{2}-7)(x-1)$ .

Find the tangent line equation to the curve $y=\frac{6 x}{x+5}$ at $x=-3$ .

Find $\lim _{x \rightarrow 7} \sqrt[3]{f(x) g(x)+9}$ given $\lim _{x \rightarrow 7} f(x)=9$ and $\lim _{x \rightarrow 7} g(x)=2$ .

Find the limit of $3f(x)$ as $x$ approaches 1, given that $f(x) = 8$ .

Find $p(0)$ if $\lim _{x \rightarrow 0} \frac{p(x)}{q(x)}=3$ and $q(0)=5$ .

Find $h^{\prime}(6)$ for $h(x)=\frac{f(x)}{g(x)}$ given $f(6)=2, f^{\prime}(6)=9, g(6)=5, g^{\prime}(6)=4$ .

Find the limit: $\lim _{x \rightarrow 3} \frac{-6 x}{\sqrt{7 x-12}}$

Find the slopes of the tangents for $g(x)=x^{2}f(x)$ and $h(x)=\frac{f(x)}{x-4}$ at $x=3$ .

Find the limit: $\lim _{x \rightarrow b} \frac{(x-b)^{60}-3 x+3 b}{x-b}$ .

Find the limit: $\lim _{x \rightarrow-4} \frac{4(2 x+1)^{2}-196}{x+4}$ .

Find the limit as $x$ approaches $b$ of $\frac{(x-b)^{60}-3x+3b}{x-b}$ or state if it doesn't exist.

Find the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ .

Simplify the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ .

Find the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ or state if it doesn't exist.

Gegeben die Funktion $f_{a}: x \mapsto e^{x}(x-a)$ , finde die Schnittpunkte mit den Achsen und das Verhalten für $x \rightarrow+\infty$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{(9+h)^{2}-81}{h}$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{2 x+23}-5}$ .

Find the general solution for the equation $\frac{d x}{d t}=\frac{\sqrt{1-x^{2}}}{\sqrt{1-t^{2}}}$ .

Evaluate the limit: $\lim _{x \rightarrow 9^{+}} \sqrt{x-9}$ .

Find $\lim _{x \rightarrow 14} E(x)$ where $E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}$ .

Find the limits of the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}+11, & x<-11 \\ \sqrt{x+11}, & x \geq-11\end{array}\right.$ at $x \rightarrow -11$ .

Find $\lim _{x \rightarrow 15} E(x)$ where $E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}$ .

Find the limit using the squeeze theorem: $\lim _{x \rightarrow 0} x \sin \left(\frac{6}{x}\right)$ .

Calculate $f^{\prime}(4)$ for the function $f(x)=\frac{2 x^{2}-2}{\sqrt{x}}$ .

Find the stationary points and their nature for the curve $y=x^{3}+5 x^{2}-8 x+1$ .

Find the limit $\lim _{x \rightarrow a} \frac{x^{5}-a^{5}}{x-a}$ using the factorization $x^{n}-a^{n}=(x-a)(\text{sum})$ .

Differentiate the function: $f(t)=\sqrt[4]{t}-\frac{1}{\sqrt[4]{t}}$

Calculate the limit: $\lim _{x \rightarrow 9^{-}} \frac{81-x^{2}}{|9-x|}$ .

Find the derivative of $y=\frac{f(x)+1}{f(x)+x}$ at $x=2$ given $f(2)=-3$ and $f'(2)=3$ .

Find the derivative of $y=\frac{f(x)+1}{f(x)+x}$ at $x=2$ , given $f(2)=-3$ and $f^{\prime}(2)=3$ .

Find the limit: $\lim _{x \rightarrow 9^{+}} \frac{x^{2}-81}{|9-x|}$ .

Find relative maxima and minima for the function $f(x)=x^{3}-3 x^{2}+1$ . Options: A, B, C, D.

Berechnen Sie $U_4$ , $O_4$ , $U_8$ und $O_8$ für $f(x)=2-x$ über das Intervall $I=[0; 2]$ .

Find the derivative of $y$ at $x=5$ given $f(5)=-3$ and $f^{\prime}(5)=3$ where $y=\frac{f(x)+1}{f(x)+x}$ .

Can Rolle's Theorem be applied to $f(x)=x^{2/3}-1$ on $[-8,8]$ ? If yes, find $c$ where $f'(c)=0$ . Enter NA if not applicable.

Find the limit as $x$ approaches 5 from the right: $\lim _{x \rightarrow 5^{+}} \frac{(x-6)^{3}(x+2)}{(x^{2}+8)(x^{2}-3 x-10)}$

Can Rolle's Theorem be applied to $f(x)=\frac{x^{2}-3 x-10}{x+4}$ on $[-2,5]$ ? If yes, find $c$ in $(a,b)$ .

Berechnen Sie $U_{4}$ , $O_{4}$ , $U_{8}$ und $O_{8}$ für $f(x)=\frac{1}{2} x^{2}$ im Intervall $I=[0; 1]$ .

A girl on a swing is 3.2 m high at the top and has a speed of $4.2 \mathrm{~m/s}$ at the bottom. What is her height at the bottom? $(2.3 \mathrm{~m})$

Find the derivative of $f(x)=\frac{2}{3}x^{3}+3x^{2}+4x$ .

Can the Mean Value Theorem apply to $f(x)=\frac{x+8}{x}$ on $[-1,2]$ ? If yes, find $c$ where $f^{\prime}(c)=\frac{f(2)-f(-1)}{3}$ .

Find the negative and positive values of $x$ where the horizontal tangents of $f(x)=2 x^{3}+12 x^{2}-126 x+22$ occur.

Berechne den Flächeninhalt zwischen dem Graphen von $\mathrm{f}$ und der X-Achse für die Intervalle: a) $I=[-1 ; 2]$ , b) $I=[-3 ; 2]$ , c) $I=[-3 ; 2]$ , d) $I=[1 ; 3]$ .

Find $f^{\prime}(-1)$ given $f(x)=x^{9} h(x)$ , $h(-1)=4$ , and $h^{\prime}(-1)=7$ .

Find the marginal revenue for the revenue function $R(q)=-6 q^{2}+200 q$ . Simplify your result. Answer: $M R(q)=$

Given the function $f(x)=2 x^{3}+3 x^{2}-36 x$ :
(a) Find the critical numbers of $f$ .
(b) Determine the intervals where $f$ is increasing or decreasing.
(c) Use the First Derivative Test to find relative extrema.

Find the derivative of $8 \sqrt{8 x^{7}+4 x^{10}}$ using the chain rule. Avoid fractional or negative exponents.

Analyze the limits of $f(x)=x^{9}$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ . What are the results?

Find the derivative of $r=(\sec \theta+\tan \theta)^{-5}$ .

Calculate the total amount and interest from a \ $3500 investment at a continuous rate of$ 7.5\%$ for 7 years.

Find the derivative of $f(x) = e^{2x} - 4x$ .

Finde den Punkt, an dem die Funktion $h(x)=e^{2 \cdot x}-4 \cdot x$ eine waagerechte Tangente hat.

Find the integral of $x^{2} e^{x / 2}$ with respect to $x$ .

Evaluate the integral: $\int \frac{\ln \left(15 x^{5}\right)}{x} d x$

Find the limit: $\lim _{t \rightarrow 0} \frac{\sqrt{2+t}-\sqrt{2-t}}{t}$ .