Calculus

Problem 30701

Leiten Sie die Funktion $g_{k}$ zweimal ab: a) $g_{k}(x)=e^{k x}$ b) $g_{k}(x)=k e^{2 x}-k$ c) $g_{k}(x)=k x+e^{k x}$ d) $g_{k}(x)=x e^{k x+1}$

See Solution

Problem 30702

Estimate the derivative $f^{\prime}(x)$ of the function $f(x)=4 \cdot 0.95^{x}$ using $h=0.001$ for $x=-3.5, -2, 0.5$ .

See Solution

Problem 30703

Find the derivative $f^{\prime}(x)$ of the function $f(x)=14x+7$ and evaluate it at $x=-9$ .

See Solution

Problem 30704

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

See Solution

Problem 30705

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

See Solution

Problem 30706

A $645 \mathrm{~N}$ diver drops from 15 m. Find speed at 5 m above water and just before hitting the water.

See Solution

Problem 30707

Find the derivative of $f(x)=5 x^{2}+5 x+6$ at $x=4$ using the limit definition of the derivative.

See Solution

Problem 30708

Find the derivative $f^{\prime}(5)$ if $f(x) = 6$ .

See Solution

Problem 30709

Find the slope of the tangent line to $f(x)=2 x^{2}$ at $x=2$ using the limit definition of the derivative.

See Solution

Problem 30710

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

See Solution

Problem 30711

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $4 x^{2}+2 y^{2}=18$ .

See Solution

Problem 30712

Differentiate the function: $f(x) = (3-x)^{4}(2+x)^{5}$ .

See Solution

Problem 30713

Find the slope of the tangent line to $y=\frac{3}{x}$ at $(8, \frac{3}{8})$ and write its equation as $y=m x+b$ .

See Solution

Problem 30714

Find the slope of the tangent line to $y=\frac{4}{x}$ at the point $\left(5, \frac{4}{5}\right)$ .

See Solution

Problem 30715

For $g(x)=x^{2}+1$ , what is the rate of change? Explain your reasoning.

See Solution

Problem 30716

Find the derivative of $f(x) = (1 - e^{-3x})^2$ .

See Solution

Problem 30717

Describe how to approximate the instantaneous rate of change at $x=a$ using average rates of change of a function.

See Solution

Problem 30718

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{2}{x^{2}}$ and evaluate it at $x=4$ .

See Solution

Problem 30719

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

See Solution

Problem 30720

Find the derivative of the function $y = \sqrt{\frac{3x - 2}{x - 3}}$ .

See Solution

Problem 30721

Find $\frac{d y}{d x}$ using implicit differentiation for $\cos (x y)=y^{2}+7$ .

See Solution

Problem 30722

Find the derivative of $y=\frac{x^{2}}{2-x}$ .

See Solution

Problem 30723

Find the function $f(x)$ and the number $a$ from the limit: $\lim_{h \rightarrow 0} \frac{6(3+h)^{4}-486}{h}$ .

See Solution

Problem 30724

Find the derivative of the function $f(x) = \frac{2}{1-e^{-4x}}$ .

See Solution

Problem 30725

What does the limit $\frac{f(4+h)-f(4)}{h}=8$ tell us about the graph of $f(x)=x^{2}$ ? Explain with a graph.

See Solution

Problem 30726

What does the limit $\lim_{h \rightarrow 0} \frac{f(4+h)-f(4)}{h} = 8$ tell us about the graph of $f(x)=x^{2}$ ? Explain with a graph.

See Solution

Problem 30727

Differentiate $(1+\ln x)^{2}$ .

See Solution

Problem 30728

Find the derivative $\frac{dy}{dx}$ for the function $y=\ln(1+2x)$ .

See Solution

Problem 30729

Gegeben ist die Funktion $f$ mit $f(x)=\frac{1}{8} x^{4}$ . Bestimme die Tangente und Normale an $f$ in den Punkten $P(2, f(2))$ und $Q(-1, f(-1))$ . Finde den Punkt, wo die Normale die Steigung $m=16$ hat.

See Solution

Problem 30730

Find the derivative of the function $f(x) = x^{2} \ln x$ .

See Solution

Problem 30731

Find the derivative of the function $f(x)=\sqrt{1-2 e^{4 x}}$ .

See Solution

Problem 30732

Differentiate the function $\frac{x^{3} \sqrt{4-x^{2}}}{5-\sqrt{x}}$ .

See Solution

Problem 30733

احسب مجال الدالة $f(x) = \frac{2x}{\ln(2x)}$ ، نقاط التقعر، ونقاط الانعطاف.

See Solution

Problem 30734

Find the maximum area of a rectangle under $f(x)=9-x^{2}$ with vertices on the graph and $x$ -axis.

See Solution

Problem 30735

Find the local maxima of the function $f(x) = -x^3 + x^2 + 2x$ .

See Solution

Problem 30736

Find the maximum area of a rectangle under $f(x)=9-x^{2}$ with vertices on the graph and $x$ -axis (in inches).

See Solution

Problem 30737

Find local minima of $f(x) = -x^{3} + x^{2} + 2x$ .

See Solution

Problem 30738

A proton ( $m=1.67 \times 10^{-27} \mathrm{~kg}$ ) enters an electric field ( $E=5000 \mathrm{~N}/\mathrm{C}$ ) with initial speed $v_0=3 \times 10^{5} \mathrm{~m/s}$ at $\alpha=30^\circ$ . Find the max vertical distance $h_{\text{max}}$ it descends. Ignore gravity.

See Solution

Problem 30739

Solve $f(x)=\ln(x^2-5x+6)=0$ and find $f'(x)=\frac{d}{dx}\ln(x^2-5x+6)$ .

See Solution

Problem 30740

Find the limit: $\lim _{n \rightarrow \infty} \frac{3 \cdot 2^{2 n}-3^{n-1}+1}{2^{n+5}+4 \cdot 3^{n}}$

See Solution

Problem 30741

Find the intervals where the function $f(x) = -x^3 + x^2 + 2x$ is decreasing.

See Solution

Problem 30742

Evaluate the integral $\int \frac{1}{x^{2}} \sqrt{1+\frac{1}{x}} d x$ using the substitution $u=1+\frac{1}{x}$ .

See Solution

Problem 30743

Differentiate $g(x)=\log_{43}(3x-2)$ to find $g'(x)$ .

See Solution

Problem 30744

Evaluate the integral $\int_{C} y^{2} dx + xy dy$ over the path $C$ bounded by $y=0$ , $y=\sqrt{x}$ , and $x=9$ .

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Problem 30745

Gegeben ist $f(x)=\frac{1}{8} x^{4}$ . Bestimme die Tangente und Normale an $P(2 \mid f(2))$ und $Q(-1 \mid f(-1))$ . Finde den Punkt, wo die Normale die Steigung $m=16$ hat.

See Solution

Problem 30746

Differentiate $g(x)=\log_{47}(7x-3)$ . Find $g'(x)=$

See Solution

Problem 30747

Find the critical points of $f(x)=1-(x+1)^{3}$ , $f^{\prime}=-3(x+1)^{2}$ , $f^{\prime \prime}=-6(x+1)$ .

See Solution

Problem 30748

Find the integral of $(5-3 x)^{10}$ with respect to $x$ .

See Solution

Problem 30749

Differentiate $y=\log_{10} x$ . What is $\frac{d}{d x} \log_{10} x=\square$ ?

See Solution

Problem 30750

Evaluate the integral: $\int \sin t \sqrt{1+\cos t} \, dt$ .

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Problem 30751

Differentiate $y=\log_{12} x$ and find $\frac{d}{dx} \log_{12} x=\square$ .

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Problem 30752

Find the derivative of $A=680(1.651)^{t}$ . What is $A^{\prime}=$ ? (Type an exact answer.)

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Problem 30753

Find the derivative of the function $y=7 \cdot 5^{x}$ . What is $y^{\prime}=$ ?

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Problem 30754

Find values of $a$ and $b$ for $f(x)=x^{3}+a x^{2}+b x-1$ with max at $x=-3$ and min at $x=-1$ .

See Solution

Problem 30755

Differentiate the function $g(t) = 27^t$ . Find $g'(t) =$ .

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Problem 30756

Differentiate the function $y=17^{x}$ . Find $\frac{d y}{d x}$ .

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Problem 30757

Find the limit expression for the area under $f(x)=x \cos (x)$ from $0$ to $\frac{\pi}{2}$ . Choices include: A. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{\pi i}{2 n} \cos \left(\frac{\pi i}{2 n}\right)\right)$ B. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n}\left(\frac{\pi}{2 n} \cos \left(\frac{\pi}{2 n}\right)\right)$ C. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n}\left(\cos \left(\frac{\pi i}{2 n}\right)\right)$ D. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n}\left(\frac{\pi i}{2 n} \cos \left(\frac{\pi i}{2 n}\right)\right)$ E. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi i}{2 n}\left(\frac{\pi i}{2 n} \cos \left(\frac{\pi i}{2 n}\right)\right)$

See Solution

Problem 30758

Differentiate the function: $y=9^{x}$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 30759

Differentiate $g(t)=29^{t}$ . Find $g^{\prime}(t)=\square$ .

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Problem 30760

Find the integral of $\frac{\sin x}{1+\cos ^{2} x}$ with respect to $x$ .

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Problem 30761

Evaluate the integral $\int \frac{\sin (\ln x)}{x} d x$ .

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Problem 30762

Differentiate $y=4^{x}$ . Find $\frac{d y}{d x}=\square$ .

See Solution

Problem 30763

Evaluate the integral $\int_{1}^{4} \frac{\sqrt{2+\sqrt{x}}}{\sqrt{x}} d x$ .

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Problem 30764

Find the derivative of the function $y=4 \cdot 5^{x}$ . What is $y^{\prime}=?$

See Solution

Problem 30765

Differentiate $g(t)=27^{t}$ . Find $g^{\prime}(t)=\square$ .

See Solution

Problem 30766

Find the derivative of $A=720(1.720)^{t}$ . What is $A^{\prime}=$ ? (Type an exact answer.)

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Problem 30767

Find the derivative of $A=470(1.731)^{t}$ . What is $A^{\prime}=$ ? (Type an exact answer.)

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Problem 30768

Differentiate the function $y=5^{x}$ . What is $\frac{d y}{d x}=?$

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Problem 30769

Estimate the total water drained in the first 3 minutes using left and right endpoint averages from rates: 60, 58, 56, 54, 52, 50, 48. Answer: \_\_\_\_\_ liters.

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Problem 30770

Bestimme den Schnittpunkt der Tangente an $f$ bei $B$ mit der $x$ -Achse für die Funktionen: a) $f(x)=0,5 x^{2}$ , b) $f(x)=\frac{1}{3} x^{3}+2 x^{2}$ , c) $f(x)=\frac{3}{x^{\prime}}$ .

See Solution

Problem 30771

Differentiate the function $y=14^{x}$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 30772

Find which area equals the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{4 n} \tan \left(\frac{i \pi}{4 n}\right)$ . Options: A, B, C, D.

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Problem 30773

Find the derivative of $y=2 \cdot 11^{x}$ . What is $y^{\prime}=?$

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Problem 30774

Differentiate the function $y=25^{x}$ . What is $\frac{d y}{d x}=?$

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Problem 30775

Differentiate the function $g(t) = 33^{t}$ . Find $g^{\prime}(t) =$ .

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Problem 30776

Find the derivative of $y=3 \cdot 7^{x}$ . What is $y^{\prime}=$ ?

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Problem 30777

Differentiate the function: $g(t)=16^{t}$ . Find $g^{\prime}(t)=$ .

See Solution

Problem 30778

Find the sea-level rise due to a $2^{\circ} \mathrm{C}$ temperature increase in a $3.8 \mathrm{~km}$ deep ocean, with $\varphi=207 \times 10^{-6}$ .

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Problem 30779

Find the derivative of $A=570(1.731)^{t}$ . What is $A^{\prime}=$ ? (Type an exact answer.)

See Solution

Problem 30780

Find the derivative of $A=670(1.662)^{t}$ . What is $A^{\prime}=$ (exact answer)?

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Problem 30781

Use the Desmos Riemann Sum Calculator to find $L_{5}, M_{5}, R_{5}, L_{25}, M_{25}, R_{25}, L_{100}, M_{100}, R_{100}$ for $f(x)=2x+1$ on $[1,4]$ . Then find the exact area under the curve.

See Solution

Problem 30782

Find $\frac{d x}{d t}$ for the function $x=9 t^{2}$ , where $19_{22}$ applies to the procedure in Examples 22.1-22.3.

See Solution

Problem 30783

Finde die Punkte, an denen die Tangente von $f$ einen Steigungswinkel von $21,8^{\circ}$ hat. a) $f(x)=5 x^{2}$ b) $f(x)=-\frac{40}{x}$ c) $f(x)=\frac{5}{6} x^{3}$ d) $f(x)=0,15$

See Solution

Problem 30784

Find the limit: $\lim _{r \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \frac{1}{k(k+1)}$

See Solution

Problem 30785

Approximate the integral $\int_{-2}^{5}(8 x-4) d x$ using the Midpoint Rule with $n$ subintervals. Find the limit as $n \rightarrow \infty$ .

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Problem 30786

Find the inflection point of the function $f(x)=x^{3}+6 x^{2}+9 x-1$ . Use $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ . Output as (x,y) or (none).

See Solution

Problem 30787

Find values of $a$ and $b$ for $f(x)=x^{3}+a x^{2}+b x-1$ with max at $x=-3$ and min at $x=-1$ .

See Solution

Problem 30788

Find the area under $f(x)=x^{3}$ from $x=0$ to $x=1$ . Choose the correct limit expression and evaluate it.

See Solution

Problem 30789

Find the relative minimum point of the function $f(x)=-x^{3}+3 x^{2}-4$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 30790

Find the relative maximum point of $f(x) = \frac{x^{2}-2}{x^{2}-1}$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 30791

Calculate the area between $f(x)=x^{2}$ and the $x$ -axis from $x=1$ to $x=10$ using right Riemann sums.

See Solution

Problem 30792

Find the limit: $\lim _{x \rightarrow 4} \frac{x-4}{x^{2}-5 x+4}$ . What is $f(4)$ ?

See Solution

Problem 30793

Calculate the area between $f(x)=x^{2}$ and the $x$ -axis on $[1,10]$ using right-endpoint Riemann sums.
a. Find $\Delta x$ in terms of $n$ : $\Delta x=\frac{9}{n}$ b. Find right endpoints $x_{1}, x_{2}, x_{3}$ in terms of $n$ : $x_{1}, x_{2}, x_{3}=\frac{1}{n}, \frac{2}{n}, \frac{3}{n}$ c. General expression for $x_{k}$ : $x_{k}=\frac{k}{n}$ d. Find $f\left(x_{k}\right)$ : $f\left(x_{k}\right)=\left(\frac{k}{n}\right)^{2}$ e. Find $f\left(x_{k}\right) \Delta x$ : $f\left(x_{k}\right) \Delta x=\left(\frac{k}{n}\right)^{2} \cdot \frac{9}{n}$ f. Find the right-endpoint Riemann sum value in terms of $n$ .

See Solution

Problem 30794

If $f(x)$ has a positive, decreasing derivative on interval $\mathrm{I}$ , what can we say about $f(x)$ ? Choose: a, b, c, d, or e.

See Solution

Problem 30795

Determine if the series $\sum_{n=1}^{\infty}\left(\frac{n+a}{n+b}\right)^{n^{2}}$ converges or diverges for real numbers $a, b$ with $b<a$ .

See Solution

Problem 30796

Find the relative minimum point of the function $f(x)=\frac{x^{2}-3}{x-2}$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 30797

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

See Solution

Problem 30798

Find the relative minimum point of the function $f(x)=x^{3}+6x^{2}+9x-1$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 30799

Find the relative minimum point for the function $f(x)=1-(x+1)^{3}$ with $f^{\prime}=-3(x+1)^{2}$ and $f^{\prime \prime}=-6(x+1)$ .

See Solution

Problem 30800

Find $f(x_k) \Delta x$ in terms of $k$ and $n$ , then the right-endpoint Riemann sum, and its limit as $n \to \infty$ .

See Solution

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Calculus

Problem 30701

Leiten Sie die Funktion gkg_{k}gk​ zweimal ab: a) gk(x)=ekxg_{k}(x)=e^{k x}gk​(x)=ekx b) gk(x)=ke2x−kg_{k}(x)=k e^{2 x}-kgk​(x)=ke2x−k c) gk(x)=kx+ekxg_{k}(x)=k x+e^{k x}gk​(x)=kx+ekx d) gk(x)=xekx+1g_{k}(x)=x e^{k x+1}gk​(x)=xekx+1

Problem 30702

Estimate the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=4⋅0.95xf(x)=4 \cdot 0.95^{x}f(x)=4⋅0.95x using h=0.001h=0.001h=0.001 for x=−3.5,−2,0.5x=-3.5, -2, 0.5x=−3.5,−2,0.5.

Problem 30703

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=14x+7f(x)=14x+7f(x)=14x+7 and evaluate it at x=−9x=-9x=−9.

Problem 30704

Find the derivative of the function y=x3(3−x)2y = x^{3}(3-x)^{2}y=x3(3−x)2.

Problem 30705

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

Problem 30706

A 645 N645 \mathrm{~N}645 N diver drops from 15 m. Find speed at 5 m above water and just before hitting the water.

Problem 30707

Find the derivative of f(x)=5x2+5x+6f(x)=5 x^{2}+5 x+6f(x)=5x2+5x+6 at x=4x=4x=4 using the limit definition of the derivative.

Problem 30708

Find the derivative f′(5)f^{\prime}(5)f′(5) if f(x)=6f(x) = 6f(x)=6.

Problem 30709

Find the slope of the tangent line to f(x)=2x2f(x)=2 x^{2}f(x)=2x2 at x=2x=2x=2 using the limit definition of the derivative.

Problem 30710

Solve the equation dydxx3+3xy2=3x3\frac{d y}{d x} x^{3}+3 x y^{2}=3 x^{3}dxdy​x3+3xy2=3x3 for yyy.

Problem 30711

Find dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the equation 4x2+2y2=184 x^{2}+2 y^{2}=184x2+2y2=18.

Problem 30712

Differentiate the function: f(x)=(3−x)4(2+x)5f(x) = (3-x)^{4}(2+x)^{5}f(x)=(3−x)4(2+x)5.

Problem 30713

Find the slope of the tangent line to y=3xy=\frac{3}{x}y=x3​ at (8,38)(8, \frac{3}{8})(8,83​) and write its equation as y=mx+by=m x+by=mx+b.

Problem 30714

Find the slope of the tangent line to y=4xy=\frac{4}{x}y=x4​ at the point (5,45)\left(5, \frac{4}{5}\right)(5,54​).

Problem 30715

For g(x)=x2+1g(x)=x^{2}+1g(x)=x2+1, what is the rate of change? Explain your reasoning.

Problem 30716

Find the derivative of f(x)=(1−e−3x)2f(x) = (1 - e^{-3x})^2f(x)=(1−e−3x)2.

Problem 30717

Describe how to approximate the instantaneous rate of change at x=ax=ax=a using average rates of change of a function.

Problem 30718

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=2x2f(x)=\frac{2}{x^{2}}f(x)=x22​ and evaluate it at x=4x=4x=4.

Problem 30719

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=(1+ex)2f(x)=(1+e^{x})^{2}f(x)=(1+ex)2.

Problem 30720

Find the derivative of the function y=3x−2x−3y = \sqrt{\frac{3x - 2}{x - 3}}y=x−33x−2​​.

Problem 30721

Find dydx\frac{d y}{d x}dxdy​ using implicit differentiation for cos⁡(xy)=y2+7\cos (x y)=y^{2}+7cos(xy)=y2+7.

Problem 30722

Find the derivative of y=x22−x y=\frac{x^{2}}{2-x} y=2−xx2​.

Problem 30723

Find the function f(x)f(x)f(x) and the number aaa from the limit: lim⁡h→06(3+h)4−486h\lim_{h \rightarrow 0} \frac{6(3+h)^{4}-486}{h}limh→0​h6(3+h)4−486​.

Problem 30724

Find the derivative of the function f(x)=21−e−4xf(x) = \frac{2}{1-e^{-4x}}f(x)=1−e−4x2​.

Problem 30725

What does the limit f(4+h)−f(4)h=8\frac{f(4+h)-f(4)}{h}=8hf(4+h)−f(4)​=8 tell us about the graph of f(x)=x2f(x)=x^{2}f(x)=x2? Explain with a graph.

Problem 30726

What does the limit lim⁡h→0f(4+h)−f(4)h=8\lim_{h \rightarrow 0} \frac{f(4+h)-f(4)}{h} = 8limh→0​hf(4+h)−f(4)​=8 tell us about the graph of f(x)=x2f(x)=x^{2}f(x)=x2? Explain with a graph.

Problem 30727

Differentiate (1+ln⁡x)2(1+\ln x)^{2}(1+lnx)2.

Problem 30728

Find the derivative dydx\frac{dy}{dx}dxdy​ for the function y=ln⁡(1+2x)y=\ln(1+2x)y=ln(1+2x).

Problem 30729

Gegeben ist die Funktion fff mit f(x)=18x4f(x)=\frac{1}{8} x^{4}f(x)=81​x4. Bestimme die Tangente und Normale an fff in den Punkten P(2,f(2))P(2, f(2))P(2,f(2)) und Q(−1,f(−1))Q(-1, f(-1))Q(−1,f(−1)). Finde den Punkt, wo die Normale die Steigung m=16m=16m=16 hat.

Problem 30730

Find the derivative of the function f(x)=x2ln⁡xf(x) = x^{2} \ln xf(x)=x2lnx.

Problem 30731

Find the derivative of the function f(x)=1−2e4xf(x)=\sqrt{1-2 e^{4 x}}f(x)=1−2e4x​.

Problem 30732

Differentiate the function x34−x25−x\frac{x^{3} \sqrt{4-x^{2}}}{5-\sqrt{x}}5−x​x34−x2​​.

Problem 30733

احسب مجال الدالة f(x)=2xln⁡(2x)f(x) = \frac{2x}{\ln(2x)}f(x)=ln(2x)2x​، نقاط التقعر، ونقاط الانعطاف.

Problem 30734

Find the maximum area of a rectangle under f(x)=9−x2f(x)=9-x^{2}f(x)=9−x2 with vertices on the graph and xxx-axis.

Problem 30735

Find the local maxima of the function f(x)=−x3+x2+2xf(x) = -x^3 + x^2 + 2xf(x)=−x3+x2+2x.

Problem 30736

Find the maximum area of a rectangle under f(x)=9−x2f(x)=9-x^{2}f(x)=9−x2 with vertices on the graph and xxx-axis (in inches).

Problem 30737

Find local minima of f(x)=−x3+x2+2xf(x) = -x^{3} + x^{2} + 2xf(x)=−x3+x2+2x.

Problem 30738

Problem 30739

Solve f(x)=ln⁡(x2−5x+6)=0f(x)=\ln(x^2-5x+6)=0f(x)=ln(x2−5x+6)=0 and find f′(x)=ddxln⁡(x2−5x+6)f'(x)=\frac{d}{dx}\ln(x^2-5x+6)f′(x)=dxd​ln(x2−5x+6).

Problem 30740

Find the limit: lim⁡n→∞3⋅22n−3n−1+12n+5+4⋅3n\lim _{n \rightarrow \infty} \frac{3 \cdot 2^{2 n}-3^{n-1}+1}{2^{n+5}+4 \cdot 3^{n}}limn→∞​2n+5+4⋅3n3⋅22n−3n−1+1​

Leiten Sie die Funktion $g_{k}$ zweimal ab: a) $g_{k}(x)=e^{k x}$ b) $g_{k}(x)=k e^{2 x}-k$ c) $g_{k}(x)=k x+e^{k x}$ d) $g_{k}(x)=x e^{k x+1}$

Estimate the derivative $f^{\prime}(x)$ of the function $f(x)=4 \cdot 0.95^{x}$ using $h=0.001$ for $x=-3.5, -2, 0.5$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=14x+7$ and evaluate it at $x=-9$ .

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

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

A $645 \mathrm{~N}$ diver drops from 15 m. Find speed at 5 m above water and just before hitting the water.

Find the derivative of $f(x)=5 x^{2}+5 x+6$ at $x=4$ using the limit definition of the derivative.

Find the derivative $f^{\prime}(5)$ if $f(x) = 6$ .

Find the slope of the tangent line to $f(x)=2 x^{2}$ at $x=2$ using the limit definition of the derivative.

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

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $4 x^{2}+2 y^{2}=18$ .

Differentiate the function: $f(x) = (3-x)^{4}(2+x)^{5}$ .

Find the slope of the tangent line to $y=\frac{3}{x}$ at $(8, \frac{3}{8})$ and write its equation as $y=m x+b$ .

Find the slope of the tangent line to $y=\frac{4}{x}$ at the point $\left(5, \frac{4}{5}\right)$ .

For $g(x)=x^{2}+1$ , what is the rate of change? Explain your reasoning.

Find the derivative of $f(x) = (1 - e^{-3x})^2$ .

Describe how to approximate the instantaneous rate of change at $x=a$ using average rates of change of a function.

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{2}{x^{2}}$ and evaluate it at $x=4$ .

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

Find the derivative of the function $y = \sqrt{\frac{3x - 2}{x - 3}}$ .

Find $\frac{d y}{d x}$ using implicit differentiation for $\cos (x y)=y^{2}+7$ .

Find the derivative of $y=\frac{x^{2}}{2-x}$ .

Find the function $f(x)$ and the number $a$ from the limit: $\lim_{h \rightarrow 0} \frac{6(3+h)^{4}-486}{h}$ .

Find the derivative of the function $f(x) = \frac{2}{1-e^{-4x}}$ .

What does the limit $\frac{f(4+h)-f(4)}{h}=8$ tell us about the graph of $f(x)=x^{2}$ ? Explain with a graph.

What does the limit $\lim_{h \rightarrow 0} \frac{f(4+h)-f(4)}{h} = 8$ tell us about the graph of $f(x)=x^{2}$ ? Explain with a graph.

Differentiate $(1+\ln x)^{2}$ .

Find the derivative $\frac{dy}{dx}$ for the function $y=\ln(1+2x)$ .

Gegeben ist die Funktion $f$ mit $f(x)=\frac{1}{8} x^{4}$ . Bestimme die Tangente und Normale an $f$ in den Punkten $P(2, f(2))$ und $Q(-1, f(-1))$ . Finde den Punkt, wo die Normale die Steigung $m=16$ hat.

Find the derivative of the function $f(x) = x^{2} \ln x$ .

Find the derivative of the function $f(x)=\sqrt{1-2 e^{4 x}}$ .

Differentiate the function $\frac{x^{3} \sqrt{4-x^{2}}}{5-\sqrt{x}}$ .

احسب مجال الدالة $f(x) = \frac{2x}{\ln(2x)}$ ، نقاط التقعر، ونقاط الانعطاف.

Find the maximum area of a rectangle under $f(x)=9-x^{2}$ with vertices on the graph and $x$ -axis.

Find the local maxima of the function $f(x) = -x^3 + x^2 + 2x$ .

Find the maximum area of a rectangle under $f(x)=9-x^{2}$ with vertices on the graph and $x$ -axis (in inches).

Find local minima of $f(x) = -x^{3} + x^{2} + 2x$ .

Solve $f(x)=\ln(x^2-5x+6)=0$ and find $f'(x)=\frac{d}{dx}\ln(x^2-5x+6)$ .

Find the limit: $\lim _{n \rightarrow \infty} \frac{3 \cdot 2^{2 n}-3^{n-1}+1}{2^{n+5}+4 \cdot 3^{n}}$

Find the intervals where the function $f(x) = -x^3 + x^2 + 2x$ is decreasing.

Evaluate the integral $\int \frac{1}{x^{2}} \sqrt{1+\frac{1}{x}} d x$ using the substitution $u=1+\frac{1}{x}$ .

Differentiate $g(x)=\log_{43}(3x-2)$ to find $g'(x)$ .

Evaluate the integral $\int_{C} y^{2} dx + xy dy$ over the path $C$ bounded by $y=0$ , $y=\sqrt{x}$ , and $x=9$ .

Gegeben ist $f(x)=\frac{1}{8} x^{4}$ . Bestimme die Tangente und Normale an $P(2 \mid f(2))$ und $Q(-1 \mid f(-1))$ . Finde den Punkt, wo die Normale die Steigung $m=16$ hat.

Differentiate $g(x)=\log_{47}(7x-3)$ . Find $g'(x)=$

Find the critical points of $f(x)=1-(x+1)^{3}$ , $f^{\prime}=-3(x+1)^{2}$ , $f^{\prime \prime}=-6(x+1)$ .

Find the integral of $(5-3 x)^{10}$ with respect to $x$ .

Differentiate $y=\log_{10} x$ . What is $\frac{d}{d x} \log_{10} x=\square$ ?

Evaluate the integral: $\int \sin t \sqrt{1+\cos t} \, dt$ .

Differentiate $y=\log_{12} x$ and find $\frac{d}{dx} \log_{12} x=\square$ .

Find the derivative of $A=680(1.651)^{t}$ . What is $A^{\prime}=$ ? (Type an exact answer.)

Find the derivative of the function $y=7 \cdot 5^{x}$ . What is $y^{\prime}=$ ?

Find values of $a$ and $b$ for $f(x)=x^{3}+a x^{2}+b x-1$ with max at $x=-3$ and min at $x=-1$ .

Differentiate the function $g(t) = 27^t$ . Find $g'(t) =$ .

Differentiate the function $y=17^{x}$ . Find $\frac{d y}{d x}$ .

Differentiate the function: $y=9^{x}$ . Find $\frac{d y}{d x}$ .

Differentiate $g(t)=29^{t}$ . Find $g^{\prime}(t)=\square$ .

Find the integral of $\frac{\sin x}{1+\cos ^{2} x}$ with respect to $x$ .

Evaluate the integral $\int \frac{\sin (\ln x)}{x} d x$ .

Differentiate $y=4^{x}$ . Find $\frac{d y}{d x}=\square$ .

Evaluate the integral $\int_{1}^{4} \frac{\sqrt{2+\sqrt{x}}}{\sqrt{x}} d x$ .

Find the derivative of the function $y=4 \cdot 5^{x}$ . What is $y^{\prime}=?$

Differentiate $g(t)=27^{t}$ . Find $g^{\prime}(t)=\square$ .

Find the derivative of $A=720(1.720)^{t}$ . What is $A^{\prime}=$ ? (Type an exact answer.)

Find the derivative of $A=470(1.731)^{t}$ . What is $A^{\prime}=$ ? (Type an exact answer.)

Differentiate the function $y=5^{x}$ . What is $\frac{d y}{d x}=?$

Bestimme den Schnittpunkt der Tangente an $f$ bei $B$ mit der $x$ -Achse für die Funktionen: a) $f(x)=0,5 x^{2}$ , b) $f(x)=\frac{1}{3} x^{3}+2 x^{2}$ , c) $f(x)=\frac{3}{x^{\prime}}$ .

Differentiate the function $y=14^{x}$ . Find $\frac{d y}{d x}$ .

Find which area equals the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{4 n} \tan \left(\frac{i \pi}{4 n}\right)$ . Options: A, B, C, D.

Find the derivative of $y=2 \cdot 11^{x}$ . What is $y^{\prime}=?$

Differentiate the function $y=25^{x}$ . What is $\frac{d y}{d x}=?$