Calculus

Problem 28701

Find the derivatives of these functions: a. $y=e^{x^{2}}$ b. $y=2 e^{2 x^{2}+3 x+1}$ c. $y=3 e^{2 x}+4 x\left(5^{x^{3}}\right)$

See Solution

Problem 28702

Berechnen Sie die Halbwertszeit von Cobalt-60 mit dem Zerfallsgesetz $N(t)=N_{0} \cdot e^{-0,13149 \cdot t}$ .

See Solution

Problem 28703

Bestimme die Ableitungsfunktion $f'$ für die folgenden Funktionen: a) $f(x)=\frac{1}{x}+3$ b) $f(x)=\frac{3}{x}-2$ c) $f(x)=2 x^{2}-3 x$ d) $f(x)=-4 x^{2}+2 x$

See Solution

Problem 28704

Find the derivatives of these functions: a. $y=e^{x^{2}}$ b. $y=2 e^{2 x^{2}+3 x+1}$ c. $y=3 e^{2 x}+4 x\left(5^{x^{3}}\right)$
Mark: 6 Find the derivatives of these functions: a. $y=3 x^{3} e^{4 x^{2}}$ b. $y=\frac{5 x^{2}}{10^{2 x+8}}$ c. $y=2 x\left(3 e^{x^{5}}\right)^{3}$

See Solution

Problem 28705

Find the surface area generated by revolving $x=\frac{y^{3}}{3}$ , $0 \leq y \leq 1$ , about the y-axis.

See Solution

Problem 28706

Find the length of the curve $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} \, dt$ from $y=-\frac{\pi}{3}$ to $y=\frac{\pi}{6}$ .

See Solution

Problem 28707

Find the derivatives of these functions: a. $y=3 x^{3} e^{4 x^{2}}$ b. $y=\frac{5 x^{2}}{10^{2 x+8}}$ c. $y=2 \times(3 e^{x^{5}})^{3}$

See Solution

Problem 28708

Find the derivative using first principles for: a) $f(x)=3 x^{2}+4 x+1$ b) $f(x)=-2 x^{3}+2 x+4$ .

See Solution

Problem 28709

Calculate the work done by a 50-N force along $(2,1,5)$ moving from A(2,1,5) to B(3,-1,2) in meters.

See Solution

Problem 28710

Determine if the series $\sum_{n=0}^{\infty} \frac{n^{50}}{3^{n}}$ converges or diverges.

See Solution

Problem 28711

Calculate the limit: $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ . Does it exist?

See Solution

Problem 28712

Calculate the limit: $\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}$ or state if it does not exist.

See Solution

Problem 28713

Differentiate these functions: (a) $y=\sin^{2}(e^{3x})$ (b) $y=\sqrt{1+\cos t+\sin^{2} t}$ (c) $y=(\sin^{2} x)(\tan^{2} x)$

See Solution

Problem 28714

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}$ . If it doesn't exist, state that.

See Solution

Problem 28715

Find the derivative $f^{\prime}(x)$ for these functions: a) $f(x)=\sin(x^2+3x+1)$ , b) $f(x)=\cos(\sin(2x))$ , c) $f(x)=2^{\sin(2x+3\cos(x))}$ .

See Solution

Problem 28716

Is the function $f(x)=\begin{cases}3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0\end{cases}$ continuous on R? (YES / NO) Does it have an inverse? (YES / NO)

See Solution

Problem 28717

Given the function $f(x)=\frac{x^{2}}{1+x+x^{2}}$ , determine if it has a max (YES/NO), min (YES/NO), and where it decreases.

See Solution

Problem 28718

Find the interval where $f(x)=3$ using the Intermediate Value Theorem, given $f(-2)=0$ , $f(0)=5$ , $f(2)=8$ , $f(4)=15$ .

See Solution

Problem 28719

Find the limit as $t$ approaches 0 for the expression $\frac{-8}{t}-4$ .

See Solution

Problem 28720

Find the stationary points of the curve $y=x^{3}-3 x^{2}-9 x+14$ and determine their nature.

See Solution

Problem 28721

Find the value of $a$ if $f(x)=\frac{6 x^{2}-2 x}{2 x}$ is continuous at $x=0$ and $f(0)=a$ .

See Solution

Problem 28722

Find the values of $x \in \mathbb{R}$ for which $f(x) = \sum_{n=0}^{\infty} e^{-2 n x}$ is defined and compute $f^{\circ}(x)$ .

See Solution

Problem 28723

Determine which statements about the function $f(x)=\left\{\begin{array}{ll} x^{2}-1, & \text { if } x \leq 2 \\ 3 & \text { if } x>2 \end{array}\right.$ are TRUE: I. $\lim _{x \rightarrow 2} f(x)$ exists II. $f(2)$ exists III. $f$ is continuous at $x=2$ .

See Solution

Problem 28724

Zeige, dass $f(x)=x^{3}-6 x^{2}+5 x$ bei $x=2$ eine Wendestelle hat und finde die Koordinaten des Wendepunkts W.

See Solution

Problem 28725

Zeige, dass die Funktion $f(x)=x^{3}-6 x^{2}+5 x$ an $x=2$ eine Wendestelle hat und finde die Koordinaten des Wendepunkts.

See Solution

Problem 28726

Finde die Punkte auf dem Graphen von $f$ , wo die Tangente parallel zu $y=-4x-2$ ist. a) $f(x)=x^{2}$ b) $f(x)=\frac{1}{x}$

See Solution

Problem 28727

Find the dimensions of a rectangular prism with volume $600 \mathrm{~m}^{3}$ that minimize the surface area.

See Solution

Problem 28728

A 13 ft ladder leans against a wall, top sliding down at 6 ft/sec. How fast is the base moving when it's 5 ft from the wall?

See Solution

Problem 28729

Find the slope of the secant line for $f(x) = 21 \cos x$ on the interval $\left[\frac{\pi}{2}, \pi\right]$ .

See Solution

Problem 28730

Find slopes of secant lines for $f(x)=21 \cos x$ at $x=\frac{\pi}{2}$ . Conjecture the tangent slope.

See Solution

Problem 28731

Bestimmen Sie die Hoch-, Tief- und Sattelpunkte von $f(x)=x^{3}-2x-5$ durch Ableitungen.

See Solution

Problem 28732

Find $x$ where the average rate of change of $f(x)=\sin x$ over $[0, \pi]$ equals the instantaneous rate on $(0, \pi)$ .

See Solution

Problem 28733

Finde die Stellen, an denen $f$ die Ableitung 2 hat, und bestimme $f^{\prime}(2)$ für die Funktionen: a) $f(x)=6 x^{2}-10 x+5$ b) $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-10 x+4,5$

See Solution

Problem 28734

Prove that the function $f(x)=\frac{1}{3} x^{3}-2 x^{2}+9 x$ has no relative extrema.

See Solution

Problem 28735

Find the derivative of $z=\frac{1}{x^{3}+1}$ at $x=4$ . Provide the exact answer using fractions.

See Solution

Problem 28736

Bestimme die Ableitung der Funktion $f(x) = -x^{2} + 4$ .

See Solution

Problem 28737

Find the derivative of the function $f(x) = -x^{2} + 4$ . What is $f'(x)$ ?

See Solution

Problem 28738

Find the local maximum and minimum points of the function $f(x)=5 x^{3}-15 x$ using the first derivative test.

See Solution

Problem 28739

Show that $f(x)=\frac{2}{3} x^{3}-2 x^{2}+7 x$ has no relative extrema.

See Solution

Problem 28740

Leite die Funktion $f(x)=\frac{1}{3} x^{3}+2 x^{2}+3 x$ ab.

See Solution

Problem 28741

Bestimme die Stammfunktion von $f(x)=\frac{1}{3} x^{3}+2 x^{2}+3 x$ .

See Solution

Problem 28742

Find the relative maximum and minimum points of the function $f(x)=x^{3}+9 x^{2}+5$ using the first-derivative test.

See Solution

Problem 28743

Berechne die Stammfunktion für die Funktion $f(x)=x^{4}-x^{3}$ .

See Solution

Problem 28744

Calculate the integral from 1 to 8 of $x \, dx$ .

See Solution

Problem 28745

Berechne die Stammfunktion von $f(x)=\frac{1}{4} x^{2}-1$ .

See Solution

Problem 28746

Calculate the integral: $2 \int_{0}^{6}(x+1) \, dx$

See Solution

Problem 28747

Calculate $H^{\prime}(3)$ for $H(x)=\frac{x}{g(x) f(x)}$ given $f(3)=6$ , $f^{\prime}(3)=-9$ , $g(3)=6$ , $g^{\prime}(3)=-2$ .

See Solution

Problem 28748

Find the derivative of the function $R(x) = 20000 + 100x - x^2$ .

See Solution

Problem 28749

Calculate the integral from -1 to 4 of the function $2x + 3$ .

See Solution

Problem 28750

Bestimme die Stammfunktion von $f(x)=\frac{1}{2} x^{3}-2 x$ .

See Solution

Problem 28751

Find the limit as $t$ approaches 0 for the expression $\frac{t}{t}$ .

See Solution

Problem 28752

Evaluate the integral: $3 \quad \int_{-1}^{4}(2 x+3) d x$ .

See Solution

Problem 28753

Find the rate of change: it decreases by 4 over 2 years. What is the rate of change in \$ per year?

See Solution

Problem 28754

Define critical points of a function and find them for $y=x^{3}-6 x^{2}$ , $y=x^{4}-8 x^{2}$ , $f(x)=\frac{2 x}{x^{2}+9}$ , $y=x^{3}+3 x^{2}+1$ . Use the first derivative test.

See Solution

Problem 28755

Bestimme die Tangentengleichung an den Punkten B für $f(x)=-2 x^{2}+4 x$ und $f(x)=\frac{3}{x^{2}}$ .

See Solution

Problem 28756

Bestimme die Stammfunktion von $f(x) = \frac{1}{3} x^{3} - 3x$ .

See Solution

Problem 28757

Calculate the integral from 2 to 3 of the function $(x^{2}-x)$ .

See Solution

Problem 28758

Berechne die Steigung der Tangente an $f(x)=\frac{3 x^{2}+4}{5} \cdot 2 x^{2}$ bei $x=-1$ .

See Solution

Problem 28759

Find the limits of the function $f(x)$ at the following points:
1. $\lim _{x \rightarrow -3^{-}} f(x)$
2. $\lim _{x \rightarrow -3^{+}} f(x)$
3. $\lim _{x \rightarrow -3} f(x)$
4. $\lim _{x \rightarrow 1^{+}} f(x)$

See Solution

Problem 28760

Die Funktion $g(x)=2 \cdot f(x)+5 x-1$ hat an der Stelle $x=5$ die Steigung $k_{g}=-1$ . Zeige dies rechnerisch.

See Solution

Problem 28761

Berechne den Flächeninhalt der Funktion $f(x)=\frac{1}{4} x^{4}-\frac{1}{4}$ im Intervall $[0 ; 2]$ .

See Solution

Problem 28762

Berechne den Flächeninhalt zwischen der x-Achse und $f(x)=\frac{1}{4} x^{4}-\frac{1}{4}$ im Intervall $[0 ; 2]$ .

See Solution

Problem 28763

Berechne die Stammfunktion von $f(x) = \frac{1}{4} x^{4} - \frac{1}{4}$ .

See Solution

Problem 28764

Bestimmen Sie das Verhalten der Funktion $g$ für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ für die gegebenen $g(x)$ .

See Solution

Problem 28765

Berechne die Ableitung von $f(x)=\frac{4}{2 x^{2}-1}$ .

See Solution

Problem 28766

Berechnen Sie die mittlere Änderungsrate für die Funktion $f$ in den Intervallen $I=[1;3]$ , $J=[-3;-1]$ , $K=[-1;2]$ für a) $f(x)=x^{2}$ , b) $f(x)=x^{3}+2x$ , c) $f(x)=2x^{2}-x$ .

See Solution

Problem 28767

Find critical points and use the first derivative test for local max/min for: a. $y=x^{4}-8 x^{2}$ , b. $f(x)=\frac{2 x}{x^{2}+9}$ , c. $y=x^{3}+3 x^{2}+1$ .

See Solution

Problem 28768

Bestimme die waagrechten und schrägen Asymptoten für die Funktionen $f(x)$ , $g(x)$ , $h(x)$ und $l(x)$ . Analysiere den Zusammenhang zwischen Grad der Polynome und Art der Asymptote.

See Solution

Problem 28769

Find the average velocity of the rocket for intervals $[4.499,4.5]$ and $[4.5,4.501]$ , then approximate the instantaneous velocity at $t=4.5$ .

See Solution

Problem 28770

Find the derivative of the function $\left(\frac{e^{x}}{x}\right)$ .

See Solution

Problem 28771

Differentiate the function $(x-1)^{2} e^{x}$ .

See Solution

Problem 28772

Calculate the integral: $\int\left(\frac{1}{x}+\frac{1}{x^{2}+1}\right) d x$

See Solution

Problem 28773

Finde die Nullstellen von $f''(x) = 3x^2 + 2x - 4 = 0$ und überprüfe die Wendepunkte bei $x_1 = 1$ und $x_2 = -\frac{5}{3}$ .

See Solution

Problem 28774

Bestimme die Stammfunktion von $f(x)=\frac{1}{3} x^{2}-\frac{4}{3}$ .

See Solution

Problem 28775

Finde die Stellen, an denen die Funktion $f$ die Ableitung 3 hat für die folgenden Funktionen: a) $f(x)=-x^{2}+x+6$ b) $f(x)=-\frac{12}{x}$ c) $f(x)=0,4 x^{5}-29 x$ d) $f(x)=\frac{1}{3} x^{3}+3,5 x^{2}+9 x-11$

See Solution

Problem 28776

Find the integral of $(\cos (x)+\sec (x))^{2}$ with respect to $x$ : $\int(\cos (x)+\sec (x))^{2} d x$ .

See Solution

Problem 28777

Find the integral of $\csc ^{2}(x)-2 e^{x}$ with respect to $x$ : $\int\left(\csc ^{2}(x)-2 e^{x}\right) d x$ .

See Solution

Problem 28778

Bestimme die Tangentengleichung an $f$ im Punkt B für die Funktionen: a) $5x^2-3x$ , b) $\frac{3}{x}$ , c) $-x^3+8x$ , d) $\sqrt{x}$ .

See Solution

Problem 28779

Find the limit as $n$ approaches infinity for $\frac{81}{20} \left(1+\frac{1}{n}\right)^2$ .

See Solution

Problem 28780

Calculate the integral: $\int \left(\frac{1}{4} x^{2} + x - \frac{5}{4}\right) dx$

See Solution

Problem 28781

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{2}}{\sqrt{x^{2}+12}}$ .

See Solution

Problem 28782

Leiten Sie die Funktionen ab und geben Sie $f^{\prime}(t)$ an: a) $f(x)=2 t x^{3}-\frac{3}{2} t^{2} x^{2}$ , b) $f(x)=(t-x)^{2}+\frac{t}{4} x^{4}$ .

See Solution

Problem 28783

Sketch the piecewise function $f(x)=\left\{\begin{array}{c}2, x<1 \\ 3, x=1 \\ x+1, x>1\end{array}\right.$ and find $\lim _{x \rightarrow 1} f(x)$ .

See Solution

Problem 28784

Evaluate the integrals using trigonometric substitution: a) $\int \sqrt{1-9 x^{2}} d x$ b) $\int \frac{d x}{x^{2} \sqrt{x^{2}+1}}$

See Solution

Problem 28785

Find $f^{(6)}(0)$ for the Taylor polynomial $p(x)=\sum_{n=0}^{10} \frac{2 x^{n}}{(n-1)!}$ .

See Solution

Problem 28786

Find the surface area of the curve $x=\sqrt{y}$ from $y=2$ to $y=6$ revolved around the $y$ -axis.

See Solution

Problem 28787

Evaluate the integral using partial fractions: $\int \frac{x+3}{2 x^{3}-8 x} d x$

See Solution

Problem 28788

Estimate the slope of the tangent line to $f(x)=4-3x^{3}$ at $P(2,-20)$ using secant lines with $x$ values near 2.

See Solution

Problem 28789

Solve the differential equation: $y^{\prime \prime}-2 y^{\prime}+5 y=\cos 2 x \cdot e^{x}$ .

See Solution

Problem 28790

Berechne die folgenden Integrale mit dem Hauptsatz der Differenzial- und Integralrechnung: a) $\int_{1}^{2} 4 x^{3}$ b) $\int_{-1}^{1}(9 x^{2}-1)$ c) $\int_{0}^{\frac{\pi}{2}} \sin (x)$ d) $\int_{1}^{3} \frac{1}{x^{2}}$ e) $\int_{0}^{4} x(x-1)$ f) $\int_{1}^{25} \frac{1}{\sqrt{x}}$

See Solution

Problem 28791

Berechne die folgenden Integrale mit dem Hauptsatz der Differenzial- und Integralrechnung: a) $\int_{1}^{2} 4 x^{3}$ , b) $\int_{-1}^{1}(9 x^{2}-1)$ , c) $\int_{0}^{\frac{\pi}{2}} \sin (x)$ , d) $\int_{1}^{3} \frac{1}{x^{2}}$ , e) $\int_{0}^{4} x(x-1)$ , f) $\int_{1}^{25} \frac{1}{\sqrt{x}}$ .

See Solution

Problem 28792

Find the surface area from revolving $y=\frac{x^{3}}{9}$ , for $0 \leq x \leq 2$ , around the x-axis.

See Solution

Problem 28793

Given piecewise functions, find the limits as $x$ approaches specific points for each function.

See Solution

Problem 28794

Find the limit: $\lim _{x \rightarrow 9} \frac{\sqrt{x}-3}{x-9}$ .

See Solution

Problem 28795

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x^{3}-1}$ .

See Solution

Problem 28796

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (5 x)}{\sin (4 x)}$ .

See Solution

Problem 28797

Berechnen Sie die mittlere Änderungsrate für $f(x)=x^{3}+2x$ in $I=[1;3]$ , $J=[-3;-1]$ , $K=[-1;2]$ und für $f(x)=2x^{2}-x$ .

See Solution

Problem 28798

Find the midpoint of the interval [1,4] using 6 subintervals. What is the width of each subinterval?

See Solution

Problem 28799

Given $f(x)=2 \tan \frac{5 \pi}{3}(x+1)-4$ , which statement is false: A) $(-0.4,-4)$ is an inflection point, B) asymptote $x=0.5$ , C) $f(0)=-2 \sqrt{3}-4$ , D) zero at $\frac{\pi}{5}$ ?

See Solution

Problem 28800

Find the derivative $\frac{d y}{d x}$ for the function $y = x^{2} - \csc(x) + 4$ .

See Solution

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Calculus

Problem 28701

Find the derivatives of these functions: a. y=ex2y=e^{x^{2}}y=ex2 b. y=2e2x2+3x+1y=2 e^{2 x^{2}+3 x+1}y=2e2x2+3x+1 c. y=3e2x+4x(5x3)y=3 e^{2 x}+4 x\left(5^{x^{3}}\right)y=3e2x+4x(5x3)

Problem 28702

Berechnen Sie die Halbwertszeit von Cobalt-60 mit dem Zerfallsgesetz N(t)=N0⋅e−0,13149⋅tN(t)=N_{0} \cdot e^{-0,13149 \cdot t}N(t)=N0​⋅e−0,13149⋅t.

Problem 28703

Bestimme die Ableitungsfunktion f′f'f′ für die folgenden Funktionen: a) f(x)=1x+3f(x)=\frac{1}{x}+3f(x)=x1​+3 b) f(x)=3x−2f(x)=\frac{3}{x}-2f(x)=x3​−2 c) f(x)=2x2−3xf(x)=2 x^{2}-3 xf(x)=2x2−3x d) f(x)=−4x2+2xf(x)=-4 x^{2}+2 xf(x)=−4x2+2x

Problem 28704

Problem 28705

Find the surface area generated by revolving x=y33x=\frac{y^{3}}{3}x=3y3​, 0≤y≤10 \leq y \leq 10≤y≤1, about the y-axis.

Problem 28706

Find the length of the curve x=∫0ysec⁡4t−1 dtx=\int_{0}^{y} \sqrt{\sec ^{4} t-1} \, dtx=∫0y​sec4t−1​dt from y=−π3y=-\frac{\pi}{3}y=−3π​ to y=π6y=\frac{\pi}{6}y=6π​.

Problem 28707

Find the derivatives of these functions: a. y=3x3e4x2y=3 x^{3} e^{4 x^{2}}y=3x3e4x2 b. y=5x2102x+8y=\frac{5 x^{2}}{10^{2 x+8}}y=102x+85x2​ c. y=2×(3ex5)3y=2 \times(3 e^{x^{5}})^{3}y=2×(3ex5)3

Problem 28708

Find the derivative using first principles for: a) f(x)=3x2+4x+1f(x)=3 x^{2}+4 x+1f(x)=3x2+4x+1 b) f(x)=−2x3+2x+4f(x)=-2 x^{3}+2 x+4f(x)=−2x3+2x+4.

Problem 28709

Calculate the work done by a 50-N force along (2,1,5)(2,1,5)(2,1,5) moving from A(2,1,5) to B(3,-1,2) in meters.

Problem 28710

Determine if the series ∑n=0∞n503n\sum_{n=0}^{\infty} \frac{n^{50}}{3^{n}}∑n=0∞​3nn50​ converges or diverges.

Problem 28711

Calculate the limit: lim⁡x→−∞exsin⁡(x2)\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)limx→−∞​exsin(x2). Does it exist?

Problem 28712

Calculate the limit: lim⁡x→−1−e−11−x2\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}x→−1−lim​e1−x2−1​ or state if it does not exist.

Problem 28713

Differentiate these functions: (a) y=sin⁡2(e3x)y=\sin^{2}(e^{3x})y=sin2(e3x) (b) y=1+cos⁡t+sin⁡2ty=\sqrt{1+\cos t+\sin^{2} t}y=1+cost+sin2t​ (c) y=(sin⁡2x)(tan⁡2x)y=(\sin^{2} x)(\tan^{2} x)y=(sin2x)(tan2x)

Problem 28714

Calculate the limit: lim⁡x→0sin⁡(3x)tan⁡(7x)\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}limx→0​tan(7x)sin(3x)​. If it doesn't exist, state that.

Problem 28715

Find the derivative f′(x)f^{\prime}(x)f′(x) for these functions: a) f(x)=sin⁡(x2+3x+1)f(x)=\sin(x^2+3x+1)f(x)=sin(x2+3x+1), b) f(x)=cos⁡(sin⁡(2x))f(x)=\cos(\sin(2x))f(x)=cos(sin(2x)), c) f(x)=2sin⁡(2x+3cos⁡(x))f(x)=2^{\sin(2x+3\cos(x))}f(x)=2sin(2x+3cos(x)).

Problem 28716

Is the function f(x)={3ln⁡(1+x)x≥0x5+xx<0f(x)=\begin{cases}3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0\end{cases}f(x)={3ln(1+x)x5+x​x≥0x<0​ continuous on R? (YES / NO) Does it have an inverse? (YES / NO)

Problem 28717

Given the function f(x)=x21+x+x2f(x)=\frac{x^{2}}{1+x+x^{2}}f(x)=1+x+x2x2​, determine if it has a max (YES/NO), min (YES/NO), and where it decreases.

Problem 28718

Find the interval where f(x)=3f(x)=3f(x)=3 using the Intermediate Value Theorem, given f(−2)=0f(-2)=0f(−2)=0, f(0)=5f(0)=5f(0)=5, f(2)=8f(2)=8f(2)=8, f(4)=15f(4)=15f(4)=15.

Problem 28719

Find the limit as ttt approaches 0 for the expression −8t−4\frac{-8}{t}-4t−8​−4.

Problem 28720

Find the stationary points of the curve y=x3−3x2−9x+14y=x^{3}-3 x^{2}-9 x+14y=x3−3x2−9x+14 and determine their nature.

Problem 28721

Find the value of a a a if f(x)=6x2−2x2x f(x)=\frac{6 x^{2}-2 x}{2 x} f(x)=2x6x2−2x​ is continuous at x=0 x=0 x=0 and f(0)=a f(0)=a f(0)=a.

Problem 28722

Find the values of x∈Rx \in \mathbb{R}x∈R for which f(x)=∑n=0∞e−2nxf(x) = \sum_{n=0}^{\infty} e^{-2 n x}f(x)=∑n=0∞​e−2nx is defined and compute f∘(x)f^{\circ}(x)f∘(x).

Problem 28723

Problem 28724

Zeige, dass f(x)=x3−6x2+5xf(x)=x^{3}-6 x^{2}+5 xf(x)=x3−6x2+5x bei x=2x=2x=2 eine Wendestelle hat und finde die Koordinaten des Wendepunkts W.

Problem 28725

Zeige, dass die Funktion f(x)=x3−6x2+5xf(x)=x^{3}-6 x^{2}+5 xf(x)=x3−6x2+5x an x=2x=2x=2 eine Wendestelle hat und finde die Koordinaten des Wendepunkts.

Problem 28726

Finde die Punkte auf dem Graphen von fff, wo die Tangente parallel zu y=−4x−2y=-4x-2y=−4x−2 ist. a) f(x)=x2f(x)=x^{2}f(x)=x2 b) f(x)=1xf(x)=\frac{1}{x}f(x)=x1​

Problem 28727

Find the dimensions of a rectangular prism with volume 600 m3600 \mathrm{~m}^{3}600 m3 that minimize the surface area.

Problem 28728

A 13 ft ladder leans against a wall, top sliding down at 6 ft/sec. How fast is the base moving when it's 5 ft from the wall?

Problem 28729

Find the slope of the secant line for f(x)=21cos⁡xf(x) = 21 \cos xf(x)=21cosx on the interval [π2,π]\left[\frac{\pi}{2}, \pi\right][2π​,π].

Problem 28730

Find slopes of secant lines for f(x)=21cos⁡xf(x)=21 \cos xf(x)=21cosx at x=π2x=\frac{\pi}{2}x=2π​. Conjecture the tangent slope.

Problem 28731

Bestimmen Sie die Hoch-, Tief- und Sattelpunkte von f(x)=x3−2x−5f(x)=x^{3}-2x-5f(x)=x3−2x−5 durch Ableitungen.

Problem 28732

Find xxx where the average rate of change of f(x)=sin⁡xf(x)=\sin xf(x)=sinx over [0,π][0, \pi][0,π] equals the instantaneous rate on (0,π)(0, \pi)(0,π).

Problem 28733

Finde die Stellen, an denen fff die Ableitung 2 hat, und bestimme f′(2)f^{\prime}(2)f′(2) für die Funktionen: a) f(x)=6x2−10x+5f(x)=6 x^{2}-10 x+5f(x)=6x2−10x+5 b) f(x)=13x3+12x2−10x+4,5f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-10 x+4,5f(x)=31​x3+21​x2−10x+4,5

Problem 28734

Prove that the function f(x)=13x3−2x2+9xf(x)=\frac{1}{3} x^{3}-2 x^{2}+9 xf(x)=31​x3−2x2+9x has no relative extrema.

Problem 28735

Find the derivative of z=1x3+1z=\frac{1}{x^{3}+1}z=x3+11​ at x=4x=4x=4. Provide the exact answer using fractions.

Problem 28736

Bestimme die Ableitung der Funktion f(x)=−x2+4f(x) = -x^{2} + 4f(x)=−x2+4.

Problem 28737

Find the derivative of the function f(x)=−x2+4f(x) = -x^{2} + 4f(x)=−x2+4. What is f′(x)f'(x)f′(x)?

Problem 28738

Find the local maximum and minimum points of the function f(x)=5x3−15xf(x)=5 x^{3}-15 xf(x)=5x3−15x using the first derivative test.

Problem 28739

Show that f(x)=23x3−2x2+7xf(x)=\frac{2}{3} x^{3}-2 x^{2}+7 xf(x)=32​x3−2x2+7x has no relative extrema.

Problem 28740

Leite die Funktion f(x)=13x3+2x2+3xf(x)=\frac{1}{3} x^{3}+2 x^{2}+3 xf(x)=31​x3+2x2+3x ab.

Problem 28741

Find the derivatives of these functions: a. $y=e^{x^{2}}$ b. $y=2 e^{2 x^{2}+3 x+1}$ c. $y=3 e^{2 x}+4 x\left(5^{x^{3}}\right)$

Berechnen Sie die Halbwertszeit von Cobalt-60 mit dem Zerfallsgesetz $N(t)=N_{0} \cdot e^{-0,13149 \cdot t}$ .

Bestimme die Ableitungsfunktion $f'$ für die folgenden Funktionen: a) $f(x)=\frac{1}{x}+3$ b) $f(x)=\frac{3}{x}-2$ c) $f(x)=2 x^{2}-3 x$ d) $f(x)=-4 x^{2}+2 x$

Find the surface area generated by revolving $x=\frac{y^{3}}{3}$ , $0 \leq y \leq 1$ , about the y-axis.

Find the length of the curve $x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} \, dt$ from $y=-\frac{\pi}{3}$ to $y=\frac{\pi}{6}$ .

Find the derivatives of these functions: a. $y=3 x^{3} e^{4 x^{2}}$ b. $y=\frac{5 x^{2}}{10^{2 x+8}}$ c. $y=2 \times(3 e^{x^{5}})^{3}$

Find the derivative using first principles for: a) $f(x)=3 x^{2}+4 x+1$ b) $f(x)=-2 x^{3}+2 x+4$ .

Calculate the work done by a 50-N force along $(2,1,5)$ moving from A(2,1,5) to B(3,-1,2) in meters.

Determine if the series $\sum_{n=0}^{\infty} \frac{n^{50}}{3^{n}}$ converges or diverges.

Calculate the limit: $\lim _{x \rightarrow-\infty} e^{x} \sin \left(x^{2}\right)$ . Does it exist?

Calculate the limit: $\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}$ or state if it does not exist.

Differentiate these functions: (a) $y=\sin^{2}(e^{3x})$ (b) $y=\sqrt{1+\cos t+\sin^{2} t}$ (c) $y=(\sin^{2} x)(\tan^{2} x)$

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\tan (7 x)}$ . If it doesn't exist, state that.

Find the derivative $f^{\prime}(x)$ for these functions: a) $f(x)=\sin(x^2+3x+1)$ , b) $f(x)=\cos(\sin(2x))$ , c) $f(x)=2^{\sin(2x+3\cos(x))}$ .

Is the function $f(x)=\begin{cases}3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0\end{cases}$ continuous on R? (YES / NO) Does it have an inverse? (YES / NO)

Given the function $f(x)=\frac{x^{2}}{1+x+x^{2}}$ , determine if it has a max (YES/NO), min (YES/NO), and where it decreases.

Find the interval where $f(x)=3$ using the Intermediate Value Theorem, given $f(-2)=0$ , $f(0)=5$ , $f(2)=8$ , $f(4)=15$ .

Find the limit as $t$ approaches 0 for the expression $\frac{-8}{t}-4$ .

Find the stationary points of the curve $y=x^{3}-3 x^{2}-9 x+14$ and determine their nature.

Find the value of $a$ if $f(x)=\frac{6 x^{2}-2 x}{2 x}$ is continuous at $x=0$ and $f(0)=a$ .

Find the values of $x \in \mathbb{R}$ for which $f(x) = \sum_{n=0}^{\infty} e^{-2 n x}$ is defined and compute $f^{\circ}(x)$ .

Zeige, dass $f(x)=x^{3}-6 x^{2}+5 x$ bei $x=2$ eine Wendestelle hat und finde die Koordinaten des Wendepunkts W.

Zeige, dass die Funktion $f(x)=x^{3}-6 x^{2}+5 x$ an $x=2$ eine Wendestelle hat und finde die Koordinaten des Wendepunkts.

Finde die Punkte auf dem Graphen von $f$ , wo die Tangente parallel zu $y=-4x-2$ ist. a) $f(x)=x^{2}$ b) $f(x)=\frac{1}{x}$

Find the dimensions of a rectangular prism with volume $600 \mathrm{~m}^{3}$ that minimize the surface area.

Find the slope of the secant line for $f(x) = 21 \cos x$ on the interval $\left[\frac{\pi}{2}, \pi\right]$ .

Find slopes of secant lines for $f(x)=21 \cos x$ at $x=\frac{\pi}{2}$ . Conjecture the tangent slope.

Bestimmen Sie die Hoch-, Tief- und Sattelpunkte von $f(x)=x^{3}-2x-5$ durch Ableitungen.

Find $x$ where the average rate of change of $f(x)=\sin x$ over $[0, \pi]$ equals the instantaneous rate on $(0, \pi)$ .

Finde die Stellen, an denen $f$ die Ableitung 2 hat, und bestimme $f^{\prime}(2)$ für die Funktionen: a) $f(x)=6 x^{2}-10 x+5$ b) $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-10 x+4,5$

Prove that the function $f(x)=\frac{1}{3} x^{3}-2 x^{2}+9 x$ has no relative extrema.

Find the derivative of $z=\frac{1}{x^{3}+1}$ at $x=4$ . Provide the exact answer using fractions.

Bestimme die Ableitung der Funktion $f(x) = -x^{2} + 4$ .

Find the derivative of the function $f(x) = -x^{2} + 4$ . What is $f'(x)$ ?

Find the local maximum and minimum points of the function $f(x)=5 x^{3}-15 x$ using the first derivative test.

Show that $f(x)=\frac{2}{3} x^{3}-2 x^{2}+7 x$ has no relative extrema.

Leite die Funktion $f(x)=\frac{1}{3} x^{3}+2 x^{2}+3 x$ ab.

Bestimme die Stammfunktion von $f(x)=\frac{1}{3} x^{3}+2 x^{2}+3 x$ .

Find the relative maximum and minimum points of the function $f(x)=x^{3}+9 x^{2}+5$ using the first-derivative test.

Berechne die Stammfunktion für die Funktion $f(x)=x^{4}-x^{3}$ .

Calculate the integral from 1 to 8 of $x \, dx$ .

Berechne die Stammfunktion von $f(x)=\frac{1}{4} x^{2}-1$ .

Calculate the integral: $2 \int_{0}^{6}(x+1) \, dx$

Calculate $H^{\prime}(3)$ for $H(x)=\frac{x}{g(x) f(x)}$ given $f(3)=6$ , $f^{\prime}(3)=-9$ , $g(3)=6$ , $g^{\prime}(3)=-2$ .

Find the derivative of the function $R(x) = 20000 + 100x - x^2$ .

Calculate the integral from -1 to 4 of the function $2x + 3$ .

Bestimme die Stammfunktion von $f(x)=\frac{1}{2} x^{3}-2 x$ .

Find the limit as $t$ approaches 0 for the expression $\frac{t}{t}$ .

Evaluate the integral: $3 \quad \int_{-1}^{4}(2 x+3) d x$ .

Define critical points of a function and find them for $y=x^{3}-6 x^{2}$ , $y=x^{4}-8 x^{2}$ , $f(x)=\frac{2 x}{x^{2}+9}$ , $y=x^{3}+3 x^{2}+1$ . Use the first derivative test.

Bestimme die Tangentengleichung an den Punkten B für $f(x)=-2 x^{2}+4 x$ und $f(x)=\frac{3}{x^{2}}$ .

Bestimme die Stammfunktion von $f(x) = \frac{1}{3} x^{3} - 3x$ .

Calculate the integral from 2 to 3 of the function $(x^{2}-x)$ .

Berechne die Steigung der Tangente an $f(x)=\frac{3 x^{2}+4}{5} \cdot 2 x^{2}$ bei $x=-1$ .

Die Funktion $g(x)=2 \cdot f(x)+5 x-1$ hat an der Stelle $x=5$ die Steigung $k_{g}=-1$ . Zeige dies rechnerisch.

Berechne den Flächeninhalt der Funktion $f(x)=\frac{1}{4} x^{4}-\frac{1}{4}$ im Intervall $[0 ; 2]$ .

Berechne den Flächeninhalt zwischen der x-Achse und $f(x)=\frac{1}{4} x^{4}-\frac{1}{4}$ im Intervall $[0 ; 2]$ .

Berechne die Stammfunktion von $f(x) = \frac{1}{4} x^{4} - \frac{1}{4}$ .

Bestimmen Sie das Verhalten der Funktion $g$ für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ für die gegebenen $g(x)$ .

Berechne die Ableitung von $f(x)=\frac{4}{2 x^{2}-1}$ .

Berechnen Sie die mittlere Änderungsrate für die Funktion $f$ in den Intervallen $I=[1;3]$ , $J=[-3;-1]$ , $K=[-1;2]$ für a) $f(x)=x^{2}$ , b) $f(x)=x^{3}+2x$ , c) $f(x)=2x^{2}-x$ .

Find critical points and use the first derivative test for local max/min for: a. $y=x^{4}-8 x^{2}$ , b. $f(x)=\frac{2 x}{x^{2}+9}$ , c. $y=x^{3}+3 x^{2}+1$ .

Bestimme die waagrechten und schrägen Asymptoten für die Funktionen $f(x)$ , $g(x)$ , $h(x)$ und $l(x)$ . Analysiere den Zusammenhang zwischen Grad der Polynome und Art der Asymptote.

Find the average velocity of the rocket for intervals $[4.499,4.5]$ and $[4.5,4.501]$ , then approximate the instantaneous velocity at $t=4.5$ .

Find the derivative of the function $\left(\frac{e^{x}}{x}\right)$ .

Differentiate the function $(x-1)^{2} e^{x}$ .

Calculate the integral: $\int\left(\frac{1}{x}+\frac{1}{x^{2}+1}\right) d x$

Finde die Nullstellen von $f''(x) = 3x^2 + 2x - 4 = 0$ und überprüfe die Wendepunkte bei $x_1 = 1$ und $x_2 = -\frac{5}{3}$ .

Bestimme die Stammfunktion von $f(x)=\frac{1}{3} x^{2}-\frac{4}{3}$ .