Calculus

Problem 32901

Determine values of $a$ for which $\lim _{x \rightarrow a} f(x)$ exists based on the piecewise function described.

See Solution

Problem 32902

Find the local maximum of $f(x)=-0.8 x^{3}+3 x^{2}-1$ between $x=2$ and $3$ . Options: A) (2.5,-5.25) B) (2.5,5.25) C) (0,-1) D) (-1,0)

See Solution

Problem 32903

Bestimmen Sie die Stammfunktion von $f(x) = 2 \sin\left(\frac{\pi}{2} x\right)$ : $\int 2 \sin\left(\frac{\pi}{2} x\right) \, dx$

See Solution

Problem 32904

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

See Solution

Problem 32905

What value do the fractions approach as the pattern continues? $\frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \ldots$

See Solution

Problem 32906

Determine where the function $y=\frac{x+4}{5 \sin x}$ is continuous. Choose A, B, or C and fill in any blanks.

See Solution

Problem 32907

Bestimmen Sie die Uhrzeit mit dem größten Besucherandrang, die Zuschauerzahl beim Anpfiff und den Durchschnitt von 16 bis 18 Uhr.

See Solution

Problem 32908

Bei einer Virusepidemie beschreibt $k(t)=0,04 t^{4}-1,25 t^{3}+9 t^{2}$ die Neuinfektionen. Analysiere den Graphen und berechne die Infektionen in den ersten 4 Wochen und in der 5. bis 8. Woche. Bestimme die Gesamtzahl der Erkrankten.

See Solution

Problem 32909

Bestimme die Ableitung $f^{\prime}(x)$ für die Funktionen: a) $(x+1)e^{x}$ , b) $x^{2}e^{-0,25x}$ , c) $(x+1)e^{-2x}$ , d) $(3-2x)e^{-0,5x}$ , e) $(x+2)e^{2x}$ , f) $(x+3)e^{4x-3}$ .

See Solution

Problem 32910

Nach einem Brand steigt die PFT-Konzentration im See. Berechnen Sie:
a) Maximalen Wert und Zeitpunkt von $k(x)=250 x \cdot e^{-0.5 x}+20$ . b) Zeitpunkt, wann $k(x)<50 \frac{\sqrt{38}}{4}$ . c) Zeitpunkt der stärksten Abnahme. d) Langfristige PFT-Konzentration.

See Solution

Problem 32911

Evaluate the limit as $\theta$ approaches $\frac{\pi}{2}$ for $5 \theta \sin \theta$ .

See Solution

Problem 32912

Find the limit: $\lim _{h \rightarrow 0} \frac{3(a+h)^{2}-3 a^{2}}{h}$ .

See Solution

Problem 32913

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{8}{a+h}-\frac{8}{a}}{h}$ .

See Solution

Problem 32914

Zeigen Sie, dass $F$ eine Stammfunktion von $f$ ist und finden Sie drei weitere Stammfunktionen für die gegebenen $f$ .

See Solution

Problem 32915

Find the limit: $\lim _{x \rightarrow 10} \frac{x-10}{x^{3}-1000}$ .

See Solution

Problem 32916

Bestimmen Sie eine Vermutung für die vierte Ableitung von $f(x)=x e^{-x}$ .

See Solution

Problem 32917

Find the limit: $\lim _{h \rightarrow \infty} \frac{\sqrt{5(a+h)}-\sqrt{5 a}}{h}$ .

See Solution

Problem 32918

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{5(a+h)}-\sqrt{5 a}}{h}$ .

See Solution

Problem 32919

Evaluate $f(x)=\frac{x^{2}-1}{|x-1|}$ for $x=0.9, 0.99, 0.999, 0.9999, 1.1, 1.01, 1.001, 1.0001$ . What does $\lim _{x \rightarrow 1} f(x)$ indicate?

See Solution

Problem 32920

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (4 x)}{9 x}=$

See Solution

Problem 32921

Find the limit: $\lim _{x \rightarrow-9} \frac{x^{2}-81}{2 x^{2}+22 x+36}$ .

See Solution

Problem 32922

Find the limit: $\lim _{x \rightarrow-9} \frac{x^{2}-81}{x^{2}+22 x+36}$

See Solution

Problem 32923

Evaluate the integral $\int_{-2}^{3}(x-5) \, dx$ .

See Solution

Problem 32924

Calculate the integral from 0 to 1 of the function 3x, written as $\int_{0}^{1} 3 x \, dx$ .

See Solution

Problem 32925

Find the derivative $g^{\prime}(2)$ for the function $g(x)=\int_{\pi}^{x} \frac{1}{1+t^{4}} d t$ .

See Solution

Problem 32926

Berechnen Sie die folgenden bestimmten Integrale: a) $\int_{1}^{3}(x+1) d x$ b) $\int_{-2}^{2}(4-x^{2}) d x$ c) $\int_{-2}^{2} 0,25 x^{3} d x$ d) $\int_{0}^{2}(x^{2}-2 x+1) d x$ e) $\int_{0}^{4}(4-x) d x$ f) $\int_{1}^{3}(0,5 x-1)^{2} d x$ g) $\int_{0}^{2}(x-2)(x+2) d x$ h) $\int_{0}^{2}(x-1)^{3} d x$

See Solution

Problem 32927

Plutonium-239 hat eine Halbwertszeit von 24110 Jahren. Berechne nach 6000 Jahren die verbleibende Menge von $1 \mathrm{~kg}$ und die Zeit, bis $80 \%$ zerfallen sind.

See Solution

Problem 32928

Evaluate the integral from 1 to 9 of $\frac{x-2}{\sqrt{x}}$ with respect to $x$ .

See Solution

Problem 32929

Calculate the indefinite integral: $\int \frac{\cos ^{5} t}{\sqrt{\sin t}} d t$ (Use $C$ for the constant).

See Solution

Problem 32930

Evaluate the integral: $\int_{1}^{9}\left(\frac{x-2}{\sqrt{x}}\right) d x$

See Solution

Problem 32931

Calculate the integral $\int_{0}^{3}|x-2| dx$ .

See Solution

Problem 32932

Berechnen Sie die folgenden bestimmten Integrale mit dem festen Parameter a: a) $\int_{1}^{a}(2 x+1) d x$ b) $\int_{a}^{2 a}\left(4-x^{2}\right) d x$ c) $\int_{0}^{a^{2}}\left(3 x^{2}-3\right) d x$ d) $\int_{0}^{3 a}\left(x^{2}-a x\right) d x$

See Solution

Problem 32933

Find $f^{\prime}(1)$ if $f(x)=\int_{-1}^{x^{2}} \sqrt{t^{3}+3} dt$ .

See Solution

Problem 32934

Find $f^{\prime}(2)$ if $f(x)=\ln(x^{2}-3)$ . Choices: 2, 4, 1, e.

See Solution

Problem 32935

Find $\frac{d y}{d x}$ if $e^{y}=6 x+3 y$ . Choose one of the following options.

See Solution

Problem 32936

Find the derivative of $y=\sin \left(e^{x^{2}}\right)$ , i.e., calculate $\frac{d y}{d x}$ .

See Solution

Problem 32937

Find the derivative of the function $f(x)=\ln(e^{-x}+1)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32938

A Lamborghini Aventador accelerates at $11 \mathrm{~m/s^2}$ . How far does it travel in 6 seconds?

See Solution

Problem 32939

For the function $y=\frac{1}{x+3}$ , find the secant slope at $x=1$ , tangent slope at $x=1$ , and tangent line equation.

See Solution

Problem 32940

Find the value of $x$ that maximizes the magnitude of the instantaneous rate of change of $g(x)=-x^{2}+3x-4$ . Options: a. -2, b. -1, c. 0, d. 6.

See Solution

Problem 32941

Find the derivative of $y=(\ln x)^{\ln x}$ with respect to $x$ : $\frac{d y}{d x}$ .

See Solution

Problem 32942

Find the derivative of the function $y=1+2^{x^{2}}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 32943

At $x=2$ , if the slope is negative before, zero at, and positive after, what do you have? Minimum, Maximum, None, or Both?

See Solution

Problem 32944

Find the root of $e^{x} = 4 - 3x$ in (0,1) using the Intermediate Value Theorem. Calculate $f(0)$ and $f(1)$ .

See Solution

Problem 32945

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-6 x^{4}$ .

See Solution

Problem 32946

Find the local maxima and minima of $f(x) = -x^{3} + x^{2} + 2x$ by finding its critical points.

See Solution

Problem 32947

Find the average rate of change of $g(x)=4 x^{2}+x-1$ from $x=0.5$ to $x=1$ . Options: a. 3.5, b. 5, c. 7, d. 14.

See Solution

Problem 32948

Find $f^{\prime}(x)$ for $f(x)=2x^{2}-9x+10$ and calculate $f^{\prime}(1)$ , $f^{\prime}(5)$ , $f^{\prime}(7)$ .

See Solution

Problem 32949

Find the $x$ where $f$ is discontinuous and check if it's continuous from the right, left, or neither for:
$f(x)=\left\{\begin{array}{ll} 1+x^{2} & \text { if } x \leq 0 \\ 2-x & \text { if } 0<x \leq 2 \\ (x-2)^{2} & \text { if } x>2 \end{array}\right.$
$x=$

See Solution

Problem 32950

Find the limit as $x$ approaches 5 for $x(x-3)$ . Is it A. $\lim _{x \rightarrow 5} x(x-3)=\square$ or B. Does not exist?

See Solution

Problem 32951

Find the instantaneous rate of change of $f(x)=-2 x^{4}-3 x+1$ at $x=5$ . Choose from: a. 1003, b. 1.003, c. 0.0013, d. -1003.

See Solution

Problem 32952

Find the intervals of increase and decrease for $f(x) = -x^3 + x^2 + 2x$ , and its end behavior as $x \to -\infty$ and $x \to \infty$ .

See Solution

Problem 32953

Find the marginal cost function for $C(x)=178+0.9 x$ . What is $C^{\prime}(x)=\square$ ?

See Solution

Problem 32954

Find the limits for $f(x)=\frac{x^{2}-3 x+2}{x+2}$ as $x \to -2^{-}, -2^{+}, -2$ .

See Solution

Problem 32955

Find the instantaneous rate of change for $f(x)=\frac{x-4}{x-2}$ at $x=2$ .

See Solution

Problem 32956

Find the first 4 non-zero terms and the general term of the Maclaurin polynomial for $f(x)=\cos (x)$ .

See Solution

Problem 32957

Find the end behavior of $f(x)=8 x^{4}+3 x^{5}+11$ by calculating $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ .

See Solution

Problem 32958

Find the horizontal asymptotes of $f(x)=\frac{4x-2x^{2}}{x^{2}-2}$ . Options: $y=-2$ , $y=4$ , $y=2$ .

See Solution

Problem 32959

Find the Maclaurin series for the function $f(x) = \cos(x)$ .

See Solution

Problem 32960

Find the average rate of change of $g(x)=4 x^{2}+x-1$ from $x=0.5$ to $x=1$ . Options: a. 3.5 b. 5 c. 7 d. 14

See Solution

Problem 32961

True or False: The slope of the secant line through $(-3,35)$ and $(b, f(b))$ equals the tangent slope at $x=-3$ as $b \to -3$ .

See Solution

Problem 32962

Evaluate the limit: $\lim_{h \to 0} \frac{(1+h)^{2} - 1}{h}$

See Solution

Problem 32963

Find the limit as $t$ approaches 0 for the expression $\frac{7}{t^{3}} - 2$ .

See Solution

Problem 32964

Estimate the instantaneous rate of change of daily receipts for "Avatar" at week 16. Round to four decimal places.

See Solution

Problem 32965

Find the limit as $x$ approaches 3 for $7(8x + 8)^3$ .

See Solution

Problem 32966

Determine the convergence of the series $\sum_{n=1}^{\infty}\left(1_{n}+\frac{2}{n}\right)^{n}$ using tests.

See Solution

Problem 32967

Find a suitable $\delta$ for $\lim _{x \rightarrow 2} 3 x+2=8$ with $\varepsilon=0.4$ . Options: $\delta=0.1333$ , $\delta=0.2667$ , $\delta=0.0178$ , $\delta=0.0444$ .

See Solution

Problem 32968

Find the limits: 1) $\lim _{t \rightarrow-\infty}\left(\frac{7}{t^{-5}}-2\right)$ , 2) $\lim _{x \rightarrow-\infty}\left(\frac{14 x^{5}-7 x^{3}+9 x}{-2 x^{5}-4 x^{2}-8}\right)$ .

See Solution

Problem 32969

Evaluate these limits: (a) $\lim _{x \rightarrow \frac{11}{2}^{+}}\left(\frac{-15 x}{11-2 x}\right)$ , (b) $\lim _{x \rightarrow \frac{11}{2}^{-}}\left(\frac{-15 x}{11-2 x}\right)$ .

See Solution

Problem 32970

Given the piecewise function $f(x)=\left\{\begin{array}{lll}2 x+15 & \text { if } & x<-6 \\ \sqrt{x+15} & \text { if } & x>-6 \\ 2 & \text { if } & x=-6\end{array}\right.$ , evaluate the statements about $f(-6)$ and its limits.

See Solution

Problem 32971

Find the limit as $z$ approaches -1 for the expression $5z + 4$ .

See Solution

Problem 32972

Evaluate the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-8x+15}{x-3}$ .

See Solution

Problem 32973

Find the number of turning points for the function $f(x)=(x-2)^{3}(x^{2}+2x-3)$ .

See Solution

Problem 32974

Find the instantaneous rate of change of $g(t)=\frac{2}{t+5}$ at $t=-3$ , rounded to 3 decimal places.

See Solution

Problem 32975

Find the limit as $a$ approaches 7 for the expression $\frac{\frac{1}{a}-\frac{1}{7}}{a-7}$ .

See Solution

Problem 32976

Find the derivative of $f(x)=4 x^{7}-5 x^{6}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 32977

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{3}{x^{2}}$ and evaluate it at $x=2$ .

See Solution

Problem 32978

Find the derivative of $f(x)=8 \sqrt{x}-\frac{1}{x^{8}}$ and write it without fractional or negative exponents.

See Solution

Problem 32979

Find the derivative $f^{\prime}(x)$ of the function $f(x)=4+3x-2x^{2}$ and evaluate it at $x=4$ .

See Solution

Problem 32980

Find $f^{\prime}(-3)$ for $f(x)=2 x^{2}-4 x+6$ and the tangent line at $(-3,36)$ in the form $y=m x+b$ .

See Solution

Problem 32981

Find the derivative $f'(x)$ for the function $f(x) = \ln \left(\frac{2x+1}{1-2x}\right)$ .

See Solution

Problem 32982

Find the global extreme values of the function $f(x)=3 x^{4}-4 x^{3}$ on the interval $[-1,2]$ .

See Solution

Problem 32983

Find the derivative of the function $f(x)=7 x^{5} \sqrt{x}+\frac{2}{x^{3} \sqrt{x}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32984

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2 x^{2}-4 x+6$ and calculate $f^{\prime}(-3)$ .

See Solution

Problem 32985

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

See Solution

Problem 32986

Find the horizontal asymptotes of $f(x)=\frac{4 x-2 x^{2}}{x^{2}-2}$ . Options: $x=-2$ , $y=2$ , $y=4$ , None, $y=-2$ .

See Solution

Problem 32987

Find $f^{\prime}(-1)$ for $f(x)=x^{6} h(x)$ , given $h(-1)=2$ and $h^{\prime}(-1)=5$ . Use product and power rules.

See Solution

Problem 32988

Given $f(t)=(t^{2}+4 t+5)(5 t^{2}+4)$ , find $f^{\prime}(t)$ and then calculate $f^{\prime}(4)$ .

See Solution

Problem 32989

Find the derivative $f'(x)$ of the function $f(x)=\frac{7x+7}{2x+6}$ and evaluate $f'(2)$ .

See Solution

Problem 32990

Find the equation of the tangent line to $f(x)=x^{2}-x-12$ parallel to the secant line between $(-2, f(-2))$ and $(4, f(4))$ .

See Solution

Problem 32991

Find the derivative of the function $f(x)=\frac{-2 x^{5}-8 x^{4}-5 x^{3}}{x^{4}}$ .

See Solution

Problem 32992

Identify the correct limit statements for the end behaviors of the exponential function $g$ :
(A) $\lim _{x \rightarrow-\infty} g(x)=-\infty$ , $\lim _{x \rightarrow \infty} g(x)=0$ (B) $\lim _{x \rightarrow-\infty} g(x)=\infty$ , $\lim _{x \rightarrow \infty} g(x)=\infty$ (C) $\lim _{x \rightarrow-\infty} g(x)=\emptyset$ , $\lim _{x \rightarrow \infty} g(x)=\infty$ (D) $\lim _{x \rightarrow-\infty} g(x)=\infty$ , $\lim _{x \rightarrow \infty} g(x)=0$

See Solution

Problem 32993

Identify the correct limit statements for the exponential function $g$ that increases at a decreasing rate. Options: (A), (B), (C), (D).

See Solution

Problem 32994

Find the derivative of the function $f(x)=\frac{4-x^{2}}{6+x^{2}}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 32995

Find the derivative of $f(x)=\frac{\sqrt{x}-3}{\sqrt{x}+3}$ and calculate $f^{\prime}(4)$ .

See Solution

Problem 32996

Find the derivative $f^{\prime}(t)$ of the function $f(t)=(t^{2}+4 t+5)(5 t^{2}+4)$ .

See Solution

Problem 32997

Find the derivative $f'(x)$ of $f(x)=\frac{4x^2+8x+5}{\sqrt{x}}$ and evaluate $f'(3)$ .

See Solution

Problem 32998

Find $x$ such that $f'(x)=5$ for $f(x)=\frac{x}{x+6}$ . Provide exact values.

See Solution

Problem 32999

Find the velocity function $v(t)$ from $s(t)=6 t^{3}-45 t^{2}+108 t$ . Where is $v(t)=0$ ? Factor out GCF. Find $a(t)$ .

See Solution

Problem 33000

True or False: Do "rate of change", "marginal profit", "marginal cost", "marginal revenue", "population growth rate", and "acceleration" all relate to the first derivative?

See Solution

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Calculus

Problem 32901

Determine values of aaa for which lim⁡x→af(x)\lim _{x \rightarrow a} f(x)limx→a​f(x) exists based on the piecewise function described.

Problem 32902

Find the local maximum of f(x)=−0.8x3+3x2−1f(x)=-0.8 x^{3}+3 x^{2}-1f(x)=−0.8x3+3x2−1 between x=2x=2x=2 and 333. Options: A) (2.5,-5.25) B) (2.5,5.25) C) (0,-1) D) (-1,0)

Problem 32903

Bestimmen Sie die Stammfunktion von f(x)=2sin⁡(π2x)f(x) = 2 \sin\left(\frac{\pi}{2} x\right)f(x)=2sin(2π​x): ∫2sin⁡(π2x) dx\int 2 \sin\left(\frac{\pi}{2} x\right) \, dx∫2sin(2π​x)dx

Problem 32904

Finde die Punkte, an denen die Tangente von fff den Steigungswinkel 21,8∘21,8^{\circ}21,8∘ hat. a) f(x)=5x2f(x)=5 x^{2}f(x)=5x2 b) f(x)=−40xf(x)=-\frac{40}{x}f(x)=−x40​ c) f(x)=56x3f(x)=\frac{5}{6} x^{3}f(x)=65​x3 d) f(x)=0,15x2−0,2xf(x)=0,15 x^{2}-0,2 xf(x)=0,15x2−0,2x

Problem 32905

What value do the fractions approach as the pattern continues? 13,25,37,49,… \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \ldots 31​,52​,73​,94​,…

Problem 32906

Determine where the function y=x+45sin⁡xy=\frac{x+4}{5 \sin x}y=5sinxx+4​ is continuous. Choose A, B, or C and fill in any blanks.

Problem 32907

Bestimmen Sie die Uhrzeit mit dem größten Besucherandrang, die Zuschauerzahl beim Anpfiff und den Durchschnitt von 16 bis 18 Uhr.

Problem 32908

Bei einer Virusepidemie beschreibt k(t)=0,04t4−1,25t3+9t2k(t)=0,04 t^{4}-1,25 t^{3}+9 t^{2}k(t)=0,04t4−1,25t3+9t2 die Neuinfektionen. Analysiere den Graphen und berechne die Infektionen in den ersten 4 Wochen und in der 5. bis 8. Woche. Bestimme die Gesamtzahl der Erkrankten.

Problem 32909

Problem 32910

Problem 32911

Evaluate the limit as θ\thetaθ approaches π2\frac{\pi}{2}2π​ for 5θsin⁡θ5 \theta \sin \theta5θsinθ.

Problem 32912

Find the limit: lim⁡h→03(a+h)2−3a2h\lim _{h \rightarrow 0} \frac{3(a+h)^{2}-3 a^{2}}{h}limh→0​h3(a+h)2−3a2​.

Problem 32913

Find the limit: lim⁡h→08a+h−8ah\lim _{h \rightarrow 0} \frac{\frac{8}{a+h}-\frac{8}{a}}{h}limh→0​ha+h8​−a8​​.

Problem 32914

Zeigen Sie, dass FFF eine Stammfunktion von fff ist und finden Sie drei weitere Stammfunktionen für die gegebenen fff.

Problem 32915

Find the limit: lim⁡x→10x−10x3−1000\lim _{x \rightarrow 10} \frac{x-10}{x^{3}-1000}limx→10​x3−1000x−10​.

Problem 32916

Bestimmen Sie eine Vermutung für die vierte Ableitung von f(x)=xe−xf(x)=x e^{-x}f(x)=xe−x.

Problem 32917

Find the limit: lim⁡h→∞5(a+h)−5ah\lim _{h \rightarrow \infty} \frac{\sqrt{5(a+h)}-\sqrt{5 a}}{h}limh→∞​h5(a+h)​−5a​​.

Problem 32918

Find the limit: lim⁡h→05(a+h)−5ah\lim _{h \rightarrow 0} \frac{\sqrt{5(a+h)}-\sqrt{5 a}}{h}limh→0​h5(a+h)​−5a​​.

Problem 32919

Problem 32920

Find the limit: lim⁡x→0sin⁡(4x)9x=\lim _{x \rightarrow 0} \frac{\sin (4 x)}{9 x}=limx→0​9xsin(4x)​=

Problem 32921

Find the limit: lim⁡x→−9x2−812x2+22x+36\lim _{x \rightarrow-9} \frac{x^{2}-81}{2 x^{2}+22 x+36}limx→−9​2x2+22x+36x2−81​.

Problem 32922

Find the limit: lim⁡x→−9x2−81x2+22x+36\lim _{x \rightarrow-9} \frac{x^{2}-81}{x^{2}+22 x+36}limx→−9​x2+22x+36x2−81​

Problem 32923

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

Problem 32924

Calculate the integral from 0 to 1 of the function 3x, written as ∫013x dx\int_{0}^{1} 3 x \, dx∫01​3xdx.

Problem 32925

Find the derivative g′(2)g^{\prime}(2)g′(2) for the function g(x)=∫πx11+t4dtg(x)=\int_{\pi}^{x} \frac{1}{1+t^{4}} d tg(x)=∫πx​1+t41​dt.

Problem 32926

Problem 32927

Plutonium-239 hat eine Halbwertszeit von 24110 Jahren. Berechne nach 6000 Jahren die verbleibende Menge von 1 kg1 \mathrm{~kg}1 kg und die Zeit, bis 80%80 \%80% zerfallen sind.

Problem 32928

Evaluate the integral from 1 to 9 of x−2x\frac{x-2}{\sqrt{x}}x​x−2​ with respect to xxx.

Problem 32929

Calculate the indefinite integral: ∫cos⁡5tsin⁡tdt\int \frac{\cos ^{5} t}{\sqrt{\sin t}} d t∫sint​cos5t​dt (Use CCC for the constant).

Problem 32930

Evaluate the integral: ∫19(x−2x)dx\int_{1}^{9}\left(\frac{x-2}{\sqrt{x}}\right) d x∫19​(x​x−2​)dx

Problem 32931

Calculate the integral ∫03∣x−2∣dx\int_{0}^{3}|x-2| dx∫03​∣x−2∣dx.

Problem 32932

Problem 32933

Find f′(1)f^{\prime}(1)f′(1) if f(x)=∫−1x2t3+3dtf(x)=\int_{-1}^{x^{2}} \sqrt{t^{3}+3} dtf(x)=∫−1x2​t3+3​dt.

Problem 32934

Find f′(2)f^{\prime}(2)f′(2) if f(x)=ln⁡(x2−3)f(x)=\ln(x^{2}-3)f(x)=ln(x2−3). Choices: 2, 4, 1, e.

Problem 32935

Find dydx\frac{d y}{d x}dxdy​ if ey=6x+3ye^{y}=6 x+3 yey=6x+3y. Choose one of the following options.

Problem 32936

Find the derivative of y=sin⁡(ex2)y=\sin \left(e^{x^{2}}\right)y=sin(ex2), i.e., calculate dydx\frac{d y}{d x}dxdy​.

Problem 32937

Find the derivative of the function f(x)=ln⁡(e−x+1)f(x)=\ln(e^{-x}+1)f(x)=ln(e−x+1). What is f′(x)f^{\prime}(x)f′(x)?

Problem 32938

A Lamborghini Aventador accelerates at 11 m/s211 \mathrm{~m/s^2}11 m/s2. How far does it travel in 6 seconds?

Problem 32939

For the function y=1x+3y=\frac{1}{x+3}y=x+31​, find the secant slope at x=1x=1x=1, tangent slope at x=1x=1x=1, and tangent line equation.

Problem 32940

Find the value of xxx that maximizes the magnitude of the instantaneous rate of change of g(x)=−x2+3x−4g(x)=-x^{2}+3x-4g(x)=−x2+3x−4. Options: a. -2, b. -1, c. 0, d. 6.

Problem 32941

Find the derivative of y=(ln⁡x)ln⁡xy=(\ln x)^{\ln x}y=(lnx)lnx with respect to xxx: dydx\frac{d y}{d x}dxdy​.

Problem 32942

Find the derivative of the function y=1+2x2y=1+2^{x^{2}}y=1+2x2. What is dydx\frac{d y}{d x}dxdy​?

Determine values of $a$ for which $\lim _{x \rightarrow a} f(x)$ exists based on the piecewise function described.

Find the local maximum of $f(x)=-0.8 x^{3}+3 x^{2}-1$ between $x=2$ and $3$ . Options: A) (2.5,-5.25) B) (2.5,5.25) C) (0,-1) D) (-1,0)

Bestimmen Sie die Stammfunktion von $f(x) = 2 \sin\left(\frac{\pi}{2} x\right)$ : $\int 2 \sin\left(\frac{\pi}{2} x\right) \, dx$

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

What value do the fractions approach as the pattern continues? $\frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \ldots$

Determine where the function $y=\frac{x+4}{5 \sin x}$ is continuous. Choose A, B, or C and fill in any blanks.

Bei einer Virusepidemie beschreibt $k(t)=0,04 t^{4}-1,25 t^{3}+9 t^{2}$ die Neuinfektionen. Analysiere den Graphen und berechne die Infektionen in den ersten 4 Wochen und in der 5. bis 8. Woche. Bestimme die Gesamtzahl der Erkrankten.

Evaluate the limit as $\theta$ approaches $\frac{\pi}{2}$ for $5 \theta \sin \theta$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{3(a+h)^{2}-3 a^{2}}{h}$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{8}{a+h}-\frac{8}{a}}{h}$ .

Zeigen Sie, dass $F$ eine Stammfunktion von $f$ ist und finden Sie drei weitere Stammfunktionen für die gegebenen $f$ .

Find the limit: $\lim _{x \rightarrow 10} \frac{x-10}{x^{3}-1000}$ .

Bestimmen Sie eine Vermutung für die vierte Ableitung von $f(x)=x e^{-x}$ .

Find the limit: $\lim _{h \rightarrow \infty} \frac{\sqrt{5(a+h)}-\sqrt{5 a}}{h}$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{5(a+h)}-\sqrt{5 a}}{h}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (4 x)}{9 x}=$

Find the limit: $\lim _{x \rightarrow-9} \frac{x^{2}-81}{2 x^{2}+22 x+36}$ .

Find the limit: $\lim _{x \rightarrow-9} \frac{x^{2}-81}{x^{2}+22 x+36}$

Evaluate the integral $\int_{-2}^{3}(x-5) \, dx$ .

Calculate the integral from 0 to 1 of the function 3x, written as $\int_{0}^{1} 3 x \, dx$ .

Find the derivative $g^{\prime}(2)$ for the function $g(x)=\int_{\pi}^{x} \frac{1}{1+t^{4}} d t$ .

Plutonium-239 hat eine Halbwertszeit von 24110 Jahren. Berechne nach 6000 Jahren die verbleibende Menge von $1 \mathrm{~kg}$ und die Zeit, bis $80 \%$ zerfallen sind.

Evaluate the integral from 1 to 9 of $\frac{x-2}{\sqrt{x}}$ with respect to $x$ .

Calculate the indefinite integral: $\int \frac{\cos ^{5} t}{\sqrt{\sin t}} d t$ (Use $C$ for the constant).

Evaluate the integral: $\int_{1}^{9}\left(\frac{x-2}{\sqrt{x}}\right) d x$

Calculate the integral $\int_{0}^{3}|x-2| dx$ .

Find $f^{\prime}(1)$ if $f(x)=\int_{-1}^{x^{2}} \sqrt{t^{3}+3} dt$ .

Find $f^{\prime}(2)$ if $f(x)=\ln(x^{2}-3)$ . Choices: 2, 4, 1, e.

Find $\frac{d y}{d x}$ if $e^{y}=6 x+3 y$ . Choose one of the following options.

Find the derivative of $y=\sin \left(e^{x^{2}}\right)$ , i.e., calculate $\frac{d y}{d x}$ .

Find the derivative of the function $f(x)=\ln(e^{-x}+1)$ . What is $f^{\prime}(x)$ ?

A Lamborghini Aventador accelerates at $11 \mathrm{~m/s^2}$ . How far does it travel in 6 seconds?

For the function $y=\frac{1}{x+3}$ , find the secant slope at $x=1$ , tangent slope at $x=1$ , and tangent line equation.

Find the value of $x$ that maximizes the magnitude of the instantaneous rate of change of $g(x)=-x^{2}+3x-4$ . Options: a. -2, b. -1, c. 0, d. 6.

Find the derivative of $y=(\ln x)^{\ln x}$ with respect to $x$ : $\frac{d y}{d x}$ .

Find the derivative of the function $y=1+2^{x^{2}}$ . What is $\frac{d y}{d x}$ ?

At $x=2$ , if the slope is negative before, zero at, and positive after, what do you have? Minimum, Maximum, None, or Both?

Find the root of $e^{x} = 4 - 3x$ in (0,1) using the Intermediate Value Theorem. Calculate $f(0)$ and $f(1)$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-6 x^{4}$ .

Find the local maxima and minima of $f(x) = -x^{3} + x^{2} + 2x$ by finding its critical points.

Find the average rate of change of $g(x)=4 x^{2}+x-1$ from $x=0.5$ to $x=1$ . Options: a. 3.5, b. 5, c. 7, d. 14.

Find $f^{\prime}(x)$ for $f(x)=2x^{2}-9x+10$ and calculate $f^{\prime}(1)$ , $f^{\prime}(5)$ , $f^{\prime}(7)$ .

Find the limit as $x$ approaches 5 for $x(x-3)$ . Is it A. $\lim _{x \rightarrow 5} x(x-3)=\square$ or B. Does not exist?

Find the instantaneous rate of change of $f(x)=-2 x^{4}-3 x+1$ at $x=5$ . Choose from: a. 1003, b. 1.003, c. 0.0013, d. -1003.

Find the intervals of increase and decrease for $f(x) = -x^3 + x^2 + 2x$ , and its end behavior as $x \to -\infty$ and $x \to \infty$ .

Find the marginal cost function for $C(x)=178+0.9 x$ . What is $C^{\prime}(x)=\square$ ?

Find the limits for $f(x)=\frac{x^{2}-3 x+2}{x+2}$ as $x \to -2^{-}, -2^{+}, -2$ .

Find the instantaneous rate of change for $f(x)=\frac{x-4}{x-2}$ at $x=2$ .

Find the first 4 non-zero terms and the general term of the Maclaurin polynomial for $f(x)=\cos (x)$ .

Find the end behavior of $f(x)=8 x^{4}+3 x^{5}+11$ by calculating $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ .

Find the horizontal asymptotes of $f(x)=\frac{4x-2x^{2}}{x^{2}-2}$ . Options: $y=-2$ , $y=4$ , $y=2$ .

Find the Maclaurin series for the function $f(x) = \cos(x)$ .

Find the average rate of change of $g(x)=4 x^{2}+x-1$ from $x=0.5$ to $x=1$ . Options: a. 3.5 b. 5 c. 7 d. 14

True or False: The slope of the secant line through $(-3,35)$ and $(b, f(b))$ equals the tangent slope at $x=-3$ as $b \to -3$ .

Evaluate the limit: $\lim_{h \to 0} \frac{(1+h)^{2} - 1}{h}$

Find the limit as $t$ approaches 0 for the expression $\frac{7}{t^{3}} - 2$ .

Find the limit as $x$ approaches 3 for $7(8x + 8)^3$ .

Determine the convergence of the series $\sum_{n=1}^{\infty}\left(1_{n}+\frac{2}{n}\right)^{n}$ using tests.

Find a suitable $\delta$ for $\lim _{x \rightarrow 2} 3 x+2=8$ with $\varepsilon=0.4$ . Options: $\delta=0.1333$ , $\delta=0.2667$ , $\delta=0.0178$ , $\delta=0.0444$ .

Find the limit as $z$ approaches -1 for the expression $5z + 4$ .

Evaluate the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-8x+15}{x-3}$ .

Find the number of turning points for the function $f(x)=(x-2)^{3}(x^{2}+2x-3)$ .

Find the instantaneous rate of change of $g(t)=\frac{2}{t+5}$ at $t=-3$ , rounded to 3 decimal places.