Calculus

Problem 12801

Find the marginal cost function for the total-cost function $c=\frac{7 q^{2}}{\sqrt{q^{2}+2}}+5000$ .

See Solution

Problem 12802

Differentiate the function $y=\frac{5 x^{2}-9 x+7}{3 x+8}$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 12803

Differentiate the function $y=(9 x^{2}-6)(3 x^{2}-5 x+3)$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 12804

Find the derivative of the function $f(x)=x^{3}+4.5 x^{2}-12 x-3$ .

See Solution

Problem 12805

Given that $f(x)$ is positive and concave up on interval $I$ , answer these:
a.) $f^{\prime \prime}(x)>\square$ on $I$ . b.) $g^{\prime \prime}(x)=2\left(A^{2}+B f^{\prime \prime}(x)\right)$ , where $A=\square$ and $B=$ c.) $g^{\prime \prime}(x)>\square$ on $I$ . d.) $g(x)$ is $\square$ on $I$ .

See Solution

Problem 12806

Hopfenwachstum: Analysiere $w(t)=\frac{1}{3000} t^{3}-\frac{180}{3000} t^{2}+\frac{8100}{3000} t$ für $0 \leq t \leq 90$ . Aufgaben a) bis j).

See Solution

Problem 12807

Untersuchen Sie die Funktion $f(x) = \frac{1}{5}(x^{4}-8x^{3}+18x^{2})$ auf Symmetrie und Nullstellen, und bestimmen Sie die Wendepunkte.

See Solution

Problem 12808

Find the sum of the series: $\sum_{k=0}^{\infty} \frac{(2 x-3)^{k}}{4^{2 k}}$

See Solution

Problem 12809

Gegeben ist $f(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}$ . Finde waagerechte Tangenten, parallele zu $g(x)=-2x+5$ und den Berührpunkt der Tangente durch $P(0,\frac{9}{8})$ und $N(\frac{1}{2},0)$ .

See Solution

Problem 12810

Find the min and max of $f(x)=x e^{-9 x}$ for $0 \leq x \leq 2$ . Use the derivative to find local extrema and check boundaries.

See Solution

Problem 12811

Indicate T or F for these statements about continuous and differentiable functions on given intervals:
1. Continuous on $(-1,0]$ has a max.
2. Continuous on $[1,3]$ has max and min.
3. Differentiable on $(-2,1)$ has max and min.
4. Differentiable on $(-4,-3]$ has a min.

See Solution

Problem 12812

Gegeben ist die Funktion $f_{t}(x)=x \cdot(x-t)^{2}$ . Bestimme $\int_{0}^{t} f_{t}(x) d x$ und finde $t$ für $\int_{0}^{t} f_{t}(x) d x=108$ .

See Solution

Problem 12813

Wachstum einer Hopfenpflanze: Analysiere $w(t)=\frac{1}{3000} t^{3}-\frac{180}{3000} t^{2}+\frac{8100}{3000} t$ für $0 \leq t \leq 90$ . Aufgaben: a) Verlauf beschreiben, b) $w(20)$ berechnen, c) Nullstellen finden, d) max. Wachstumsrate bestimmen, e) stärkste Änderung, f) Höhe nach 60 Tagen, g) Höhe $h(t)=\frac{1}{12000} t^{4}-\frac{1}{50} t^{3}+\frac{27}{20} t^{2}$ zeigen, h) Höhe beim max. Wachstum, i) Zeit für 7m Höhe, j) durchschnittliches Wachstum.

See Solution

Problem 12814

Find the first and second derivatives of the function $f(x)=\frac{2}{x^{2}+1}$ .

See Solution

Problem 12815

Berechnen Sie das Integral $\int_{-1}^{1} e^{x} \, dx$

See Solution

Problem 12816

Calculate the integral $\int_{-2}^{3}(3 x-5) d x$ .

See Solution

Problem 12817

Show that if $y=\left(x+\sqrt{1+x^{2}}\right)^{\frac{1}{2}}$ , then $\left(1+x^{2}\right)\left(\frac{d y}{d x}\right)^{2}=\frac{1}{4} y^{2}$ and $\left(1+x^{2}\right)\left(\frac{d^{2} y}{d x^{2}}\right)+x \frac{d y}{d x}-\frac{1}{4} y=0$ .

See Solution

Problem 12818

Verpackung eines Halbzylinders:
a) Bestimme Zielgröße, Nebenbedingung und Zielfunktion für $V = 400 \mathrm{~cm}^{3}$ . Finde $r$ und $h$ für minimalen Materialverbrauch.
b) Vergleiche $r$ und Materialverbrauch für $h = 2r$ mit optimaler Lösung aus a).
c) Interpretiere den Schnittpunkt der Funktionen $M_{r}(V)=\pi \cdot r^{2}+\frac{2(\pi+2)}{\pi \cdot r} V$ .

See Solution

Problem 12819

Find the third derivative $f'''(x)$ of $f(x)$ : $f'''(x)=(-0.5x-0.875)(-0.125 e^{-0.125x})+(-0.5)e^{-0.125x}$ . Define $u$ , $v$ , $u'$ , and $v'$ .

See Solution

Problem 12820

Calculate the integral from -2 to 3 of the function $4 x^{2}-3 x+5$ .

See Solution

Problem 12821

Bestimme die Tangentengleichung von $f(x)=x^{3}-4x$ bei $x_{0}=1$ .

See Solution

Problem 12822

Bestimmen Sie die Tangentengleichung $f(x) = mx + b$ für $f(x) = 3x^2$ am Punkt $A(0, f(0))$ .

See Solution

Problem 12823

Schreibe die Funktion um, um einen konstanten Faktor zu erhalten, und leite dann ab. a) $f(x)=\frac{x^{4}}{2}$ b) $f(x)=\frac{7}{x}$ c) $f(x)=-\frac{3}{x^{2}}$ d) $f(x)=\frac{5}{3 x^{3}}$ e) $f(x)=(3 x)^{3}$ f) $f(x)=\sqrt{9 x}$

See Solution

Problem 12824

Calculate the integral $\int_{1}^{5} \frac{3}{x} d x$ .

See Solution

Problem 12825

Calculate the integral $\int_{0}^{4}(2 x-6) d x$ .

See Solution

Problem 12826

Calculate the integral: $\int_{0}^{9} \frac{2}{5} \sqrt{x} \, dx$

See Solution

Problem 12827

Berechnen Sie die folgenden Integrale: a) $\int_{0}^{2}(2+x)^{3} d x$ b) $\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) d x$ c) $\int_{0}^{2} \frac{1}{(x+1)^{2}} d x$ d) $\int_{-0.5}^{0} e^{2 x+1} d x$ e) $\int_{0}^{\pi} \sin (3 x-\pi) d x$ f) $\int_{-1}^{1} \frac{1}{5} e^{\frac{1}{2} x} d x$

See Solution

Problem 12828

Use the intermediate value theorem to check if $f(x)=x^{4}-4x^{2}-5$ has a zero between $a=4$ and $b=5$ .

See Solution

Problem 12829

Estimate the area under the curve for points $(-5,6)$ to $(1,1)$ using the left-endpoint method with 6 rectangles.

See Solution

Problem 12830

Find $\int y \, dx$ for: (i) $y=\frac{x^{5}+x^{6}}{x^{4}}$ , (ii) $y=\frac{(x^{2}+4x)^{2}}{x}$ , (iii) $y=\frac{8x^{5}-9x^{3}}{2x^{2}}$ , (iv) $y=\frac{(x^{2}+x)(x^{3}-x)}{x}$ .

See Solution

Problem 12831

Approximate the area under $f(x)=-\frac{3}{2} x^{2}+18 x+8$ from $[4,5]$ using a right Riemann sum with $n=8$ . Is it an underestimate or overestimate?

See Solution

Problem 12832

Approximate the area under $f(x)=-\frac{1}{2} x^{2}+6 x+2$ on $[4,5]$ using a right Riemann sum with $n=3$ . Is it an underestimate or overestimate?

See Solution

Problem 12833

For the function $y=\frac{x^{2}+1}{x}$ , find where it increases/decreases and identify local maxima/minima.

See Solution

Problem 12834

Find the slope of the tangent line to $\cos (\pi x)=x^{7} y^{2}$ at the point $(-1,1)$ .

See Solution

Problem 12835

Find the slope of the tangent line to $cos(\pi x) = x^{7} y^{2}$ at the point $(-1, 1)$ .

See Solution

Problem 12836

Find the indefinite integrals: (i) $\int(5 t+3)^{2} t^{3} d t$ (ii) $\int(y+1)^{3} d y$ (iii) $\int(2 x-3)(x+1)^{2} d x$ (iv) $\int \frac{x^{2}-1}{x+1} d x$

See Solution

Problem 12837

Approximate the area under $f(x)=\frac{x^{2}}{10}+1$ from $-6$ to $0$ using $n=3$ rectangles.
Area $\approx \square$ unit $^{2}$

See Solution

Problem 12838

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation of the circle $x^{2}+y^{2}=25$ .

See Solution

Problem 12839

Find the derivative of $y = x \sin x$ .

See Solution

Problem 12840

Approximate the area under $f(x)=\frac{x^{2}}{10}+1$ from $x=-7$ to $x=1$ using $n=4$ rectangles (left-endpoint).

See Solution

Problem 12841

Use a left Riemann sum with $n=7$ for $f(x)=-\frac{3}{2} x^{2}+9 x+2$ on $[4,5]$ . Is it an underestimate or overestimate?

See Solution

Problem 12842

Gravel is dumped at $18 \mathrm{~m}^{3} / \mathrm{t}$ . Find the height increase rate (in $\mathrm{m} / \mathrm{h}$ ) when height is $9 \mathrm{~m}$ .

See Solution

Problem 12843

Find if the left Riemann sum with $n=5$ for $f(x)=-3 x^{2}+24 x+4$ on $[2,3]$ is an underestimate or overestimate.

See Solution

Problem 12844

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

See Solution

Problem 12845

Berechne die Fläche zwischen dem Graphen von $f$ und der $x$ -Achse für die Intervalle: a) $f(x)=2 x^{2}+1$ , $I=[1 ; 2]$ ; b) $f(x)=x^{3}+x+1$ , $I=[2 ; 3]$ ; c) $f(x)=(2-x)^{2}$ , $I=[1 ; 3]$ .

See Solution

Problem 12846

Find the simplest form of $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=3x-4x^{2}$ .

See Solution

Problem 12847

Bestimmen Sie die Fläche unter $f(x)=\sqrt[3]{x}$ im Intervall $[0 ; 2]$ mit der Umkehrfunktion von $f$ .

See Solution

Problem 12848

Gegeben sind die Funktionen $f(x)=\frac{1}{8} x^{4}+\frac{9}{4} x^{2}-\frac{41}{4}$ und $g(x)=-\frac{9}{4} x^{2}-\frac{47}{8}$ .
Untersuchen Sie Nullstellen, Symmetrie, Extrema, Wendepunkte, und zeichnen Sie die Graphen für $-3,5 < x < 3,5$ . Bestimmen Sie Schnittpunkte und Parallelität der Graphen.

See Solution

Problem 12849

Eine Landefähre hat die Höhe $h(t)=-0,01 t^{3}+1,1 t^{2}-30 t+250$ für $0 \leq t \leq 50$ .
a) Skizzieren Sie $h(t)$ für $t=0$ bis $50$ mit Schrittweite 10. b) Finden Sie $v(t)$ , die vertikale Geschwindigkeit. c) Bestimmen Sie die Anfangsgeschwindigkeit und deren Bedeutung. d) Höhe und Geschwindigkeit bei $t=25 \mathrm{~s}$ ? Fällt oder steigt sie? e) Wann ist der minimale Abstand zur Oberfläche und wie groß ist er?

See Solution

Problem 12850

A car decelerates at $1.5 \mathrm{~m/s}^2$ to a final velocity of $18 \mathrm{~m/s}$ at $11$ seconds. Find $t$ .

See Solution

Problem 12851

Berechne die Fläche, die der Graph von $f$ mit der $x$ -Achse einschließt für: a) $f(x)=0,5 x^{2}-3 x$ , b) $f(x)=(x-1)^{2}-1$ , c) $f(x)=x^{4}-4 x^{2}$ .

See Solution

Problem 12852

Evaluate the integral $\int_{-1}^{1} \frac{\cos x}{1+e^{1 / x}} d x$ .

See Solution

Problem 12853

Bestimme die Ableitungsfunktion für die Funktionen a) $f(x)=\frac{1}{3} \cdot x$ , b) $f(x)=2 x-3$ , c) $f(x)=3 x^{2}$ , d) $f(x)=2 x^{3}$ und die Ableitung an der Stelle $x_{0}$ .

See Solution

Problem 12854

Bestimme die Ableitungsfunktion als Grenzwert des Differenzenquotienten für: a) $f(x)=\frac{1}{3} \cdot x$ , b) $f(x)=2 x-3$ , c) $f(x)=3 x^{2}$ , d) $f(x)=2 x^{3}$ .

See Solution

Problem 12855

Calculate the area between the curve $y=\frac{8}{x}$ and the x-axis on the interval $[1,5]$ .

See Solution

Problem 12856

Berechnen Sie die Fläche zwischen den Funktionen $f$ und $g$ für die Intervalle a) $[-1, 1]$ und b) $[0, 1]$ .

See Solution

Problem 12857

Bestimme die Ableitung der Funktionen an den angegebenen Stellen: a) $f(x)=3 x^{2}$ , $x_{0}=2$ b) $f(x)=x^{2}-2 x$ , $x_{0}=1$ c) $f(x)=5 x-1$ , $x_{0}=6$ d) $f(x)=x^{3}-5 x^{2}+7 x$ , $x_{0}=0$ e) $f(x)=5$ , $x_{0}=27$ f) $f(x)=4 x^{2}-5 x+2$ , $x_{0}=-2$

See Solution

Problem 12858

Find the area between the curve $f(x)=x^{2}-45$ and the $x$ -axis over the interval $[-2,1]$ .

See Solution

Problem 12859

Find the water rising rate in a cone (56 ft deep, 28 ft wide) filled at 15 ft³/min when water is 1 ft deep.

See Solution

Problem 12860

Identify the hypotheses and conclusions for Rolle's Theorem and the Mean Value Theorem based on the given lines.

See Solution

Problem 12861

Bestimme die Steigung von $f(x)=(2x-1)^{3}$ bei $P(1, f(1))$ , finde $a$ mit $g'(a)=-2$ für $g(x)=\frac{1}{(x-1)^{2}}$ und prüfe, ob $h(x)=\sqrt{3x+5}$ waagerechte Tangenten hat.

See Solution

Problem 12862

Check if these functions meet the Mean Value Theorem conditions: a) $f(x)=e^{3 x}$ on $[-1,2]$ b) $f(x)=|x-2|$ on $[0,3]$ c) $f(x)=\frac{1}{1+x}$ on $[-2,3]$ d) $f(x)=\frac{1}{1+x}$ on $[0,1]$

See Solution

Problem 12863

Find how $\frac{d V}{d t}$ relates to $\frac{d r}{d t}$ , $\frac{d h}{d t}$ for a cylinder with $V=\pi r^{2} h$ .

See Solution

Problem 12864

Find the rate of change of the distance from the particle at $(2,16)$ to the origin as $x$ increases at 5 units/sec on $y=4\sqrt{5x+6}$ .

See Solution

Problem 12865

A particle moves on the curve $y=4 \sqrt{5 x+6}$ . At $(2,16)$ , if $x$ increases at 5 units/sec, find the distance rate to the origin.

See Solution

Problem 12866

Bestimmen Sie die Ableitungen der Funktionen: a) $f(x)=(x^{2}-5)^{2}$ , b) $g(x)=(x^{2}+5x+1)^{2}$ , c) $h(x)=\sqrt{x}(x^{2}-3x)$ , d) $k(x)=(x^{2}-5)(x^{2}+5)$ .

See Solution

Problem 12867

Find $d u$ for the substitution $u=5 x$ in the integral $\int \cos(5 x) d x$ . Rewrite and evaluate the integral.

See Solution

Problem 12868

Find the difference quotient for $f(x)=\frac{4}{x}$ : $\frac{f(x+h)-f(x)}{h} =$

See Solution

Problem 12869

Berechne das Integral: $\int_{2}^{5} 4 x^{2}+2 x \, dx$

See Solution

Problem 12870

Given $g(x) = f(x^2 - x)$ , find $g'(3)$ using values from the table for $f(x)$ and $f'(x)$ .

See Solution

Problem 12871

Evaluate the integral using the substitution $u=z-8$ : $\int z \sqrt{z-8} d z$ . Use $C$ for the constant of integration.

See Solution

Problem 12872

Find the drug concentration $C=50\cdot\left(\frac{t}{9+t^{2}}\right)$ after 2 hours and when it drops to $0.5\%$ . What is the end behavior as $t \rightarrow \infty$ ?

See Solution

Problem 12873

Show that $\frac{d y}{d x}=\frac{3 y-2 x}{8 y-3 x}$ for the equation $x^{2}+4 y^{2}=7+3 x y$ .

See Solution

Problem 12874

Find $d u$ for the integral $\int \frac{x^{4}}{x^{5}-7} d x$ with $u=x^{5}-7$ . Rewrite and evaluate the integral.

See Solution

Problem 12875

Given the drug concentration formula $C=5 \cdot\left(\frac{t}{3+t^{2}}\right)$ , find: a. $C$ after 3 hours (2 decimal places). b. Time until $C$ drops to $0.6\%$ (2 decimal places). c. End behavior as $t \rightarrow \infty$ .

See Solution

Problem 12876

Evaluate the triple integral in cylindrical coordinates: $\int_{-2}^{2} \int_{0}^{\sqrt{4-x^{2}}} \int_{0}^{x^{2}+y^{2}} z \, dz \, dy \, dx$ . Provide a numerical answer.

See Solution

Problem 12877

The drug concentration $C=50 \cdot\left(\frac{t}{30+t^{2}}\right)$ . Find $C$ after 1 hour, time until $C=0.7\%$ , and end behavior as $t \to \infty$ .

See Solution

Problem 12878

Rewrite the integral $\int t^{7} e^{-t^{8}} d t$ using substitution to express it as $-\frac{1}{8} e^{u} d u$ . Find $u$ and $d u$ . Then evaluate the integral with constant $C$ .

See Solution

Problem 12879

Rewrite the integral $\int t^{7} e^{-t^{8}} d t$ using substitution to express it as $-\frac{1}{8} e^{u} d u$ and evaluate.

See Solution

Problem 12880

Leiten Sie folgende Funktionen mit der Produktregel ab: a) $f(x)=x \cdot e^{x}$ , b) $f(x)=3 x \cdot e^{x}$ , c) $f(x)=x^{2} \cdot e^{x}$ , d) $f(x)=(3 x^{3}+x^{2}) \cdot e^{x}$ , e) $f(x)=-x^{4} \cdot e^{x}$ , f) $f(x)=(x^{2}+x+1) \cdot (x^{3}+1)$ , g) $f(x)=\frac{2}{x} \cdot e^{x}$ , h) $f(x)=\sqrt{x} \cdot e^{x}$ , i) $f(x)=(3 x+2) \cdot \sqrt{x}$ , j) $f(x)=(2 \sqrt{x}+3) \cdot e^{x}$ , k) $f(x)=\sqrt{x} \cdot \frac{1}{x}$ , l) $f(x)=e^{x} \cdot \left(\frac{1}{x^{3}}+x\right)$ .

See Solution

Problem 12881

Find the differential $d \underline{u}$ in terms of $d \underline{x}$ using the Jacobian of the transformation $F(x,y,z)$ .

See Solution

Problem 12882

Evaluate the integral: $\int \sec^{2}(9 \theta) d \theta$ . Use $C$ for the constant of integration.

See Solution

Problem 12883

Evaluate the integral: $\int \frac{d x}{2 x+5}$ , using absolute values and $C$ for the constant of integration.

See Solution

Problem 12884

Evaluate the integral: $\int \frac{(\ln (x))^{42}}{x} d x$ (Use $C$ for the constant of integration.)

See Solution

Problem 12885

Find the indefinite integral: $\int \sec ^{2}(\theta) \tan ^{8}(\theta) d \theta$ (use $C$ for the constant).

See Solution

Problem 12886

Evaluate the integral: $\int_{\pi / 3}^{2 \pi / 3} \csc ^{2}\left(\frac{1}{2} t\right) d t$ .

See Solution

Problem 12887

If $\underline{u} = (u, v, w) = (xy + 3, x - z, y - xz)$ , find $d\underline{u}$ in terms of $d\underline{x}$ .

See Solution

Problem 12888

Find the function $f(x)$ if $f^{\prime \prime \prime \prime}(x) = \ln x$ . Integrate four times to find $f(x)$ .

See Solution

Problem 12889

How long does it take for an object to fall from 100 meters? Round to the nearest tenth of a second.

See Solution

Problem 12890

Berechne das Integral: $\int_{2}^{4} 4 e^{2 x} d x$

See Solution

Problem 12891

Analyze the curve on the $xy$ plane and determine where it is increasing, decreasing, and concave up/down.
(a) Increasing: $(1,3) \cup(4,6)$ (b) Decreasing: $(0,1) \cup(3,4)$ (c) Concave up: $(1,3) \cup(4,6)$ (d) Concave down: $(0,1) \cup(3,4)$ (e) Inflection points: $(1,2), (3,4), (4,1)$

See Solution

Problem 12892

Find the derivative of the function $f(x)=\frac{x^{2}+1}{3 x}$ .

See Solution

Problem 12893

Find the derivative of $f(x)=\frac{x^{2}+1}{3 x}$ using the power rule.

See Solution

Problem 12894

Berechne das Integral: $\int_{-5}^{5}(2 x-4)^{3} d x$

See Solution

Problem 12895

Evaluate the integral $\int t^{7} e^{-t^{8}} d t$ using the substitution $u=-t^{8}$ and find the constant of integration $\mathrm{C}$ .

See Solution

Problem 12896

Which integral calculates the moment of inertia for a unit-density spherical shell of radius $R$ about the $z$ -axis?

See Solution

Problem 12897

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

See Solution

Problem 12898

Evaluate the integral $\int \frac{e^{u}}{(8-e^{u})^{2}} du$ using substitution to express it as $-\frac{1}{x^{2}} dx$ . Find $x$ and $dx$ .

See Solution

Problem 12899

Find the derivative of $y$ with respect to $x$ for $y=\frac{\ln x}{4+5 \ln x}$ .

See Solution

Problem 12900

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

See Solution

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Calculus

Problem 12801

Find the marginal cost function for the total-cost function c=7q2q2+2+5000c=\frac{7 q^{2}}{\sqrt{q^{2}+2}}+5000c=q2+2​7q2​+5000.

Problem 12802

Differentiate the function y=5x2−9x+73x+8y=\frac{5 x^{2}-9 x+7}{3 x+8}y=3x+85x2−9x+7​. Find dydx\frac{d y}{d x}dxdy​.

Problem 12803

Differentiate the function y=(9x2−6)(3x2−5x+3)y=(9 x^{2}-6)(3 x^{2}-5 x+3)y=(9x2−6)(3x2−5x+3). What is dydx\frac{d y}{d x}dxdy​?

Problem 12804

Find the derivative of the function f(x)=x3+4.5x2−12x−3f(x)=x^{3}+4.5 x^{2}-12 x-3f(x)=x3+4.5x2−12x−3.

Problem 12805

Problem 12806

Hopfenwachstum: Analysiere w(t)=13000t3−1803000t2+81003000tw(t)=\frac{1}{3000} t^{3}-\frac{180}{3000} t^{2}+\frac{8100}{3000} tw(t)=30001​t3−3000180​t2+30008100​t für 0≤t≤900 \leq t \leq 900≤t≤90. Aufgaben a) bis j).

Problem 12807

Untersuchen Sie die Funktion f(x)=15(x4−8x3+18x2)f(x) = \frac{1}{5}(x^{4}-8x^{3}+18x^{2})f(x)=51​(x4−8x3+18x2) auf Symmetrie und Nullstellen, und bestimmen Sie die Wendepunkte.

Problem 12808

Find the sum of the series: ∑k=0∞(2x−3)k42k\sum_{k=0}^{\infty} \frac{(2 x-3)^{k}}{4^{2 k}}∑k=0∞​42k(2x−3)k​

Problem 12809

Gegeben ist f(x)=13x3−32x2f(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}f(x)=31​x3−23​x2. Finde waagerechte Tangenten, parallele zu g(x)=−2x+5g(x)=-2x+5g(x)=−2x+5 und den Berührpunkt der Tangente durch P(0,98)P(0,\frac{9}{8})P(0,89​) und N(12,0)N(\frac{1}{2},0)N(21​,0).

Problem 12810

Find the min and max of f(x)=xe−9xf(x)=x e^{-9 x}f(x)=xe−9x for 0≤x≤20 \leq x \leq 20≤x≤2. Use the derivative to find local extrema and check boundaries.

Problem 12811

Problem 12812

Gegeben ist die Funktion ft(x)=x⋅(x−t)2f_{t}(x)=x \cdot(x-t)^{2}ft​(x)=x⋅(x−t)2. Bestimme ∫0tft(x)dx\int_{0}^{t} f_{t}(x) d x∫0t​ft​(x)dx und finde ttt für ∫0tft(x)dx=108\int_{0}^{t} f_{t}(x) d x=108∫0t​ft​(x)dx=108.

Problem 12813

Problem 12814

Find the first and second derivatives of the function f(x)=2x2+1f(x)=\frac{2}{x^{2}+1}f(x)=x2+12​.

Problem 12815

Berechnen Sie das Integral ∫−11ex dx\int_{-1}^{1} e^{x} \, dx∫−11​exdx

Problem 12816

Calculate the integral ∫−23(3x−5)dx\int_{-2}^{3}(3 x-5) d x∫−23​(3x−5)dx.

Problem 12817

Problem 12818

Problem 12819

Problem 12820

Calculate the integral from -2 to 3 of the function 4x2−3x+54 x^{2}-3 x+54x2−3x+5.

Problem 12821

Bestimme die Tangentengleichung von f(x)=x3−4xf(x)=x^{3}-4xf(x)=x3−4x bei x0=1x_{0}=1x0​=1.

Problem 12822

Bestimmen Sie die Tangentengleichung f(x)=mx+bf(x) = mx + bf(x)=mx+b für f(x)=3x2f(x) = 3x^2f(x)=3x2 am Punkt A(0,f(0))A(0, f(0))A(0,f(0)).

Problem 12823

Problem 12824

Calculate the integral ∫153xdx\int_{1}^{5} \frac{3}{x} d x∫15​x3​dx.

Problem 12825

Calculate the integral ∫04(2x−6)dx\int_{0}^{4}(2 x-6) d x∫04​(2x−6)dx.

Problem 12826

Calculate the integral: ∫0925x dx\int_{0}^{9} \frac{2}{5} \sqrt{x} \, dx∫09​52​x​dx

Problem 12827

Problem 12828

Use the intermediate value theorem to check if f(x)=x4−4x2−5f(x)=x^{4}-4x^{2}-5f(x)=x4−4x2−5 has a zero between a=4a=4a=4 and b=5b=5b=5.

Problem 12829

Estimate the area under the curve for points (−5,6)(-5,6)(−5,6) to (1,1)(1,1)(1,1) using the left-endpoint method with 6 rectangles.

Problem 12830

Find ∫y dx\int y \, dx∫ydx for: (i) y=x5+x6x4y=\frac{x^{5}+x^{6}}{x^{4}}y=x4x5+x6​, (ii) y=(x2+4x)2xy=\frac{(x^{2}+4x)^{2}}{x}y=x(x2+4x)2​, (iii) y=8x5−9x32x2y=\frac{8x^{5}-9x^{3}}{2x^{2}}y=2x28x5−9x3​, (iv) y=(x2+x)(x3−x)xy=\frac{(x^{2}+x)(x^{3}-x)}{x}y=x(x2+x)(x3−x)​.

Problem 12831

Approximate the area under f(x)=−32x2+18x+8f(x)=-\frac{3}{2} x^{2}+18 x+8f(x)=−23​x2+18x+8 from [4,5][4,5][4,5] using a right Riemann sum with n=8n=8n=8. Is it an underestimate or overestimate?

Problem 12832

Approximate the area under f(x)=−12x2+6x+2f(x)=-\frac{1}{2} x^{2}+6 x+2f(x)=−21​x2+6x+2 on [4,5][4,5][4,5] using a right Riemann sum with n=3n=3n=3. Is it an underestimate or overestimate?

Problem 12833

For the function y=x2+1xy=\frac{x^{2}+1}{x}y=xx2+1​, find where it increases/decreases and identify local maxima/minima.

Problem 12834

Find the slope of the tangent line to cos⁡(πx)=x7y2\cos (\pi x)=x^{7} y^{2}cos(πx)=x7y2 at the point (−1,1)(-1,1)(−1,1).

Problem 12835

Find the slope of the tangent line to cos(πx)=x7y2cos(\pi x) = x^{7} y^{2}cos(πx)=x7y2 at the point (−1,1)(-1, 1)(−1,1).

Problem 12836

Find the indefinite integrals: (i) ∫(5t+3)2t3dt\int(5 t+3)^{2} t^{3} d t∫(5t+3)2t3dt (ii) ∫(y+1)3dy\int(y+1)^{3} d y∫(y+1)3dy (iii) ∫(2x−3)(x+1)2dx\int(2 x-3)(x+1)^{2} d x∫(2x−3)(x+1)2dx (iv) ∫x2−1x+1dx\int \frac{x^{2}-1}{x+1} d x∫x+1x2−1​dx

Problem 12837

Approximate the area under f(x)=x210+1f(x)=\frac{x^{2}}{10}+1f(x)=10x2​+1 from −6-6−6 to 000 using n=3n=3n=3 rectangles. Area ≈□\approx \square≈□ unit2^{2}2

Problem 12838

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for the equation of the circle x2+y2=25x^{2}+y^{2}=25x2+y2=25.

Problem 12839

Find the derivative of y=xsin⁡xy = x \sin xy=xsinx.

Problem 12840

Approximate the area under f(x)=x210+1f(x)=\frac{x^{2}}{10}+1f(x)=10x2​+1 from x=−7x=-7x=−7 to x=1x=1x=1 using n=4n=4n=4 rectangles (left-endpoint).

Problem 12841

Use a left Riemann sum with n=7n=7n=7 for f(x)=−32x2+9x+2f(x)=-\frac{3}{2} x^{2}+9 x+2f(x)=−23​x2+9x+2 on [4,5][4,5][4,5]. Is it an underestimate or overestimate?

Problem 12842

Gravel is dumped at 18 m3/t18 \mathrm{~m}^{3} / \mathrm{t}18 m3/t. Find the height increase rate (in m/h\mathrm{m} / \mathrm{h}m/h) when height is 9 m9 \mathrm{~m}9 m.

Problem 12843

Find if the left Riemann sum with n=5n=5n=5 for f(x)=−3x2+24x+4f(x)=-3 x^{2}+24 x+4f(x)=−3x2+24x+4 on [2,3][2,3][2,3] is an underestimate or overestimate.

Problem 12844

Find the marginal cost function for the total-cost function $c=\frac{7 q^{2}}{\sqrt{q^{2}+2}}+5000$ .

Differentiate the function $y=\frac{5 x^{2}-9 x+7}{3 x+8}$ . Find $\frac{d y}{d x}$ .

Differentiate the function $y=(9 x^{2}-6)(3 x^{2}-5 x+3)$ . What is $\frac{d y}{d x}$ ?

Find the derivative of the function $f(x)=x^{3}+4.5 x^{2}-12 x-3$ .

Hopfenwachstum: Analysiere $w(t)=\frac{1}{3000} t^{3}-\frac{180}{3000} t^{2}+\frac{8100}{3000} t$ für $0 \leq t \leq 90$ . Aufgaben a) bis j).

Untersuchen Sie die Funktion $f(x) = \frac{1}{5}(x^{4}-8x^{3}+18x^{2})$ auf Symmetrie und Nullstellen, und bestimmen Sie die Wendepunkte.

Find the sum of the series: $\sum_{k=0}^{\infty} \frac{(2 x-3)^{k}}{4^{2 k}}$

Gegeben ist $f(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}$ . Finde waagerechte Tangenten, parallele zu $g(x)=-2x+5$ und den Berührpunkt der Tangente durch $P(0,\frac{9}{8})$ und $N(\frac{1}{2},0)$ .

Find the min and max of $f(x)=x e^{-9 x}$ for $0 \leq x \leq 2$ . Use the derivative to find local extrema and check boundaries.

Gegeben ist die Funktion $f_{t}(x)=x \cdot(x-t)^{2}$ . Bestimme $\int_{0}^{t} f_{t}(x) d x$ und finde $t$ für $\int_{0}^{t} f_{t}(x) d x=108$ .

Find the first and second derivatives of the function $f(x)=\frac{2}{x^{2}+1}$ .

Berechnen Sie das Integral $\int_{-1}^{1} e^{x} \, dx$

Calculate the integral $\int_{-2}^{3}(3 x-5) d x$ .

Calculate the integral from -2 to 3 of the function $4 x^{2}-3 x+5$ .

Bestimme die Tangentengleichung von $f(x)=x^{3}-4x$ bei $x_{0}=1$ .

Bestimmen Sie die Tangentengleichung $f(x) = mx + b$ für $f(x) = 3x^2$ am Punkt $A(0, f(0))$ .

Calculate the integral $\int_{1}^{5} \frac{3}{x} d x$ .

Calculate the integral $\int_{0}^{4}(2 x-6) d x$ .

Calculate the integral: $\int_{0}^{9} \frac{2}{5} \sqrt{x} \, dx$

Use the intermediate value theorem to check if $f(x)=x^{4}-4x^{2}-5$ has a zero between $a=4$ and $b=5$ .

Estimate the area under the curve for points $(-5,6)$ to $(1,1)$ using the left-endpoint method with 6 rectangles.

Find $\int y \, dx$ for: (i) $y=\frac{x^{5}+x^{6}}{x^{4}}$ , (ii) $y=\frac{(x^{2}+4x)^{2}}{x}$ , (iii) $y=\frac{8x^{5}-9x^{3}}{2x^{2}}$ , (iv) $y=\frac{(x^{2}+x)(x^{3}-x)}{x}$ .

Approximate the area under $f(x)=-\frac{3}{2} x^{2}+18 x+8$ from $[4,5]$ using a right Riemann sum with $n=8$ . Is it an underestimate or overestimate?

Approximate the area under $f(x)=-\frac{1}{2} x^{2}+6 x+2$ on $[4,5]$ using a right Riemann sum with $n=3$ . Is it an underestimate or overestimate?

For the function $y=\frac{x^{2}+1}{x}$ , find where it increases/decreases and identify local maxima/minima.

Find the slope of the tangent line to $\cos (\pi x)=x^{7} y^{2}$ at the point $(-1,1)$ .

Find the slope of the tangent line to $cos(\pi x) = x^{7} y^{2}$ at the point $(-1, 1)$ .

Find the indefinite integrals: (i) $\int(5 t+3)^{2} t^{3} d t$ (ii) $\int(y+1)^{3} d y$ (iii) $\int(2 x-3)(x+1)^{2} d x$ (iv) $\int \frac{x^{2}-1}{x+1} d x$

Approximate the area under $f(x)=\frac{x^{2}}{10}+1$ from $-6$ to $0$ using $n=3$ rectangles.
Area $\approx \square$ unit $^{2}$

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation of the circle $x^{2}+y^{2}=25$ .

Find the derivative of $y = x \sin x$ .

Approximate the area under $f(x)=\frac{x^{2}}{10}+1$ from $x=-7$ to $x=1$ using $n=4$ rectangles (left-endpoint).

Use a left Riemann sum with $n=7$ for $f(x)=-\frac{3}{2} x^{2}+9 x+2$ on $[4,5]$ . Is it an underestimate or overestimate?

Gravel is dumped at $18 \mathrm{~m}^{3} / \mathrm{t}$ . Find the height increase rate (in $\mathrm{m} / \mathrm{h}$ ) when height is $9 \mathrm{~m}$ .

Find if the left Riemann sum with $n=5$ for $f(x)=-3 x^{2}+24 x+4$ on $[2,3]$ is an underestimate or overestimate.

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

Berechne die Fläche zwischen dem Graphen von $f$ und der $x$ -Achse für die Intervalle: a) $f(x)=2 x^{2}+1$ , $I=[1 ; 2]$ ; b) $f(x)=x^{3}+x+1$ , $I=[2 ; 3]$ ; c) $f(x)=(2-x)^{2}$ , $I=[1 ; 3]$ .

Find the simplest form of $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=3x-4x^{2}$ .

Bestimmen Sie die Fläche unter $f(x)=\sqrt[3]{x}$ im Intervall $[0 ; 2]$ mit der Umkehrfunktion von $f$ .

A car decelerates at $1.5 \mathrm{~m/s}^2$ to a final velocity of $18 \mathrm{~m/s}$ at $11$ seconds. Find $t$ .

Berechne die Fläche, die der Graph von $f$ mit der $x$ -Achse einschließt für: a) $f(x)=0,5 x^{2}-3 x$ , b) $f(x)=(x-1)^{2}-1$ , c) $f(x)=x^{4}-4 x^{2}$ .

Evaluate the integral $\int_{-1}^{1} \frac{\cos x}{1+e^{1 / x}} d x$ .

Bestimme die Ableitungsfunktion für die Funktionen a) $f(x)=\frac{1}{3} \cdot x$ , b) $f(x)=2 x-3$ , c) $f(x)=3 x^{2}$ , d) $f(x)=2 x^{3}$ und die Ableitung an der Stelle $x_{0}$ .

Bestimme die Ableitungsfunktion als Grenzwert des Differenzenquotienten für: a) $f(x)=\frac{1}{3} \cdot x$ , b) $f(x)=2 x-3$ , c) $f(x)=3 x^{2}$ , d) $f(x)=2 x^{3}$ .

Calculate the area between the curve $y=\frac{8}{x}$ and the x-axis on the interval $[1,5]$ .

Berechnen Sie die Fläche zwischen den Funktionen $f$ und $g$ für die Intervalle a) $[-1, 1]$ und b) $[0, 1]$ .

Find the area between the curve $f(x)=x^{2}-45$ and the $x$ -axis over the interval $[-2,1]$ .

Bestimme die Steigung von $f(x)=(2x-1)^{3}$ bei $P(1, f(1))$ , finde $a$ mit $g'(a)=-2$ für $g(x)=\frac{1}{(x-1)^{2}}$ und prüfe, ob $h(x)=\sqrt{3x+5}$ waagerechte Tangenten hat.

Find how $\frac{d V}{d t}$ relates to $\frac{d r}{d t}$ , $\frac{d h}{d t}$ for a cylinder with $V=\pi r^{2} h$ .

Find the rate of change of the distance from the particle at $(2,16)$ to the origin as $x$ increases at 5 units/sec on $y=4\sqrt{5x+6}$ .

A particle moves on the curve $y=4 \sqrt{5 x+6}$ . At $(2,16)$ , if $x$ increases at 5 units/sec, find the distance rate to the origin.

Bestimmen Sie die Ableitungen der Funktionen: a) $f(x)=(x^{2}-5)^{2}$ , b) $g(x)=(x^{2}+5x+1)^{2}$ , c) $h(x)=\sqrt{x}(x^{2}-3x)$ , d) $k(x)=(x^{2}-5)(x^{2}+5)$ .

Find $d u$ for the substitution $u=5 x$ in the integral $\int \cos(5 x) d x$ . Rewrite and evaluate the integral.

Find the difference quotient for $f(x)=\frac{4}{x}$ : $\frac{f(x+h)-f(x)}{h} =$

Berechne das Integral: $\int_{2}^{5} 4 x^{2}+2 x \, dx$

Given $g(x) = f(x^2 - x)$ , find $g'(3)$ using values from the table for $f(x)$ and $f'(x)$ .

Evaluate the integral using the substitution $u=z-8$ : $\int z \sqrt{z-8} d z$ . Use $C$ for the constant of integration.

Find the drug concentration $C=50\cdot\left(\frac{t}{9+t^{2}}\right)$ after 2 hours and when it drops to $0.5\%$ . What is the end behavior as $t \rightarrow \infty$ ?

Show that $\frac{d y}{d x}=\frac{3 y-2 x}{8 y-3 x}$ for the equation $x^{2}+4 y^{2}=7+3 x y$ .

Find $d u$ for the integral $\int \frac{x^{4}}{x^{5}-7} d x$ with $u=x^{5}-7$ . Rewrite and evaluate the integral.