Calculus

Problem 27901

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

See Solution

Problem 27902

Let $g(x)$ be defined as: $g(x) = -8x + 36$ for $x \leq 4$ and $g(x) = x^2 - 3x$ for $x > 4$ . Is $g$ continuous/differentiable at $x=4$ ? Options: A, B, C, D.

See Solution

Problem 27903

Find $\frac{d y}{d x}$ at the point $(1,-1)$ for the equation $x^{3}+x y-4 y=4$ .

See Solution

Problem 27904

For $f(x)=2x+3$ , find $\delta$ in terms of $\epsilon$ such that if $|x-5|<\delta$ , then $|f(x)-13|<\epsilon$ . Also, check if $f(x)$ is discontinuous and justify.

See Solution

Problem 27905

Find the relative maximum of $y=\frac{3}{4} x^{\frac{5}{3}}-\frac{3}{4} x^{\frac{8}{3}}$ . Find $y'$ and critical points, then use the second derivative test.

See Solution

Problem 27906

Differentiate these functions without simplifying: (a) $f(x)=x^{2} \tan (x)+\sin (x) \sec (x)+\cos x \csc x+\sqrt{x} \sin x$ ; (b) $g(x)=\frac{x^{3}+\sin (x)}{1+x^{2}}$ .

See Solution

Problem 27907

Problem 4: Justify your work and state any theorems used. (a) Show $f(x)=0$ has a solution in $(-1,1)$ for $f(x)=2x^3+1$ . (b) Given $1+2x-x^3 \leq g(x) \leq 1+3x^2+x^5$ , find $\lim_{x \to 0} f(x)$ and state any theorems.

See Solution

Problem 27908

Gegeben ist die Funktion $f(x)=\frac{1}{12} x^{4}+\frac{1}{6} x^{3}-x^{2}+5 x$ . Finde die ersten drei Ableitungen und die Wendepunkte.

See Solution

Problem 27909

Find the derivative of $h(x)$ , where $h(x) = \tan^{-1}(2x - 1)$ .

See Solution

Problem 27910

Find the derivative $f^{\prime}(x)$ for $f(x)=2 x^{2}+3 x+5$ using limits, then find the tangent line at $x=0$ .

See Solution

Problem 27911

Find the time $t$ when the particle with position $x(t)=2 t^{3}-21 t^{2}+72 t-53$ is at rest.

See Solution

Problem 27912

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

See Solution

Problem 27913

Find $f^{\prime}(1)$ if the tangent line at $(1,7)$ passes through $(-2,-2)$ .

See Solution

Problem 27914

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{1+x}-1}{x}$ .

See Solution

Problem 27915

Find the limit: $\lim _{x \rightarrow 16} \frac{4-\sqrt{x}}{16-x}$ .

See Solution

Problem 27916

Find the limit as $x$ approaches infinity for $\frac{x^{2}-x+10}{x+2}$ . What is the result? A. $-\infty$ B. 0 C. $\infty$ D. 1

See Solution

Problem 27917

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{(x+3)^{2}}-\frac{1}{9}}{x}$ .

See Solution

Problem 27918

Find the slope of the tangent line to the curve $y^{3}-x y^{2}+x^{3}=5$ at the point $(1,2)$ .

See Solution

Problem 27919

Find $\frac{d y}{d x}$ if $\sin (x+y)=3 x-2 y$ . Options: (A) $\frac{3-\cos (x+y)}{2}$ , (B) $\frac{1-\cos (x+y)}{\cos (x+y)}$ , (C) $\frac{3}{2+\cos (x+y)}$ , (D) $\frac{3-\cos (x+y)}{2+\cos (x+y)}$ .

See Solution

Problem 27920

Find the limit of $y=\frac{1}{x-2}$ as $x$ approaches 2 from the right. What is the limit? A. 0 B. no limit C. $\infty$ D. $-\infty$

See Solution

Problem 27921

Find the limit: $\lim _{x \rightarrow 9} \frac{\frac{1}{x^{2}}-\frac{1}{81}}{x-9}$ .

See Solution

Problem 27922

Given $f(-2)=3$ , $f'(-2)=4$ , $g(4)=5$ , and $g'(4)=2$ , find $\frac{d y}{d x}$ at $(-2,4)$ for $f(x) g(y)=17-x-y$ . (A) -27 (B) $-\frac{11}{3}$ (C) -3 (D) $-\frac{4}{7}$

See Solution

Problem 27923

Find the tangent and normal line equations for $y^{2}=x^{3}$ at $(1,-1)$ . What is $\frac{d y}{d x}$ ? A. $3 x^{2}$ B. $\frac{3 x^{2}}{2}$ C. $\frac{3 x^{2}}{2 y}$

See Solution

Problem 27924

Find the intervals where the graph of $y=g(x)$ is concave down. Use interval notation for your answer: (a,b). $g(x)=\operatorname{abs}\left((2 x+4)(x-1)^{\wedge} 2\right)$

See Solution

Problem 27925

If $f$ is continuous and $f(x)=\frac{x^{2}-4}{x+2}$ for $x \neq -2$ , find $f(-2)$ . Also, find $\lim _{x \rightarrow 2} f(x)$ for $f(x)=\left\{\begin{array}{ll}\ln x & 0<x \leq 2 \\ x^{2} \ln 2 & 2<x \leq 4\end{array}\right.$ .

See Solution

Problem 27926

Find the slope of the curve $y=\frac{1}{x+1}$ at $x=1$ . Where does the slope equal -1?

See Solution

Problem 27927

Find the derivative $\frac{d}{d x}\left(\frac{1}{x^{3}}-\frac{1}{x}+x^{2}\right)$ at $x=-1$ and $f^{\prime}(2)$ for $f(x)=\sqrt{2 x}$ .

See Solution

Problem 27928

Find the limit: $\lim _{x \rightarrow \infty} \frac{(2 x-1)(3-x)}{(x-1)(x+3)}$ and evaluate $f(\pi / 9)$ for $f(x)=\cos (3 x)$ .

See Solution

Problem 27929

17. For which $x$ values is the graph of $y=-5 /(x-2)$ concave downward?
18. Where does the function $f(x)=x^{3}-3 x^{2}$ have a relative maximum?
19. When does $f^{\prime \prime}(x)=x(x+1)(x-2)^{2}$ have inflection points?
20. Find $\lim _{x \rightarrow \infty} \frac{(2 x-1)(3-x)}{(x-1)(x+3)}$ .
21. What is $f^{\prime}(\pi / 9)$ if $f(x)=\cos (3 x)$ ?

See Solution

Problem 27930

Calculate the time for the Apollo command module to orbit the moon at $120 \mathrm{~km}$ altitude. Moon's mass: $7.35 \times 10^{22}$ kg, radius: $1.74 \times 10^{6} \mathrm{~m}$ .

See Solution

Problem 27931

Given $f(0)=2$ , $f^{\prime}(0)=-3$ , and $f^{\prime \prime}(0)=0$ , find tangents and analyze $g$ defined by $g^{\prime}(x)=e^{-2 x}(3 f(x)+2 f^{\prime}(x))$ .

See Solution

Problem 27932

Calculate the area between the curves $y=\frac{3}{x}$ , $y=\frac{3}{x^{3}}$ , and the line $x=3$ .

See Solution

Problem 27933

Calculate the area between the curves $y=e^{\frac{x}{2}}$ , $y=x^{2}-1$ , and the lines $x=1$ , $x=-1$ .

See Solution

Problem 27934

Calculate the area between the curves $y=(x-3)^{2}$ and $y=4x-15$ .

See Solution

Problem 27935

Calculate the area between $y=\sin(x)$ , $y=4x$ , $x=\frac{\pi}{4}$ , and $x=\frac{3\pi}{4}$ .

See Solution

Problem 27936

Calculate the area of the region $R$ enclosed by $y=x^{3}-6 x^{2}-x+6$ , the $x$ -axis, $x=-1$ , and $x=5$ .

See Solution

Problem 27937

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

See Solution

Problem 27938

Find the limit: $\lim _{x \rightarrow \infty}\left(1+\frac{7}{3 x}\right)^{x-1}$ .

See Solution

Problem 27939

Berechne die Steigung der Tangente an $f(x)=0,4 x(x+2)(x-3)$ im Punkt $P=(3, f(3))$ .

See Solution

Problem 27940

Find $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x^{2}}\right)^{3 x-4}$ .

See Solution

Problem 27941

Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{1}{1+\tan^{\sqrt{2}} x} dx$ .

See Solution

Problem 27942

Evaluate the integral $\int_{-4 \pi \sqrt{2}}^{4 \pi \sqrt{2}}\left(\frac{\sin x}{1+x^{4}}+1\right) d x$ .

See Solution

Problem 27943

Find the limit as $x$ approaches infinity for $e^{\frac{3x-3}{7}}$ .

See Solution

Problem 27944

Find the limit as $x$ approaches 0 for the function $f(x) = 3 \operatorname{Ln}(x+1)$ .

See Solution

Problem 27945

Find the volume of the solid formed by revolving the area between $x=y^{2}$ , $x=6-y$ , and the $x$ -axis around the $y$ -axis using the washer method.

See Solution

Problem 27946

Zeige, dass die Funktion $f(x)=\frac{x^{4}}{8}-\frac{4 x^{3}}{3}+4 x^{2}+1$ bei $x=0$ ein lokales Minimum hat.

See Solution

Problem 27947

Find the derivative of $f(x) = \frac{x}{2x + 1}$ using the quotient rule step by step.

See Solution

Problem 27948

Evaluate the integral from 2 to 5: $\int_{2}^{5} t^{2} \ln (4 t) d t$ .

See Solution

Problem 27949

Gegeben ist die Funktion $f$ . Kreuze die richtigen Aussagen an:
1. $f^{\prime \prime}(a)=0$ → Wendestelle bei $a$ .
2. $f^{\prime}(x)<0$ für $x \in (x_{1}, x_{2})$ → linksgekümmt in $(x_{1}, x_{2})$ .
3. Nullstellen von $f$ sind Lösungen von $f^{\prime}(x)=0$ .
4. $f^{\prime}(x_{1})<0$ und $f^{\prime}(x_{2})>0$ → Wendestelle bei $a$ .
5. $f$ ist streng monoton steigend, wenn $f^{\prime}(x)>0$ für alle $x \in \mathbb{R}$ .

See Solution

Problem 27950

Find $f'(1)$ for $f(x)=\frac{x}{2x+1}$ using the definition of the derivative.

See Solution

Problem 27951

Find the volume of the solid of revolution formed by the region in the first quadrant between $x=y^{2}$ and $x=6-y$ around $y=4$ using the shell method.

See Solution

Problem 27952

Find the derivative of $(1+\sqrt{\sec x})^{2}$ with respect to $x$ .

See Solution

Problem 27953

Find the derivative of $(x+1)^{5} e^{x}$ with respect to $x$ .

See Solution

Problem 27954

Evaluate the integral of $\sin^3 r \cos^2 r \, dr$ .

See Solution

Problem 27955

Find the derivative of $\frac{\tan x}{\sqrt{x^{3}}+1}$ with respect to $x$ .

See Solution

Problem 27956

Find the derivative of the function $5^{2x}$ . What is $\frac{d}{dx} 5^{2x}$ ?

See Solution

Problem 27957

Bestimme die Gleichung der Wendetangente $t_{w}$ für die Funktion $f(x)=-3 x^{3}+9 x^{2}-4$ bei $W=(1/2)$ .

See Solution

Problem 27958

Find the tangent line equation to the curve $x y^{2}+(2 x+y)^{2}=1$ at the point $(0,1)$ .

See Solution

Problem 27959

An object moves along a line with $s(t)=t^{5/2}-5t^{3/2}, t \geq 0$ . Find: a) velocity at $t=1$ , b) when moving right, c) acceleration at $t=1$ .

See Solution

Problem 27960

Given functions $f$ and $g$ , compute: a) $(2 f - 5 g)^{\prime}(1)$ b) $\left(\frac{f}{f + g}\right)^{\prime}(2)$ c) $(g \circ g)^{\prime}(1)$ d) $(f g)^{\prime}(0)$ Use differentiation rules in your solutions.

See Solution

Problem 27961

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

See Solution

Problem 27962

Find the derivative of $f(x)=(x^{2}+1)^{1/x}$ using logarithmic differentiation.

See Solution

Problem 27963

Estimate $(83)^{1/4}$ using linear approximation. Identify the function and base point. Is it an overestimate or underestimate?

See Solution

Problem 27964

A balloon inflates at 1 in³/sec. Find the radius change rate when volume is 10 in³. Use $V = \frac{4}{3} \pi r^{3}$ .

See Solution

Problem 27965

Does the function $f(x)=\frac{8}{(x-2)^{4}+1}$ for $x \in[2, \infty)$ have an inverse?

See Solution

Problem 27966

Approximate the area under $h(x)=x^{3}+2$ from $x=-1$ to $x=5$ using a right Riemann sum with 3 subdivisions.

See Solution

Problem 27967

Find the surface area of revolution of the curve $y=9 x^{3}$ from $x=0$ to $x=2.7$ around the $x$ -axis.

See Solution

Problem 27968

Differentiate the following with $a$ as a constant: 1) $\tan(5 - t^2)$ 2) $\frac{y-5}{5-y^2}$ .

See Solution

Problem 27969

Find the derivative $y'$ for the equation $e^{x+y} + x - y^2 = 3$ .

See Solution

Problem 27970

Find the average rate of change of $y=\sqrt[3]{x}$ over $[2,7]$ rounded to the nearest hundredth.

See Solution

Problem 27971

Differentiate each function, treating $a$ as a constant:
1. $f(x)=(1+2x+x^3)^{1/2}$
2. $y=\frac{2t}{1+t^2}$
3. $g(t)=a t(a+t)^{4}$
4. $\sin(2\theta+5)$
5. $\left(y+\frac{1}{y}\right)^{5}$
6. $e^{x^2+x}$
7. $\ln(x \cos x)$
8. $\tan(5-t^2)$
9. $\frac{y-5}{5-y^2}$
10. Find $y^{\prime}$ for $e^{x+y}+x-y^2=3$ .

See Solution

Problem 27972

Find the tangent line equation for $y = g(x) = |-(2x + 10)(x - 4)^2|$ at each inflection point where $g$ is differentiable.

See Solution

Problem 27973

Find the volume of the solid of revolution for the region between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis using different methods.

See Solution

Problem 27974

Given the function $f(x)=\frac{4}{3} x-\tan x$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ , determine where $f$ is increasing.

See Solution

Problem 27975

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=0$ , and $x=\frac{\pi}{4}$ around the $y$ -axis.

See Solution

Problem 27976

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=0$ , and $x=\frac{\pi}{4}$ around $x=\frac{\pi}{4}$ .

See Solution

Problem 27977

Calculate the value of the integral $\int_{0}^{\frac{\pi}{6}}(\sec x+\tan x)^{2} d x$ . Choose the correct answer.

See Solution

Problem 27978

Find the area between the curves $y=5 \sqrt{x}$ and $y=5 x^{2}$ . What is the area?

See Solution

Problem 27979

Find the volume of the solid formed by rotating the area between $y=\sec x$ , $y=0$ , $x=0$ , and $x=\frac{\pi}{4}$ around $x=\frac{\pi}{4}$ .

See Solution

Problem 27980

Find the volume of a solid with a square cross-section under the curve $y=4-x^{2}$ in the first quadrant.

See Solution

Problem 27981

Find the volume change rate of a sphere when the radius $r=9$ inches and $r$ increases at 3 inches/minute.

See Solution

Problem 27982

Find the rate of change of the area of a circle as its radius $r$ increases at 4 cm/min.

See Solution

Problem 27983

A conical tank is 10 ft wide and 12 ft deep. Water flows in at 10 ft³/min. Find the depth change rate when water is 8 ft deep.

See Solution

Problem 27984

Find the area change rates of a circle when $r=8$ cm and $r=32$ cm, given $r$ increases at 4 cm/min.

See Solution

Problem 27985

Find the volume of a solid with a disk base $x^{2}+y^{2} \leq 4$ and equilateral triangle cross-sections.

See Solution

Problem 27986

A 25 ft ladder leans against a wall. Base moves away at 2 ft/s. Find top speed down when base is 7 ft, 15 ft, and 24 ft from wall.

See Solution

Problem 27987

Approximate the area under $h(x)$ from $x=0$ to $x=7$ using a left Riemann sum with 3 unequal intervals.

See Solution

Problem 27988

Approximate the area under $h(x)$ from $x=3$ to $x=13$ using a right Riemann sum with 4 unequal intervals.

See Solution

Problem 27989

Approximate the area under $f(x)$ from $x=0$ to $x=8$ using a right Riemann sum with 3 unequal intervals.

See Solution

Problem 27990

Approximate the area under $h(x)=x^{3}+2$ from $x=-1$ to $x=5$ using a right Riemann sum with 3 subdivisions.

See Solution

Problem 27991

Find the derivative of $f(x) = \sqrt{x}$ .

See Solution

Problem 27992

Bestimme die Tangentengleichung der Funktion $f(x)=2e^x+2$ am Schnittpunkt mit der $y$ -Achse.

See Solution

Problem 27993

Calculate the limit: $\lim _{x \rightarrow-\infty} \frac{x^{6}+7 x^{4}-40}{1-x-5 x^{7}}$ .

See Solution

Problem 27994

Find the derivative of the function $y=(4-x)e^{4-x}$ and identify its relative extrema.

See Solution

Problem 27995

Mr. Rooley's students measure bacteria growth with $n(t)=990 e^{0.14 t}$ .
(a) Find the growth rate in %. (b) What is the initial population at $t=0$ ? (c) How many bacteria after 3 hours?

See Solution

Problem 27996

Calculate the integral of $\frac{5}{x}$ with respect to $x$ .

See Solution

Problem 27997

Evaluate the integral $\int \frac{x^{3}-3 x^{2}+5}{x-3} \, dx$ .

See Solution

Problem 27998

Evaluate the integral: $\int \frac{x^{2}+x+1}{x^{2}+1} d x$

See Solution

Problem 27999

Calculate the integral: $\int \frac{1}{\sqrt{x+1}} d x$

See Solution

Problem 28000

Find the integral: $\int \frac{x^{3}-3 x^{2}+5}{x-3} \, dx$ .

See Solution

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Calculus

Problem 27901

Find the limit: lim⁡x→2x2+2x−8x2−2x\lim _{x \rightarrow 2} \frac{x^{2}+2 x-8}{x^{2}-2 x}limx→2​x2−2xx2+2x−8​.

Problem 27902

Let g(x)g(x)g(x) be defined as: g(x)=−8x+36g(x) = -8x + 36g(x)=−8x+36 for x≤4x \leq 4x≤4 and g(x)=x2−3xg(x) = x^2 - 3xg(x)=x2−3x for x>4x > 4x>4. Is ggg continuous/differentiable at x=4x=4x=4? Options: A, B, C, D.

Problem 27903

Find dydx\frac{d y}{d x}dxdy​ at the point (1,−1)(1,-1)(1,−1) for the equation x3+xy−4y=4x^{3}+x y-4 y=4x3+xy−4y=4.

Problem 27904

For f(x)=2x+3f(x)=2x+3f(x)=2x+3, find δ\deltaδ in terms of ϵ\epsilonϵ such that if ∣x−5∣<δ|x-5|<\delta∣x−5∣<δ, then ∣f(x)−13∣<ϵ|f(x)-13|<\epsilon∣f(x)−13∣<ϵ. Also, check if f(x)f(x)f(x) is discontinuous and justify.

Problem 27905

Find the relative maximum of y=34x53−34x83y=\frac{3}{4} x^{\frac{5}{3}}-\frac{3}{4} x^{\frac{8}{3}}y=43​x35​−43​x38​. Find y′y'y′ and critical points, then use the second derivative test.

Problem 27906

Problem 27907

Problem 27908

Gegeben ist die Funktion f(x)=112x4+16x3−x2+5xf(x)=\frac{1}{12} x^{4}+\frac{1}{6} x^{3}-x^{2}+5 xf(x)=121​x4+61​x3−x2+5x. Finde die ersten drei Ableitungen und die Wendepunkte.

Problem 27909

Find the derivative of h(x)h(x)h(x), where h(x)=tan⁡−1(2x−1)h(x) = \tan^{-1}(2x - 1)h(x)=tan−1(2x−1).

Problem 27910

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=2x2+3x+5f(x)=2 x^{2}+3 x+5f(x)=2x2+3x+5 using limits, then find the tangent line at x=0x=0x=0.

Problem 27911

Find the time ttt when the particle with position x(t)=2t3−21t2+72t−53x(t)=2 t^{3}-21 t^{2}+72 t-53x(t)=2t3−21t2+72t−53 is at rest.

Problem 27912

Find the limit: lim⁡x→∞x+x21−2x2\lim _{x \rightarrow \infty} \frac{x+x^{2}}{1-2 x^{2}}limx→∞​1−2x2x+x2​.

Problem 27913

Find f′(1)f^{\prime}(1)f′(1) if the tangent line at (1,7)(1,7)(1,7) passes through (−2,−2)(-2,-2)(−2,−2).

Problem 27914

Find the limit: lim⁡x→011+x−1x\lim _{x \rightarrow 0} \frac{\frac{1}{1+x}-1}{x}limx→0​x1+x1​−1​.

Problem 27915

Find the limit: lim⁡x→164−x16−x\lim _{x \rightarrow 16} \frac{4-\sqrt{x}}{16-x}limx→16​16−x4−x​​.

Problem 27916

Find the limit as xxx approaches infinity for x2−x+10x+2\frac{x^{2}-x+10}{x+2}x+2x2−x+10​. What is the result? A. −∞-\infty−∞ B. 0 C. ∞\infty∞ D. 1

Problem 27917

Find the limit: lim⁡x→01(x+3)2−19x\lim _{x \rightarrow 0} \frac{\frac{1}{(x+3)^{2}}-\frac{1}{9}}{x}limx→0​x(x+3)21​−91​​.

Problem 27918

Find the slope of the tangent line to the curve y3−xy2+x3=5y^{3}-x y^{2}+x^{3}=5y3−xy2+x3=5 at the point (1,2)(1,2)(1,2).

Problem 27919

Problem 27920

Find the limit of y=1x−2y=\frac{1}{x-2}y=x−21​ as xxx approaches 2 from the right. What is the limit? A. 0 B. no limit C. ∞\infty∞ D. −∞-\infty−∞

Problem 27921

Find the limit: lim⁡x→91x2−181x−9\lim _{x \rightarrow 9} \frac{\frac{1}{x^{2}}-\frac{1}{81}}{x-9}limx→9​x−9x21​−811​​.

Problem 27922

Problem 27923

Find the tangent and normal line equations for y2=x3y^{2}=x^{3}y2=x3 at (1,−1)(1,-1)(1,−1). What is dydx\frac{d y}{d x}dxdy​? A. 3x23 x^{2}3x2 B. 3x22\frac{3 x^{2}}{2}23x2​ C. 3x22y\frac{3 x^{2}}{2 y}2y3x2​

Problem 27924

Find the intervals where the graph of y=g(x)y=g(x)y=g(x) is concave down. Use interval notation for your answer: (a,b). g(x)=abs⁡((2x+4)(x−1)∧2)g(x)=\operatorname{abs}\left((2 x+4)(x-1)^{\wedge} 2\right)g(x)=abs((2x+4)(x−1)∧2)

Problem 27925

Problem 27926

Find the slope of the curve y=1x+1y=\frac{1}{x+1}y=x+11​ at x=1x=1x=1. Where does the slope equal -1?

Problem 27927

Find the derivative ddx(1x3−1x+x2)\frac{d}{d x}\left(\frac{1}{x^{3}}-\frac{1}{x}+x^{2}\right)dxd​(x31​−x1​+x2) at x=−1x=-1x=−1 and f′(2)f^{\prime}(2)f′(2) for f(x)=2xf(x)=\sqrt{2 x}f(x)=2x​.

Problem 27928

Find the limit: lim⁡x→∞(2x−1)(3−x)(x−1)(x+3)\lim _{x \rightarrow \infty} \frac{(2 x-1)(3-x)}{(x-1)(x+3)}limx→∞​(x−1)(x+3)(2x−1)(3−x)​ and evaluate f(π/9)f(\pi / 9)f(π/9) for f(x)=cos⁡(3x)f(x)=\cos (3 x)f(x)=cos(3x).

Problem 27929

Problem 27930

Calculate the time for the Apollo command module to orbit the moon at 120 km120 \mathrm{~km}120 km altitude. Moon's mass: 7.35×10227.35 \times 10^{22}7.35×1022 kg, radius: 1.74×106 m1.74 \times 10^{6} \mathrm{~m}1.74×106 m.

Problem 27931

Given f(0)=2f(0)=2f(0)=2, f′(0)=−3f^{\prime}(0)=-3f′(0)=−3, and f′′(0)=0f^{\prime \prime}(0)=0f′′(0)=0, find tangents and analyze ggg defined by g′(x)=e−2x(3f(x)+2f′(x))g^{\prime}(x)=e^{-2 x}(3 f(x)+2 f^{\prime}(x))g′(x)=e−2x(3f(x)+2f′(x)).

Problem 27932

Calculate the area between the curves y=3xy=\frac{3}{x}y=x3​, y=3x3y=\frac{3}{x^{3}}y=x33​, and the line x=3x=3x=3.

Problem 27933

Calculate the area between the curves y=ex2y=e^{\frac{x}{2}}y=e2x​, y=x2−1y=x^{2}-1y=x2−1, and the lines x=1x=1x=1, x=−1x=-1x=−1.

Problem 27934

Calculate the area between the curves y=(x−3)2y=(x-3)^{2}y=(x−3)2 and y=4x−15y=4x-15y=4x−15.

Problem 27935

Calculate the area between y=sin⁡(x)y=\sin(x)y=sin(x), y=4xy=4xy=4x, x=π4x=\frac{\pi}{4}x=4π​, and x=3π4x=\frac{3\pi}{4}x=43π​.

Problem 27936

Calculate the area of the region RRR enclosed by y=x3−6x2−x+6y=x^{3}-6 x^{2}-x+6y=x3−6x2−x+6, the xxx-axis, x=−1x=-1x=−1, and x=5x=5x=5.

Problem 27937

Berechne die Ableitung von f(x)=1x⋅ex2f(x)=\frac{1}{x} \cdot e^{x^{2}}f(x)=x1​⋅ex2.

Problem 27938

Find the limit: lim⁡x→∞(1+73x)x−1\lim _{x \rightarrow \infty}\left(1+\frac{7}{3 x}\right)^{x-1}limx→∞​(1+3x7​)x−1.

Problem 27939

Berechne die Steigung der Tangente an f(x)=0,4x(x+2)(x−3)f(x)=0,4 x(x+2)(x-3)f(x)=0,4x(x+2)(x−3) im Punkt P=(3,f(3))P=(3, f(3))P=(3,f(3)).

Problem 27940

Find lim⁡x→∞(1+1x2)3x−4\lim _{x \rightarrow \infty}\left(1+\frac{1}{x^{2}}\right)^{3 x-4}limx→∞​(1+x21​)3x−4.

Problem 27941

Evaluate the integral ∫0π211+tan⁡2xdx\int_{0}^{\frac{\pi}{2}} \frac{1}{1+\tan^{\sqrt{2}} x} dx∫02π​​1+tan2​x1​dx.

Problem 27942

Evaluate the integral ∫−4π24π2(sin⁡x1+x4+1)dx\int_{-4 \pi \sqrt{2}}^{4 \pi \sqrt{2}}\left(\frac{\sin x}{1+x^{4}}+1\right) d x∫−4π2​4π2​​(1+x4sinx​+1)dx.

Problem 27943

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

Let $g(x)$ be defined as: $g(x) = -8x + 36$ for $x \leq 4$ and $g(x) = x^2 - 3x$ for $x > 4$ . Is $g$ continuous/differentiable at $x=4$ ? Options: A, B, C, D.

Find $\frac{d y}{d x}$ at the point $(1,-1)$ for the equation $x^{3}+x y-4 y=4$ .

For $f(x)=2x+3$ , find $\delta$ in terms of $\epsilon$ such that if $|x-5|<\delta$ , then $|f(x)-13|<\epsilon$ . Also, check if $f(x)$ is discontinuous and justify.

Find the relative maximum of $y=\frac{3}{4} x^{\frac{5}{3}}-\frac{3}{4} x^{\frac{8}{3}}$ . Find $y'$ and critical points, then use the second derivative test.

Gegeben ist die Funktion $f(x)=\frac{1}{12} x^{4}+\frac{1}{6} x^{3}-x^{2}+5 x$ . Finde die ersten drei Ableitungen und die Wendepunkte.

Find the derivative of $h(x)$ , where $h(x) = \tan^{-1}(2x - 1)$ .

Find the derivative $f^{\prime}(x)$ for $f(x)=2 x^{2}+3 x+5$ using limits, then find the tangent line at $x=0$ .

Find the time $t$ when the particle with position $x(t)=2 t^{3}-21 t^{2}+72 t-53$ is at rest.

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

Find $f^{\prime}(1)$ if the tangent line at $(1,7)$ passes through $(-2,-2)$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{1+x}-1}{x}$ .

Find the limit: $\lim _{x \rightarrow 16} \frac{4-\sqrt{x}}{16-x}$ .

Find the limit as $x$ approaches infinity for $\frac{x^{2}-x+10}{x+2}$ . What is the result? A. $-\infty$ B. 0 C. $\infty$ D. 1

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{(x+3)^{2}}-\frac{1}{9}}{x}$ .

Find the slope of the tangent line to the curve $y^{3}-x y^{2}+x^{3}=5$ at the point $(1,2)$ .

Find the limit of $y=\frac{1}{x-2}$ as $x$ approaches 2 from the right. What is the limit? A. 0 B. no limit C. $\infty$ D. $-\infty$

Find the limit: $\lim _{x \rightarrow 9} \frac{\frac{1}{x^{2}}-\frac{1}{81}}{x-9}$ .

Find the tangent and normal line equations for $y^{2}=x^{3}$ at $(1,-1)$ . What is $\frac{d y}{d x}$ ? A. $3 x^{2}$ B. $\frac{3 x^{2}}{2}$ C. $\frac{3 x^{2}}{2 y}$

Find the intervals where the graph of $y=g(x)$ is concave down. Use interval notation for your answer: (a,b). $g(x)=\operatorname{abs}\left((2 x+4)(x-1)^{\wedge} 2\right)$

Find the slope of the curve $y=\frac{1}{x+1}$ at $x=1$ . Where does the slope equal -1?

Find the derivative $\frac{d}{d x}\left(\frac{1}{x^{3}}-\frac{1}{x}+x^{2}\right)$ at $x=-1$ and $f^{\prime}(2)$ for $f(x)=\sqrt{2 x}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{(2 x-1)(3-x)}{(x-1)(x+3)}$ and evaluate $f(\pi / 9)$ for $f(x)=\cos (3 x)$ .

Calculate the time for the Apollo command module to orbit the moon at $120 \mathrm{~km}$ altitude. Moon's mass: $7.35 \times 10^{22}$ kg, radius: $1.74 \times 10^{6} \mathrm{~m}$ .

Given $f(0)=2$ , $f^{\prime}(0)=-3$ , and $f^{\prime \prime}(0)=0$ , find tangents and analyze $g$ defined by $g^{\prime}(x)=e^{-2 x}(3 f(x)+2 f^{\prime}(x))$ .

Calculate the area between the curves $y=\frac{3}{x}$ , $y=\frac{3}{x^{3}}$ , and the line $x=3$ .

Calculate the area between the curves $y=e^{\frac{x}{2}}$ , $y=x^{2}-1$ , and the lines $x=1$ , $x=-1$ .

Calculate the area between the curves $y=(x-3)^{2}$ and $y=4x-15$ .

Calculate the area between $y=\sin(x)$ , $y=4x$ , $x=\frac{\pi}{4}$ , and $x=\frac{3\pi}{4}$ .

Calculate the area of the region $R$ enclosed by $y=x^{3}-6 x^{2}-x+6$ , the $x$ -axis, $x=-1$ , and $x=5$ .

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

Find the limit: $\lim _{x \rightarrow \infty}\left(1+\frac{7}{3 x}\right)^{x-1}$ .

Berechne die Steigung der Tangente an $f(x)=0,4 x(x+2)(x-3)$ im Punkt $P=(3, f(3))$ .

Find $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x^{2}}\right)^{3 x-4}$ .

Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{1}{1+\tan^{\sqrt{2}} x} dx$ .

Evaluate the integral $\int_{-4 \pi \sqrt{2}}^{4 \pi \sqrt{2}}\left(\frac{\sin x}{1+x^{4}}+1\right) d x$ .

Find the limit as $x$ approaches infinity for $e^{\frac{3x-3}{7}}$ .

Find the limit as $x$ approaches 0 for the function $f(x) = 3 \operatorname{Ln}(x+1)$ .

Find the volume of the solid formed by revolving the area between $x=y^{2}$ , $x=6-y$ , and the $x$ -axis around the $y$ -axis using the washer method.

Zeige, dass die Funktion $f(x)=\frac{x^{4}}{8}-\frac{4 x^{3}}{3}+4 x^{2}+1$ bei $x=0$ ein lokales Minimum hat.

Find the derivative of $f(x) = \frac{x}{2x + 1}$ using the quotient rule step by step.

Evaluate the integral from 2 to 5: $\int_{2}^{5} t^{2} \ln (4 t) d t$ .

Find $f'(1)$ for $f(x)=\frac{x}{2x+1}$ using the definition of the derivative.

Find the volume of the solid of revolution formed by the region in the first quadrant between $x=y^{2}$ and $x=6-y$ around $y=4$ using the shell method.

Find the derivative of $(1+\sqrt{\sec x})^{2}$ with respect to $x$ .

Find the derivative of $(x+1)^{5} e^{x}$ with respect to $x$ .

Evaluate the integral of $\sin^3 r \cos^2 r \, dr$ .

Find the derivative of $\frac{\tan x}{\sqrt{x^{3}}+1}$ with respect to $x$ .

Find the derivative of the function $5^{2x}$ . What is $\frac{d}{dx} 5^{2x}$ ?

Bestimme die Gleichung der Wendetangente $t_{w}$ für die Funktion $f(x)=-3 x^{3}+9 x^{2}-4$ bei $W=(1/2)$ .

Find the tangent line equation to the curve $x y^{2}+(2 x+y)^{2}=1$ at the point $(0,1)$ .

An object moves along a line with $s(t)=t^{5/2}-5t^{3/2}, t \geq 0$ . Find: a) velocity at $t=1$ , b) when moving right, c) acceleration at $t=1$ .

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

Find the derivative of $f(x)=(x^{2}+1)^{1/x}$ using logarithmic differentiation.

Estimate $(83)^{1/4}$ using linear approximation. Identify the function and base point. Is it an overestimate or underestimate?

A balloon inflates at 1 in³/sec. Find the radius change rate when volume is 10 in³. Use $V = \frac{4}{3} \pi r^{3}$ .

Does the function $f(x)=\frac{8}{(x-2)^{4}+1}$ for $x \in[2, \infty)$ have an inverse?

Approximate the area under $h(x)=x^{3}+2$ from $x=-1$ to $x=5$ using a right Riemann sum with 3 subdivisions.

Find the surface area of revolution of the curve $y=9 x^{3}$ from $x=0$ to $x=2.7$ around the $x$ -axis.

Differentiate the following with $a$ as a constant: 1) $\tan(5 - t^2)$ 2) $\frac{y-5}{5-y^2}$ .

Find the derivative $y'$ for the equation $e^{x+y} + x - y^2 = 3$ .

Find the average rate of change of $y=\sqrt[3]{x}$ over $[2,7]$ rounded to the nearest hundredth.

Find the tangent line equation for $y = g(x) = |-(2x + 10)(x - 4)^2|$ at each inflection point where $g$ is differentiable.

Find the volume of the solid of revolution for the region between $y=x^{3}+3$ , $y=4$ , and the $y$ -axis using different methods.

Given the function $f(x)=\frac{4}{3} x-\tan x$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ , determine where $f$ is increasing.