Calculus

Problem 7901

Find the antiderivative of $f(x)=\left(\frac{1}{5}\right)^{x}$ over the interval $I=[0, 5]$ .

See Solution

Problem 7902

Evaluate the second derivative $f^{\prime \prime}(5), f^{\prime \prime}(3)$ , and $f^{\prime \prime}(4)$ for $f(x)=\sqrt{6x-2}$ .

See Solution

Problem 7903

Find the second derivative of the function $f(x)=-2 x^{2}-5 \sqrt[3]{x}-9$ .

See Solution

Problem 7904

Find the intervals where the function $f(x)=-4 x^{3}-72 x^{2}+5 x-2$ is concave up or down.

See Solution

Problem 7905

Find the second derivative of the function $f(x)=-8 x^{3}+6 x^{2}-9 x$ .

See Solution

Problem 7906

Find the point of diminishing returns for the sales function $S(x)=123+31.5 x^{2}-0.5 x^{3}$ , where $0 \leq x \leq 42$ .

See Solution

Problem 7907

Evaluate the second derivative $f^{\prime \prime}(1)$ , $f^{\prime \prime}(5)$ , and $f^{\prime \prime}(7)$ for $f(x)=-8 x^{3}+6 x^{2}-9 x$ .

See Solution

Problem 7908

Find points of inflection for the function $f(x)=-4 x^{3}-84 x^{2}+6 x-2$ . Provide your answer as $(x, y)$ -pairs.

See Solution

Problem 7909

Find intervals where the function $f(x)=-4 x^{3}-84 x^{2}+6 x-2$ is concave up or down.

See Solution

Problem 7910

Find the function $f$ and the number $a$ in the limit that represents the derivative: $\lim _{h \rightarrow 0} \frac{(1+h)^{10}-1}{h}$ .

See Solution

Problem 7911

Find the derivative of $f(x)=-4x^{3}-84x^{2}+6x-2$ .

See Solution

Problem 7912

Find the function $f$ and the number $a$ in the limit representing the derivative: $\lim _{h \rightarrow 0} \frac{(1+h)^{10}-1}{h}$ .

See Solution

Problem 7913

Find the constant $c$ that ensures the function $f(x)$ is continuous for all $x$ in $(-\infty, \infty)$ .

See Solution

Problem 7914

Gegeben ist $f^{\prime}(x) = 3x^2 - 4$ .
a) Bestimme den Grad von $f(x)$ und begründe.
b) Zeichne den Graphen von $f(x)$ .
c) Finde die Steigung von $f(x)$ im Tiefpunkt.

See Solution

Problem 7915

Check if the function $f(x)=\frac{x^{2}-9}{x-3}$ for $x \neq 3$ and $f(3)=10$ is continuous at $a=3$ . Select all that apply.

See Solution

Problem 7916

Find the function $f$ and the number $a$ such that $\lim _{h \rightarrow 0} \sqrt[4]{\frac{4}{16+h}-2}$ is the derivative of $f$ at $a$ .
$a=$ $f=$

See Solution

Problem 7917

Find slopes of secant lines for $f(x)=19 x^{3}-x$ and conjecture the tangent slope at $x=1$ . Round as needed.

See Solution

Problem 7918

Find the function $f$ and the number $a$ such that $\lim_{h \rightarrow 0} \sqrt{16+h}-2$ represents the derivative. $a=$ $f=$

See Solution

Problem 7919

Check if the series $\sum_{n=1}^{\infty} \frac{2}{n^{2}+7 n+10}$ converges or diverges. If convergent, find the sum.

See Solution

Problem 7920

Zeigen Sie, dass die Differenzenquotienten von $f(x)=x^{2}$ in $[a ; b]$ und $[a-1 ; b+1]$ gleich sind.

See Solution

Problem 7921

Find the limit: $\lim _{x \rightarrow-2} \frac{x+2}{\sqrt{x^{2}+12}-4}$

See Solution

Problem 7922

Find which statements about the tangent line $L(x)=8(x-2)+1$ to $g(x)=x^{3}-4x+1$ at $(2,1)$ are true.

See Solution

Problem 7923

Given the equation $\frac{d y}{d t}=t-2$ , find the slope at points $(0,1)$ , $(0,5)$ , and $(525,152)$ . Which functions could be solutions? Also, what info determines the slope at a point? A. $y$ -value B. slope at $t=0$ C. $t$ -value D. None.

See Solution

Problem 7924

Find the function $f$ and the number $a$ for the limit $\lim _{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}$ where $a=\sqrt{16}$ .

See Solution

Problem 7925

Find the net change and average rate of change of $r(t)=5-\frac{1}{5} t$ for $t=5$ and $t=10$ .

See Solution

Problem 7926

Find the function $f$ and the number $a$ where $a=16$ given the limit $\lim_{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}$ .

See Solution

Problem 7927

Find the limit $\lim_{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}$ , where $a=16$ and $f=$ ?

See Solution

Problem 7928

Find the function $f$ and the number $a$ for the limit $\lim _{h \rightarrow 0} \frac{\sqrt[4]{16+h}-2}{h}$ , where $a=16$ .

See Solution

Problem 7929

Find the function $f$ and the number $a$ such that $\lim _{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}$ represents the derivative. $a = 16$ $f =$

See Solution

Problem 7930

Find the average velocity of an object with $s(3)=177$ and $s(5)=233$ over the interval $[3,5]$ .

See Solution

Problem 7931

Find the derivative $R'(1)$ for the function $R(x)=\ln \left(x^{2} \cdot \sqrt{g(x)}\right)$ , where $g(x)$ is undefined.

See Solution

Problem 7932

Find the derivative of the function $f(x)=(x^{2}+3x) \cdot \sin(x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 7933

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

See Solution

Problem 7934

Calculate the integral of the function $f(x) = e^{\frac{1}{2} x}$ .

See Solution

Problem 7935

Berechnen Sie das Integral von $e^{\frac{1}{2} x^{2}}$ .

See Solution

Problem 7936

Berechnen Sie das Integral von $f(x)=4 e^{-x}$ .

See Solution

Problem 7937

Find the derivative of $f(x)=(4-3x)^{2} \cdot \sin(x)$ .

See Solution

Problem 7938

Leiten Sie die Funktion $f$ mit der Produktregel und der Kettenregel ab. Beispiele: a) $f(x)=x \cdot \sin(3x)$ , b) $f(x)=(2x-1)^{2} \cdot \sqrt{x}$ , usw.

See Solution

Problem 7939

Berechnen Sie das Integral von $f(x) = 4x^{-3}$ .

See Solution

Problem 7940

Integrieren Sie die Funktion $f(x)=\frac{6}{x^{2}}$ .

See Solution

Problem 7941

Find the derivative of the function $f(x)=3 x^{5} \cdot \cos (2 x)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 7942

Find the derivative of $F(t)=\left(\frac{1}{9 t+1}\right)^{5}$ . What is $F^{\prime}(t)=?$

See Solution

Problem 7943

How long for a shell to hit the ground if fired from a cliff at $80 \mathrm{m/s}$ at a $40^\circ$ angle, 200 m down?

See Solution

Problem 7944

Calculate the indefinite integral: $\int e^{x+e^{x}} d x$

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Problem 7945

Calculate the integral $\int \left(\cos^{3} x\right) \sqrt[3]{\sin x} \, dx$ .

See Solution

Problem 7946

Find the integral using trigonometric substitution: $\int \frac{x^{3}}{\sqrt{4+x^{2}}} d x$

See Solution

Problem 7947

Find the indefinite integral using integration by parts: $\int x^{2} \sin 2 x \, dx$

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Problem 7948

Find the integral of the natural logarithm: $\int \ln x \, dx$ .

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Problem 7949

Find the derivative of $y=\frac{x \sin x}{1+\cos x}$ without simplifying. Show all work. [5pts]

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Problem 7950

Finde die Tangentengleichung $t_{1}$ für $f(x)=\frac{1}{3} x^{2}$ bei $x_{0}=-2$ und die orthogonale Tangente.

See Solution

Problem 7951

Evaluate the integral $\int_{0}^{\pi / 2} \cos ^{8} x \, dx$ using Wallis's Formula.

See Solution

Problem 7952

Find the derivative of $(5x + 4)^{100}$ with respect to $x$ . Do not simplify your answer. [3pts]

See Solution

Problem 7953

Zeigen Sie, dass der Punkt $T(x_{0} \mid 0)$ auch ein Tiefpunkt von $g(x)=x^{2} \cdot f(x)$ ist, wenn $f^{\prime \prime}(x_{0})>0$ .

See Solution

Problem 7954

Find the derivative of $y=\csc(e^{x})$ . Do not simplify your answer. [3pts]

See Solution

Problem 7955

Calculate the car's acceleration if it speeds from $0 \mathrm{~m/s}$ to $4 \mathrm{~m/s}$ in 0.5 seconds. I $\mathrm{m/s}^{2}$

See Solution

Problem 7956

Find the derivative of $y=\csc(e^{x})$ . Do not simplify your answer. [3pts]

See Solution

Problem 7957

Find the derivative of $y=\tan^{10}(x^{5})$ , denoted as $y'$ . Do not simplify your answer. [4pts]

See Solution

Problem 7958

Calculate the car's acceleration if its velocity changes from $0 \mathrm{~m/s}$ to $10 \mathrm{~m/s}$ in 0.5 seconds. $\mathrm{m/s}^{2}$

See Solution

Problem 7959

How much will \$2000 grow in 10 years at a continuous interest rate of 6\%? Use the formula for continuous compounding.

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Problem 7960

Differentiate implicitly: $x \sin (y) + y \sin (x) = 3$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 7961

Find the tangent line equation using implicit differentiation for $y \sin (16 x) = x \cos (2 y)$ at $(\pi / 2, \pi / 4)$ .

See Solution

Problem 7962

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ .
a) Bestimme die Steigung von $f$ bei $P(2 \mid f(2))$ . b) Gibt es Punkte mit waagerechter Tangente? c) Wo hat die Tangente die Steigung 1? Finde die Gleichung in diesen Punkten.

See Solution

Problem 7963

Find the tangent line equation $y=$ to the curve $y \sin (16 x)=x \cos (2 y)$ at $(\pi / 2, \pi / 4)$ using implicit differentiation.

See Solution

Problem 7964

Find the derivative of $f(x)=2 x^{2} x^{\cos (x)}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 7965

How much will $\$ 2000$ grow in 10 years at a continuous interest rate of $3 \%$ ?

See Solution

Problem 7966

Find $\left(f^{-1}\right)^{\prime}(x)$ for these functions using $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$ : 3) $f(x)=-5 x+1$ , 4) $f(x)=-2 x+2$ , 5) $f(x)=\sqrt{-2 x-3}$ , 6) $f(x)=-4 x^{3}-4$ .

See Solution

Problem 7967

Find the derivative of $\ln \left(\frac{(6 x+3)^{5}}{\sqrt[3]{3 x^{2}-3 x+4}}\right)$ .

See Solution

Problem 7968

Find the tangent line equation for $y=f\left(\frac{\sqrt{3}-2}{3}\right)\left(x \cdot \frac{\sqrt{3}-2}{3}\right)+f\left(\frac{\sqrt{3}-2}{3}\right)$ , where $f(x)=\frac{1}{9}(3 x+2)^{3}$ and slope is 1.

See Solution

Problem 7969

Find $\left(f^{-1}\right)^{\prime}(x)$ using $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$ for: 3) $f(x)=-5 x+1$ 4) $f(x)=-2 x+2$ 5) $f(x)=\sqrt{-2 x-3}$ 6) $f(x)=-4 x^{3}-4$

See Solution

Problem 7970

Find the horizontal asymptotes of $f(x)=\frac{3000+60x}{x}$ .

See Solution

Problem 7971

Find the rate of change of $p$ with respect to $q$ given $p$ from a lens with focal length $6 \mathrm{cms}$ and $\frac{1}{6}=\frac{1}{p}+\frac{1}{q}$ .

See Solution

Problem 7972

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin ^{3}(x)}{\sin x-\tan x}$ .

See Solution

Problem 7973

Find the limit: $L = \lim _{x \rightarrow 0} \frac{2-\cos (3 x)-\cos (4 x)}{x}$ .

See Solution

Problem 7974

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan ^{2}(3 x)+\sin \left(5 x^{2}\right)}{x^{2}}$ .

See Solution

Problem 7975

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{\sin (4 x)}$ .

See Solution

Problem 7976

A cylinder's height equals its diameter. If height increases at 2 in/s, find the volume increase rate, $V$ , when height is 4 in.

See Solution

Problem 7977

Determine concavity, inflection points, and critical points for $f(x)=-(4x+4\sin(x))$ , $0 \leq x \leq 2\pi$ .

See Solution

Problem 7978

The radius of a circle grows at $7 \mathrm{ft}/\mathrm{sec}$ . Find the area change rate in terms of circumference $C$ .

See Solution

Problem 7979

Find the rocket's velocity at $t = 10$ seconds given height $h(t) = 3t^{2}$ ft.

See Solution

Problem 7980

Find the derivative $f^{\prime}(a)$ for the function $f(x)=2 x^{2}-3 x+1$ .

See Solution

Problem 7981

Find $f^{\prime}(7)$ for the function $f(x)=\sqrt{6x+7}$ .

See Solution

Problem 7982

An object is dropped from a 600 m cliff. Find its speed after 7 seconds using $600 - 4.9 t^{2}$ .

See Solution

Problem 7983

Rocket height after $t$ seconds is $3t^{2}$ ft. Find velocity and acceleration at $t=10$ seconds.

See Solution

Problem 7984

Find the rocket's velocity and acceleration at $t = 10$ seconds, given height $h = 3t^{2}$ ft.

See Solution

Problem 7985

A rocket moves up at $900 \mathrm{~km/hr}$ . Find the angle's increase rate from a telescope $2 \mathrm{~km}$ away after $3 \mathrm{~min}$ .

See Solution

Problem 7986

Find the quantity $q$ (in thousands) that maximizes profit for $P(q)=-0.03 q^{2}+5 q-25$ . What is the max profit?

See Solution

Problem 7987

Find the number of thousands of sunglasses, $q$ , to maximize profit from $P(q)=-0.01 q^{2}+3 q-24$ . What is the max profit?

See Solution

Problem 7988

A cylinder leaks water at $3 \mathrm{ft}^3/\mathrm{s}$ . Find the height change rate when height is $10 \mathrm{ft}$ .

See Solution

Problem 7989

Find the rate of change of surface area at $t=2$ min for a sphere with $r=18$ cm and $dr/dt=70$ cm/min.

See Solution

Problem 7990

Find $G^{\prime}(a)$ for $G(x)=5x^{2}-x^{3}$ and the tangent lines at $(3,18)$ and $(4,16)$ . $G^{\prime}(a)=$ Line 1: $y_{1}(x)=$ , Line 2: $y_{2}(x)=$ .

See Solution

Problem 7991

Show that for spheres with radii in $[1,5]$ , the volume $v = 275$ cm³ exists using the Intermediate Value Theorem.

See Solution

Problem 7992

Find the slope of the tangent line for $y(t)$ at points $(0,1)$ , $(0,5)$ , and $(541,141)$ for $\frac{d y}{d t}=t-2$ . Identify possible solutions from options A-E. What info is needed for the slope?

See Solution

Problem 7993

Find the slope of the tangent line for $y(t)$ at points $(0,1)$ , $(0,5)$ , and $(541,141)$ for $\frac{d y}{d t}=t-2$ . Which functions could be solutions? What info is needed for the slope?

See Solution

Problem 7994

Find $f(3)$ and $f^{\prime}(3)$ if the tangent line to $y=f(x)$ at $(3,2)$ passes through $(0,1)$ .

See Solution

Problem 7995

Sketch the graph of $f(x)=x^{1/3}$ and show it has a vertical tangent at $x=0$ using limits.

See Solution

Problem 7996

Find the derivative of the area $A=\pi r^{2}$ with respect to the radius $r$ : compute $\frac{d A}{d r}$ .

See Solution

Problem 7997

Find the derivative $f'(4)$ for $f(x)=\sqrt{x}$ , then use it to find the tangent line and approximate $\sqrt{4.1}$ .

See Solution

Problem 7998

Find the slope of the tangent line to the parabola $y=7x-x^{2}$ at $(1,6)$ and its equation.

See Solution

Problem 7999

Find the derivative of $f(x)=\sqrt{7-x^{3}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 8000

Sketch the graph of $f(x)=x^{1/3}$ . Show that it has a vertical tangent at $x=0$ using limits.

See Solution

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Calculus

Problem 7901

Find the antiderivative of f(x)=(15)xf(x)=\left(\frac{1}{5}\right)^{x}f(x)=(51​)x over the interval I=[0,5]I=[0, 5]I=[0,5].

Problem 7902

Evaluate the second derivative f′′(5),f′′(3)f^{\prime \prime}(5), f^{\prime \prime}(3)f′′(5),f′′(3), and f′′(4)f^{\prime \prime}(4)f′′(4) for f(x)=6x−2f(x)=\sqrt{6x-2}f(x)=6x−2​.

Problem 7903

Find the second derivative of the function f(x)=−2x2−5x3−9f(x)=-2 x^{2}-5 \sqrt[3]{x}-9f(x)=−2x2−53x​−9.

Problem 7904

Find the intervals where the function f(x)=−4x3−72x2+5x−2f(x)=-4 x^{3}-72 x^{2}+5 x-2f(x)=−4x3−72x2+5x−2 is concave up or down.

Problem 7905

Find the second derivative of the function f(x)=−8x3+6x2−9xf(x)=-8 x^{3}+6 x^{2}-9 xf(x)=−8x3+6x2−9x.

Problem 7906

Find the point of diminishing returns for the sales function S(x)=123+31.5x2−0.5x3S(x)=123+31.5 x^{2}-0.5 x^{3}S(x)=123+31.5x2−0.5x3, where 0≤x≤420 \leq x \leq 420≤x≤42.

Problem 7907

Evaluate the second derivative f′′(1)f^{\prime \prime}(1)f′′(1), f′′(5)f^{\prime \prime}(5)f′′(5), and f′′(7)f^{\prime \prime}(7)f′′(7) for f(x)=−8x3+6x2−9xf(x)=-8 x^{3}+6 x^{2}-9 xf(x)=−8x3+6x2−9x.

Problem 7908

Find points of inflection for the function f(x)=−4x3−84x2+6x−2f(x)=-4 x^{3}-84 x^{2}+6 x-2f(x)=−4x3−84x2+6x−2. Provide your answer as (x,y)(x, y)(x,y)-pairs.

Problem 7909

Find intervals where the function f(x)=−4x3−84x2+6x−2f(x)=-4 x^{3}-84 x^{2}+6 x-2f(x)=−4x3−84x2+6x−2 is concave up or down.

Problem 7910

Find the function fff and the number aaa in the limit that represents the derivative: lim⁡h→0(1+h)10−1h\lim _{h \rightarrow 0} \frac{(1+h)^{10}-1}{h}h→0lim​h(1+h)10−1​.

Problem 7911

Find the derivative of f(x)=−4x3−84x2+6x−2f(x)=-4x^{3}-84x^{2}+6x-2f(x)=−4x3−84x2+6x−2.

Problem 7912

Find the function fff and the number aaa in the limit representing the derivative: lim⁡h→0(1+h)10−1h\lim _{h \rightarrow 0} \frac{(1+h)^{10}-1}{h}h→0lim​h(1+h)10−1​.

Problem 7913

Find the constant ccc that ensures the function f(x)f(x)f(x) is continuous for all xxx in (−∞,∞)(-\infty, \infty)(−∞,∞).

Problem 7914

Gegeben ist f′(x)=3x2−4f^{\prime}(x) = 3x^2 - 4f′(x)=3x2−4. a) Bestimme den Grad von f(x)f(x)f(x) und begründe. b) Zeichne den Graphen von f(x)f(x)f(x). c) Finde die Steigung von f(x)f(x)f(x) im Tiefpunkt.

Problem 7915

Check if the function f(x)=x2−9x−3f(x)=\frac{x^{2}-9}{x-3}f(x)=x−3x2−9​ for x≠3x \neq 3x=3 and f(3)=10f(3)=10f(3)=10 is continuous at a=3a=3a=3. Select all that apply.

Problem 7916

Find the function fff and the number aaa such that lim⁡h→0416+h−24\lim _{h \rightarrow 0} \sqrt[4]{\frac{4}{16+h}-2}limh→0​416+h4​−2​ is the derivative of fff at aaa. a= a= a= f= f= f=

Problem 7917

Find slopes of secant lines for f(x)=19x3−xf(x)=19 x^{3}-xf(x)=19x3−x and conjecture the tangent slope at x=1x=1x=1. Round as needed.

Problem 7918

Find the function fff and the number aaa such that lim⁡h→016+h−2\lim_{h \rightarrow 0} \sqrt{16+h}-2limh→0​16+h​−2 represents the derivative. a= a= a= f= f= f=

Problem 7919

Check if the series ∑n=1∞2n2+7n+10\sum_{n=1}^{\infty} \frac{2}{n^{2}+7 n+10}∑n=1∞​n2+7n+102​ converges or diverges. If convergent, find the sum.

Problem 7920

Zeigen Sie, dass die Differenzenquotienten von f(x)=x2f(x)=x^{2}f(x)=x2 in [a;b][a ; b][a;b] und [a−1;b+1][a-1 ; b+1][a−1;b+1] gleich sind.

Problem 7921

Find the limit: lim⁡x→−2x+2x2+12−4\lim _{x \rightarrow-2} \frac{x+2}{\sqrt{x^{2}+12}-4}limx→−2​x2+12​−4x+2​

Problem 7922

Find which statements about the tangent line L(x)=8(x−2)+1L(x)=8(x-2)+1L(x)=8(x−2)+1 to g(x)=x3−4x+1g(x)=x^{3}-4x+1g(x)=x3−4x+1 at (2,1)(2,1)(2,1) are true.

Problem 7923

Given the equation dydt=t−2\frac{d y}{d t}=t-2dtdy​=t−2, find the slope at points (0,1)(0,1)(0,1), (0,5)(0,5)(0,5), and (525,152)(525,152)(525,152). Which functions could be solutions? Also, what info determines the slope at a point? A. yyy-value B. slope at t=0t=0t=0 C. ttt-value D. None.

Problem 7924

Find the function fff and the number aaa for the limit lim⁡h→010+h4−2h\lim _{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}limh→0​h410+h​−2​ where a=16a=\sqrt{16}a=16​.

Problem 7925

Find the net change and average rate of change of r(t)=5−15tr(t)=5-\frac{1}{5} tr(t)=5−51​t for t=5t=5t=5 and t=10t=10t=10.

Problem 7926

Find the function fff and the number aaa where a=16a=16a=16 given the limit lim⁡h→010+h4−2h\lim_{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}limh→0​h410+h​−2​.

Problem 7927

Find the limit lim⁡h→010+h4−2h\lim_{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}limh→0​h410+h​−2​, where a=16a=16a=16 and f=f=f=?

Problem 7928

Find the function fff and the number aaa for the limit lim⁡h→016+h4−2h\lim _{h \rightarrow 0} \frac{\sqrt[4]{16+h}-2}{h}limh→0​h416+h​−2​, where a=16a=16a=16.

Problem 7929

Find the function fff and the number aaa such that lim⁡h→010+h4−2h\lim _{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}limh→0​h410+h​−2​ represents the derivative. a=16 a = 16 a=16 f= f = f=

Problem 7930

Find the average velocity of an object with s(3)=177s(3)=177s(3)=177 and s(5)=233s(5)=233s(5)=233 over the interval [3,5][3,5][3,5].

Problem 7931

Find the derivative R′(1)R'(1)R′(1) for the function R(x)=ln⁡(x2⋅g(x))R(x)=\ln \left(x^{2} \cdot \sqrt{g(x)}\right)R(x)=ln(x2⋅g(x)​), where g(x)g(x)g(x) is undefined.

Problem 7932

Find the derivative of the function f(x)=(x2+3x)⋅sin⁡(x)f(x)=(x^{2}+3x) \cdot \sin(x)f(x)=(x2+3x)⋅sin(x). What is f′(x)f^{\prime}(x)f′(x)?

Problem 7933

Find the derivative of f(x)=(2x−1)2⋅xf(x)=(2x-1)^{2} \cdot \sqrt{x}f(x)=(2x−1)2⋅x​.

Problem 7934

Calculate the integral of the function f(x)=e12xf(x) = e^{\frac{1}{2} x}f(x)=e21​x.

Problem 7935

Berechnen Sie das Integral von e12x2e^{\frac{1}{2} x^{2}}e21​x2.

Problem 7936

Berechnen Sie das Integral von f(x)=4e−xf(x)=4 e^{-x}f(x)=4e−x.

Problem 7937

Find the derivative of f(x)=(4−3x)2⋅sin⁡(x)f(x)=(4-3x)^{2} \cdot \sin(x)f(x)=(4−3x)2⋅sin(x).

Problem 7938

Leiten Sie die Funktion fff mit der Produktregel und der Kettenregel ab. Beispiele: a) f(x)=x⋅sin⁡(3x)f(x)=x \cdot \sin(3x)f(x)=x⋅sin(3x), b) f(x)=(2x−1)2⋅xf(x)=(2x-1)^{2} \cdot \sqrt{x}f(x)=(2x−1)2⋅x​, usw.

Problem 7939

Berechnen Sie das Integral von f(x)=4x−3f(x) = 4x^{-3}f(x)=4x−3.

Problem 7940

Find the antiderivative of $f(x)=\left(\frac{1}{5}\right)^{x}$ over the interval $I=[0, 5]$ .

Evaluate the second derivative $f^{\prime \prime}(5), f^{\prime \prime}(3)$ , and $f^{\prime \prime}(4)$ for $f(x)=\sqrt{6x-2}$ .

Find the second derivative of the function $f(x)=-2 x^{2}-5 \sqrt[3]{x}-9$ .

Find the intervals where the function $f(x)=-4 x^{3}-72 x^{2}+5 x-2$ is concave up or down.

Find the second derivative of the function $f(x)=-8 x^{3}+6 x^{2}-9 x$ .

Find the point of diminishing returns for the sales function $S(x)=123+31.5 x^{2}-0.5 x^{3}$ , where $0 \leq x \leq 42$ .

Evaluate the second derivative $f^{\prime \prime}(1)$ , $f^{\prime \prime}(5)$ , and $f^{\prime \prime}(7)$ for $f(x)=-8 x^{3}+6 x^{2}-9 x$ .

Find points of inflection for the function $f(x)=-4 x^{3}-84 x^{2}+6 x-2$ . Provide your answer as $(x, y)$ -pairs.

Find intervals where the function $f(x)=-4 x^{3}-84 x^{2}+6 x-2$ is concave up or down.

Find the function $f$ and the number $a$ in the limit that represents the derivative: $\lim _{h \rightarrow 0} \frac{(1+h)^{10}-1}{h}$ .

Find the derivative of $f(x)=-4x^{3}-84x^{2}+6x-2$ .

Find the function $f$ and the number $a$ in the limit representing the derivative: $\lim _{h \rightarrow 0} \frac{(1+h)^{10}-1}{h}$ .

Find the constant $c$ that ensures the function $f(x)$ is continuous for all $x$ in $(-\infty, \infty)$ .

Gegeben ist $f^{\prime}(x) = 3x^2 - 4$ .
a) Bestimme den Grad von $f(x)$ und begründe.
b) Zeichne den Graphen von $f(x)$ .
c) Finde die Steigung von $f(x)$ im Tiefpunkt.

Check if the function $f(x)=\frac{x^{2}-9}{x-3}$ for $x \neq 3$ and $f(3)=10$ is continuous at $a=3$ . Select all that apply.

Find the function $f$ and the number $a$ such that $\lim _{h \rightarrow 0} \sqrt[4]{\frac{4}{16+h}-2}$ is the derivative of $f$ at $a$ .
$a=$ $f=$

Find slopes of secant lines for $f(x)=19 x^{3}-x$ and conjecture the tangent slope at $x=1$ . Round as needed.

Find the function $f$ and the number $a$ such that $\lim_{h \rightarrow 0} \sqrt{16+h}-2$ represents the derivative. $a=$ $f=$

Check if the series $\sum_{n=1}^{\infty} \frac{2}{n^{2}+7 n+10}$ converges or diverges. If convergent, find the sum.

Zeigen Sie, dass die Differenzenquotienten von $f(x)=x^{2}$ in $[a ; b]$ und $[a-1 ; b+1]$ gleich sind.

Find the limit: $\lim _{x \rightarrow-2} \frac{x+2}{\sqrt{x^{2}+12}-4}$

Find which statements about the tangent line $L(x)=8(x-2)+1$ to $g(x)=x^{3}-4x+1$ at $(2,1)$ are true.

Given the equation $\frac{d y}{d t}=t-2$ , find the slope at points $(0,1)$ , $(0,5)$ , and $(525,152)$ . Which functions could be solutions? Also, what info determines the slope at a point? A. $y$ -value B. slope at $t=0$ C. $t$ -value D. None.

Find the function $f$ and the number $a$ for the limit $\lim _{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}$ where $a=\sqrt{16}$ .

Find the net change and average rate of change of $r(t)=5-\frac{1}{5} t$ for $t=5$ and $t=10$ .

Find the function $f$ and the number $a$ where $a=16$ given the limit $\lim_{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}$ .

Find the limit $\lim_{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}$ , where $a=16$ and $f=$ ?

Find the function $f$ and the number $a$ for the limit $\lim _{h \rightarrow 0} \frac{\sqrt[4]{16+h}-2}{h}$ , where $a=16$ .

Find the function $f$ and the number $a$ such that $\lim _{h \rightarrow 0} \frac{\sqrt[4]{10+h}-2}{h}$ represents the derivative. $a = 16$ $f =$

Find the average velocity of an object with $s(3)=177$ and $s(5)=233$ over the interval $[3,5]$ .

Find the derivative $R'(1)$ for the function $R(x)=\ln \left(x^{2} \cdot \sqrt{g(x)}\right)$ , where $g(x)$ is undefined.

Find the derivative of the function $f(x)=(x^{2}+3x) \cdot \sin(x)$ . What is $f^{\prime}(x)$ ?

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

Calculate the integral of the function $f(x) = e^{\frac{1}{2} x}$ .

Berechnen Sie das Integral von $e^{\frac{1}{2} x^{2}}$ .

Berechnen Sie das Integral von $f(x)=4 e^{-x}$ .

Find the derivative of $f(x)=(4-3x)^{2} \cdot \sin(x)$ .

Leiten Sie die Funktion $f$ mit der Produktregel und der Kettenregel ab. Beispiele: a) $f(x)=x \cdot \sin(3x)$ , b) $f(x)=(2x-1)^{2} \cdot \sqrt{x}$ , usw.

Berechnen Sie das Integral von $f(x) = 4x^{-3}$ .

Integrieren Sie die Funktion $f(x)=\frac{6}{x^{2}}$ .

Find the derivative of the function $f(x)=3 x^{5} \cdot \cos (2 x)$ . What is $f^{\prime}(x)$ ?

Find the derivative of $F(t)=\left(\frac{1}{9 t+1}\right)^{5}$ . What is $F^{\prime}(t)=?$

How long for a shell to hit the ground if fired from a cliff at $80 \mathrm{m/s}$ at a $40^\circ$ angle, 200 m down?

Calculate the indefinite integral: $\int e^{x+e^{x}} d x$

Calculate the integral $\int \left(\cos^{3} x\right) \sqrt[3]{\sin x} \, dx$ .

Find the integral using trigonometric substitution: $\int \frac{x^{3}}{\sqrt{4+x^{2}}} d x$

Find the indefinite integral using integration by parts: $\int x^{2} \sin 2 x \, dx$

Find the integral of the natural logarithm: $\int \ln x \, dx$ .

Find the derivative of $y=\frac{x \sin x}{1+\cos x}$ without simplifying. Show all work. [5pts]

Finde die Tangentengleichung $t_{1}$ für $f(x)=\frac{1}{3} x^{2}$ bei $x_{0}=-2$ und die orthogonale Tangente.

Evaluate the integral $\int_{0}^{\pi / 2} \cos ^{8} x \, dx$ using Wallis's Formula.

Find the derivative of $(5x + 4)^{100}$ with respect to $x$ . Do not simplify your answer. [3pts]

Zeigen Sie, dass der Punkt $T(x_{0} \mid 0)$ auch ein Tiefpunkt von $g(x)=x^{2} \cdot f(x)$ ist, wenn $f^{\prime \prime}(x_{0})>0$ .

Find the derivative of $y=\csc(e^{x})$ . Do not simplify your answer. [3pts]

Calculate the car's acceleration if it speeds from $0 \mathrm{~m/s}$ to $4 \mathrm{~m/s}$ in 0.5 seconds. I $\mathrm{m/s}^{2}$

Find the derivative of $y=\csc(e^{x})$ . Do not simplify your answer. [3pts]

Find the derivative of $y=\tan^{10}(x^{5})$ , denoted as $y'$ . Do not simplify your answer. [4pts]

Calculate the car's acceleration if its velocity changes from $0 \mathrm{~m/s}$ to $10 \mathrm{~m/s}$ in 0.5 seconds. $\mathrm{m/s}^{2}$

Differentiate implicitly: $x \sin (y) + y \sin (x) = 3$ . Find $\frac{d y}{d x}$ .

Find the tangent line equation using implicit differentiation for $y \sin (16 x) = x \cos (2 y)$ at $(\pi / 2, \pi / 4)$ .

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ .
a) Bestimme die Steigung von $f$ bei $P(2 \mid f(2))$ . b) Gibt es Punkte mit waagerechter Tangente? c) Wo hat die Tangente die Steigung 1? Finde die Gleichung in diesen Punkten.

Find the tangent line equation $y=$ to the curve $y \sin (16 x)=x \cos (2 y)$ at $(\pi / 2, \pi / 4)$ using implicit differentiation.

Find the derivative of $f(x)=2 x^{2} x^{\cos (x)}$ . What is $f^{\prime}(x)=?$

How much will $\$ 2000$ grow in 10 years at a continuous interest rate of $3 \%$ ?

Find the derivative of $\ln \left(\frac{(6 x+3)^{5}}{\sqrt[3]{3 x^{2}-3 x+4}}\right)$ .

Find the horizontal asymptotes of $f(x)=\frac{3000+60x}{x}$ .

Find the rate of change of $p$ with respect to $q$ given $p$ from a lens with focal length $6 \mathrm{cms}$ and $\frac{1}{6}=\frac{1}{p}+\frac{1}{q}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin ^{3}(x)}{\sin x-\tan x}$ .

Find the limit: $L = \lim _{x \rightarrow 0} \frac{2-\cos (3 x)-\cos (4 x)}{x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan ^{2}(3 x)+\sin \left(5 x^{2}\right)}{x^{2}}$ .