Calculus

Problem 32701

Find the position of the particle when its speed is maximum, given $x=6.0 t^{2}-1.0 t^{3}$ . Choose A, B, C, D, or E.

See Solution

Problem 32702

Find the time for a mailbag to hit the ground if a helicopter's height is given by $h=3.50 t^{3}$ at $t=1.95$ s.

See Solution

Problem 32703

Bestimme die Koordinaten des lokalen Extrempunkts der Funktion $f_{a}(x)=(\ln x)^{2}+a \cdot \ln x-\frac{3}{4} a^{2}$ .

See Solution

Problem 32704

Graph the function $f(x)$ and find: a) domain and range, b) points where $\lim f(x)$ exists, c) left-hand limit only, d) right-hand limit only.
$f(x)=\left\{\begin{array}{ll} \sqrt{49-x^{2}} & 0 \leq x<7 \\ 7 & 7 \leq x<14 \\ 14 & x=14 \end{array}\right.$

See Solution

Problem 32705

Find the limits of the piecewise function $f(x)$ as $x \to 4$ and $x \to 2^+$ . What are the correct choices?

See Solution

Problem 32706

Find $\lim f(x)$ as $x \to 4^{-}$ and $x \to 4^{+}$ . Also, determine $f(4)$ .

See Solution

Problem 32707

Given the function $f(x)=\left\{\begin{array}{ll}6-x, & x<4 \\ 3, & x=4 \\ \frac{x}{2}, & x>4\end{array}\right.$ , find $f(4)$ and determine if $\lim _{x \rightarrow 4} f(x)$ exists.

See Solution

Problem 32708

Gegeben ist die Funktion $f_{a}(x)=(\ln x)^{2}+a \cdot \ln x-\frac{3}{4} a^{2}$ . Zeigen Sie, dass $f_{a}$ einen lokalen Extrempunkt hat und bestimmen Sie dessen Koordinaten. Finden Sie $a$ , sodass der Extrempunkt eine Nullstelle ist. Bestimmen Sie die Tangente $t_{a}$ im Punkt $P_{a}(1, -\frac{3}{4} a^{2})$ und den Wert von $a$ , für den das Dreieck im IV. Quadranten gleichschenklig ist.

See Solution

Problem 32709

Evaluate the limits of the piecewise function $f(x)$ as $x$ approaches 4 from the right and left.

See Solution

Problem 32710

Gegeben ist $f_{a}(x)=(\ln x)^{2}+a \cdot \ln x-\frac{3}{4} a^{2}$ . Zeigen Sie, dass $f_{a}$ einen Extrempunkt hat und bestimmen Sie dessen Koordinaten. Bestimmen Sie $a$ , wenn Extremstelle Nullstelle ist. Finde die Gleichung der Tangente $t_{a}$ im Punkt $P_{a}(1, -\frac{3}{4} a^{2})$ und den Wert von $a$ , damit das Dreieck gleichschenklig ist.

See Solution

Problem 32711

Graph $f(x)=\begin{cases} 1-x^{2}, & x \neq 1 \\ 4, & x=1 \end{cases}$ . Find $\lim_{x \rightarrow 1^{-}} f(x)$ and $\lim_{x \rightarrow 1^{+}} f(x)$ . Does $\lim_{x \rightarrow 1} f(x)$ exist?

See Solution

Problem 32712

Evaluate the piecewise function $f(x)$ defined as: $f(x)=\begin{cases} 6-x, & x<4 \\ 3, & x=4 \\ \frac{x}{2}, & x>4 \end{cases}$ . Find the limit as $x$ approaches 4 from the right.

See Solution

Problem 32713

Find the $x$ where $f(x)=(x-1)^{2}$ has a gradient of -8.

See Solution

Problem 32714

Find the work done by the force $F=3 x^{2}+1$ moving a particle from $x=3$ to $x=10 \mathrm{~m}$ . Options: a. 785 J, b. 1758 J, c. 1466 J, d. 980 J, e. 1175 J.

See Solution

Problem 32715

Find the largest $\delta>0$ so that $|f(x)-11|<1.25$ when $0<|x-3|<\delta$ , where $f(x)=5x-4$ . The largest value of $\delta$ is $\square$ .

See Solution

Problem 32716

Find the largest interval around $c=4$ where $|f(x)-28|<0.35$ for $f(x)=5x+8$ . What is the largest $\delta>0$ ?

See Solution

Problem 32717

Find the largest $\delta > 0$ so that $|f(x) - L| < \varepsilon$ for $0 < |x - c| < \delta$ , where $f(x) = \sqrt{x}$ , $L = 2$ , $c = 4$ , $\varepsilon = \frac{1}{9}$ . The largest value of $\delta$ is $\square$ .

See Solution

Problem 32718

Gegeben ist die Funktion $f_{a}(x)=(\ln x)^{2}+a \cdot \ln x-\frac{3}{4} a^{2}$ . Zeigen Sie, dass $f_{a}$ einen lokalen Extrempunkt hat und bestimmen Sie dessen Koordinaten. Finden Sie $a$ , sodass Extremstelle Nullstelle ist. Bestimmen Sie die Tangente $t_{a}$ und den Wert von $a$ , für den das Dreieck gleichschenklig ist.

See Solution

Problem 32719

Prove that $\lim_{x \rightarrow 6} f(x)=36$ for $f(x)=\begin{cases} x^2 & x \neq 6 \\ 5 & x=6 \end{cases}$ .

See Solution

Problem 32720

Find the work done in stretching a rubber band from $x=0$ to $x=L$ with force $F=a x+b x^{2}$ . Choices: a. $a L+2 b L^{2}$ b. $\frac{1}{2} a L^{2}+\frac{1}{3} b L^{3}$ c. $a L^{2}+b L^{3}$ d. $b L$ e. $a+2 b L$

See Solution

Problem 32721

Prove that $\lim _{x \rightarrow 0}(9 x-6)=-6$ using an $\varepsilon-\delta$ argument. Choose the correct $\delta$ .

See Solution

Problem 32722

Define $\lim _{x \rightarrow 0} g(x)=k$ and choose the true statement about it from the options A, B, C, or D.

See Solution

Problem 32723

Find the largest interval around $c=5$ where $|f(x)-11|<0.07$ for $f(x)=\sqrt{7x+86}$ . Determine $\delta>0$ for $0<|x-5|<\delta$ .

See Solution

Problem 32724

Find $L=\lim_{x \rightarrow -7} \frac{x^{2}+8x+7}{x+7}$ and the largest $\delta>0$ for $|f(x)-L|<0.3$ .

See Solution

Problem 32725

Find the largest interval around $c=-8$ where $|f(x)-64|<0.1$ for $f(x)=x^{2}$ and the largest $\delta>0$ .

See Solution

Problem 32726

Find the largest interval around $c=-9$ where $|f(x)-81|<0.3$ for $f(x)=x^{2}$ . Also, determine $\delta>0$ for $0<|x+9|<\delta$ .

See Solution

Problem 32727

Prove that $\lim _{x \rightarrow 5} \frac{2}{x}=\frac{2}{5}$ using the $\varepsilon-\delta$ definition of limits.

See Solution

Problem 32728

Calculate $f(x)=\frac{x+4}{(x-1)^{2}}$ for values near $x=1$ and find $\lim _{x \rightarrow 1} f(x)$ .

See Solution

Problem 32729

Find the largest interval around $c=-7$ where $|f(x)-49|<0.35$ for $f(x)=x^2$ . Also, find $\delta$ such that $|f(x)-49|<0.35$ for $0<|x+7|<\delta$ .

See Solution

Problem 32730

Find the largest interval around $c=-4$ where $|f(x)-16|<0.35$ for $f(x)=x^{2}$ . Then, find $\delta$ such that $|f(x)-16|<0.35$ for $0<|x+4|<\delta$ .

See Solution

Problem 32731

Find the limit $L=\lim _{x \rightarrow 1} (7-2x)$ and the largest $\delta>0$ for $|f(x)-L|<0.02$ when $0<|x-1|<\delta$ . $L=\square$

See Solution

Problem 32732

Find the largest interval around $c=-8$ where $|f(x)-64|<0.35$ for $f(x)=x^{2}$ . Also, find $\delta>0$ for $0<|x+8|<\delta$ .

See Solution

Problem 32733

Gegeben ist die Funktion $f_{a}(x)=(\ln x)^{2}+a \cdot \ln x-\frac{3}{4} a^{2}$ mit $a \geq 0$ .
1. Zeigen Sie, dass $f_{a}$ einen lokalen Extrempunkt hat, geben Sie die Koordinaten an und bestimmen Sie den Parameter $a$ , bei dem die Extremstelle Nullstelle ist.
2. Bestimmen Sie die Tangentengleichung $t_{a}$ an $G_{a}$ im Punkt $P_{a}(1, -\frac{3}{4} a^{2})$ und den Parameterwert $a$ für ein gleichschenkliges Dreieck.

See Solution

Problem 32734

Prove that $\lim_{x \rightarrow 6} f(x) = 36$ for $f(x) = x^2$ if $x \neq 6$ and $f(6) = 5$ .

See Solution

Problem 32735

Find the largest interval around $c=-7$ where $|f(x)-49|<0.25$ for $f(x)=x^{2}$ and the largest $\delta>0$ .

See Solution

Problem 32736

Find the largest open interval around $c=5$ where $|f(x)-11|<0.06$ for $f(x)=\sqrt{6x+91}$ . Round to four decimal places.

See Solution

Problem 32737

Find the largest interval around $c=3$ where $|f(x)-L|<0.07$ holds for $f(x)=mx$ , $L=3m$ , $m>0$ . What is $\delta$ ?

See Solution

Problem 32738

Find the limit as $x$ approaches 5 from the right for $f(x)=\left(\frac{3 x}{x+1}\right)\left(\frac{5 x+5}{x^{2}+x}\right)$ .

See Solution

Problem 32739

Find the largest interval around $c$ where $|f(x)-L|<\varepsilon$ for $f(x)=x^{2}$ , $L=64$ , $c=-8$ , $\varepsilon=0.35$ . Also, determine the largest $\delta>0$ satisfying $0<|x-c|<\delta \rightarrow|f(x)-L|<\varepsilon$ .

See Solution

Problem 32740

Find the limit as $x$ approaches 2 from the left: $\lim _{x \rightarrow 2^{-}}\left(\frac{2}{x+1}\right)\left(\frac{x+5}{x}\right)\left(\frac{4-x}{9}\right)=\square$

See Solution

Problem 32741

Find the limit as $h$ approaches 0 from the positive side: $\lim _{h \rightarrow 0^{+}} \frac{\sqrt{h^{2}+2 h+19}-\sqrt{19}}{h}$ .

See Solution

Problem 32742

Find the limit: $\lim _{t \rightarrow 0} \frac{\sin k t}{t}$ using $\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1$ .

See Solution

Problem 32743

Find the limit: $\lim _{\theta \rightarrow 0} \frac{7 \sin \sqrt{5} \theta}{\sqrt{5} \theta}$ . A. $\square$ B. Does not exist.

See Solution

Problem 32744

Find the integral $\int \sec (5 x) d x$ . Which is the correct answer from the options given?

See Solution

Problem 32745

Find the limit: $\lim _{y \rightarrow 0} 49 y^{2}(\cot y)(\csc 7 y)$ . A. $\lim _{y \rightarrow 0} 49 y^{2}(\cot y)(\csc 7 y)=-\quad$ B. Limit does not exist.

See Solution

Problem 32746

Find the relative maximum point of the function $f(x)=-x^{3}+3x^{2}-4$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 32747

Bestimmen Sie $p$ für $f(x)=3x^{2}+p^{2}$ , sodass die Fläche im Intervall $[-1; 2]$ gleich $21 \mathrm{FE}$ ist.

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Problem 32748

Find the end behavior of $f(x) = 6 - \ln x$ . Choose A or B for the limits as $x \to \infty$ and $x \to 0^{+}$ .

See Solution

Problem 32749

Find the limit as $x$ approaches $8$ from the right for $f(x) = \left(\frac{2 x}{x+1}\right)\left(\frac{4 x+8}{x^{2}+x}\right)$ .

See Solution

Problem 32750

Find the limit as $x$ approaches 4 from the right for $f(x)=\left(\frac{8 x}{x+1}\right)\left(\frac{2 x+4}{x^{2}+x}\right)$ .

See Solution

Problem 32751

Finde die Extremstellen der Funktion $f(x)=x^{4}$ . Bestimme $f'(x)$ und $f''(x)$ , um das Extremum zu analysieren.

See Solution

Problem 32752

Graph the piecewise functions:
1. $f(x)=\{x^{3}, x \neq 1; 0, x=1\}$ . Find $\lim_{x \to 1} f(x)$ and check if it exists.
2. $f(x)=\{1-x^{2}, x \neq 1; 2, x=1\}$ . Find $\lim_{x \to 1^{+}} f(x)$ and $\lim_{x \to 1^{-}} f(x)$ , and check if it exists.

See Solution

Problem 32753

Find the limit: $\lim _{y \rightarrow 0} \frac{\sin 11 y}{5 y}$ . What to multiply by to get $\frac{\sin \theta}{\theta}$ ? Multiply by $\frac{11}{11}$ .

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Problem 32754

Find the tangent line equation to the curve at $(0,2)$ where $x=\ln(t)$ and $y=1+t^{2}$ .

See Solution

Problem 32755

Calculate the integral $\int_{1}^{2} \frac{\ln x^{3}}{x} d x$ . What is the result?

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Problem 32756

مطلوب إيجاد معادلة الخط المماس لمنحنى الدالة $f(x)=x^{4}+2 x-1$ عند النقطة $x=1$ .

See Solution

Problem 32757

Find the limit: $\lim _{n \rightarrow \infty} \frac{4 \cdot 6 + 9 \cdot 4 + 14 \cdot 5 + \cdots + (5n-1)(n+2)}{n^{3}}$

See Solution

Problem 32758

Finde die Extremstellen der Funktion $f(x) = (x+2)^3 \cdot x$ .

See Solution

Problem 32759

If $y=x^{2} e^{x}$ , find $\frac{d y}{d x}$ and show $\frac{d y}{d x}=0$ for $x=0$ and $x=-2$ .

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Problem 32760

Berechne die erste und zweite Ableitung von $h(t) = -\frac{2}{75} t^{4} + \frac{22}{15} t^{3} - \frac{80}{3} t^{2} + 170 t$ .

See Solution

Problem 32761

Solve the differential equation: $\frac{d y}{d x}=3 x \sqrt{y}$ for $y \neq 0$ .

See Solution

Problem 32762

Evaluate the limit: $\lim _{x \rightarrow 2}\left(\frac{3}{2 x^{4}+4 x^{2}-44}\right)^{2}$ .

See Solution

Problem 32763

Find a solution to the equation $\frac{d y}{d x}=(y-3)(2 x+1)$ that passes through $(-1,5)$ .

See Solution

Problem 32764

Find the limit as $b$ approaches 7: $\lim _{b \rightarrow 7} \frac{8 b}{\sqrt{5 b+29}-1}$ .

See Solution

Problem 32765

Solve the differential equation $\frac{d u}{d t}=\frac{2 t+\sec ^{2}(t)}{2 u}$ with initial condition $u(0)=-2$ .

See Solution

Problem 32766

Find $g^{\prime}(4)$ if $g(x)=2 f(x)+3$ and $f^{\prime}(4)=5$ .

See Solution

Problem 32767

Find the Taylor polynomial of degree 2 for $f(x) = \sqrt{x}$ at $a$ where $\sqrt{a} = 16$ to approximate $\sqrt{18}$ . Show true value is in $\left(\eta - \frac{1}{2048}, \eta + \frac{1}{2048}\right)$ .

See Solution

Problem 32768

Find the limit: $\lim _{x \rightarrow 3 \pi / 2} \frac{\sin ^{2} x+7 \sin x+6}{\sin x+1}$

See Solution

Problem 32769

Find where the function $h(x)=\frac{2+2 \sin x}{\cos x}$ is continuous and analyze the limits $\lim _{x \rightarrow \pi / 2^{-}} h(x)$ and $\lim _{x \rightarrow 4 \pi / 3} h(x)$ .

See Solution

Problem 32770

Find the limit as $x$ approaches 5 for $\sqrt{x^{2}+11}$ .

See Solution

Problem 32771

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

See Solution

Problem 32772

Find the intervals where the function $f(x)=(8 x+9)^{\frac{2}{7}}$ is continuous, considering endpoints.

See Solution

Problem 32773

Berechne das Integral von $f(x)=2-x^{2}$ von 0 bis 1 mit 5 und 500 Rechtecken, links für Lisa und rechts für Theo. Vergleiche die Ergebnisse.

See Solution

Problem 32774

Given the function $f(x)$ , check if it's continuous at 3, determine left/right continuity, and state intervals of continuity.
$f(x)=\left\{\begin{array}{ll} x^{2}+2 x & \text { if } x \geq 3 \\ 3 x & \text { if } x<3 \end{array}\right.$

See Solution

Problem 32775

Find if the curve $y=x^{3}+4x-10$ has a tangent with slope -2. If yes, determine the equation and point of tangency.

See Solution

Problem 32776

Find the limit: $\lim _{x \rightarrow \pi / 2^{-}} \frac{2+2 \sin x}{\cos x}=\square$ .

See Solution

Problem 32777

Find the limit: $\lim _{x \rightarrow 4 \pi / 3} \frac{2+2 \sin x}{\cos x}=\square$ (Type exact answer with radicals.)

See Solution

Problem 32778

Find the value of $a$ for which the piecewise function $g(x)$ is continuous at $x=1$ from the left, right, and overall.

See Solution

Problem 32779

Find the Taylor polynomial of degree 2 for $f(x)=\sqrt{x}$ at $a=16$ to approximate $\sqrt{18}$ . Show true value is in $\left(\eta-\frac{1}{2048}, \eta+\frac{1}{2015}\right)$ .

See Solution

Problem 32780

Find $\lim _{x \rightarrow 2} f(x)$ for the piecewise function $f(x)=\{x-1 \text{ if } x \leq 2, 2x-3 \text{ if } x>2\}$ .

See Solution

Problem 32781

Venus orbits the Sun every 225 days. Its max distance is 108.9M km and min is 107.5M km. Find the average rate of change of distance over the interval $[0, 112.5]$ days.

See Solution

Problem 32782

Find the derivative of the function $f(x)=\ln \left(e^{-x}+1\right)$ , i.e., compute $f^{\prime}(x)$ .

See Solution

Problem 32783

The tangent line to the parabola $y=x^{2}$ at $(-2,4)$ is given by $y-4=2x(x+2)$ . True or False?

See Solution

Problem 32784

If $\sin (x y)=x$ , find $\frac{d x}{d y}$ . Options: a. $\frac{\sec (x y)-y}{x}$ b. $\frac{\sec (x y)}{x}$ c. $-\frac{1+\sec (x y)}{x}$ d. $\sec (x y)$

See Solution

Problem 32785

Find $g'(3)$ if $f(g(x))=x$ , $f(1)=3$ , and $f'(1)=-4$ . Choices: (a) $\frac{1}{3}$ (b) $-\frac{1}{3}$ (c) $\frac{1}{4}$ (d) $-\frac{1}{4}$ .

See Solution

Problem 32786

Find the limit: $\lim _{h \rightarrow 0} \frac{\tan 3(x+h)-\tan 3 x}{h}$ . Options: a. $3 \cot (3 x)$ b. 0 c. $\sec ^{2}(3 x)$ d. $3 \sec ^{2}(3 x)$

See Solution

Problem 32787

Find the tangent line equation to the curve $\sin x + \cos y = 1$ at $(\pi / 2, \pi / 2)$ .

See Solution

Problem 32788

Prove that if $\lim _{x \rightarrow p} F(x)=a$ and $\lim _{x \rightarrow p} f(x)=b$ , then $a=b$ .

See Solution

Problem 32789

Evaluate the integrals: $\int_{\sigma}^{v} v \, dv$ and $\int_{sc}^{s} ac \, ds$ .

See Solution

Problem 32790

Prove that $f(x) = \frac{1}{x}$ is uniformly continuous on $[1, \infty)$ but not on $(0,1]$ .

See Solution

Problem 32791

Find $f^{\prime}(x)$ for the function $f(x)=\frac{-3 x}{\sin (x)+\cos (x)}$ at $x=\pi$ .

See Solution

Problem 32792

Find the tangent line equation at the point $(2,1)$ for the function $f(x)=\frac{8}{x^{2}+4}$ .

See Solution

Problem 32793

Find the limit as $x$ approaches 1 for the expression $\frac{\sqrt{x^{2}+3}-2}{x+1}$ .

See Solution

Problem 32794

Invest $\$ 1600$ at $4.6\%$ interest compounded continuously. Find the amount after 4 years.

See Solution

Problem 32795

حدد القيم لـ $x$ حيث $-4<x<3$ ، و $f$ متصلة ولكن غير قابلة للاشتقاق، مع وجود خطوط مماسة عمودية وأفقية.

See Solution

Problem 32796

Invest \ $4500 for 11 years at monthly compounded interest$ f $. Show there's an$ r $in$ (0,0.08) $to reach \$7000. Approximate$ r$.

See Solution

Problem 32797

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

See Solution

Problem 32798

Find the derivative of the function $f(x)=\frac{7 x^{5}+5 x^{4}-2 x^{3}}{x^{4}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32799

Find the limit: $\lim _{x \rightarrow \infty} \frac{6 x-8 x^{2}}{2 x^{2}-3}$ . Options: 0, 4, $-\infty$ , $\infty$ , $-4$ .

See Solution

Problem 32800

Prove that wavefunctions for different energy levels in a particle in a box are orthogonal.

See Solution

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Calculus

Problem 32701

Find the position of the particle when its speed is maximum, given x=6.0t2−1.0t3x=6.0 t^{2}-1.0 t^{3}x=6.0t2−1.0t3. Choose A, B, C, D, or E.

Problem 32702

Find the time for a mailbag to hit the ground if a helicopter's height is given by h=3.50t3h=3.50 t^{3}h=3.50t3 at t=1.95t=1.95t=1.95 s.

Problem 32703

Bestimme die Koordinaten des lokalen Extrempunkts der Funktion fa(x)=(ln⁡x)2+a⋅ln⁡x−34a2f_{a}(x)=(\ln x)^{2}+a \cdot \ln x-\frac{3}{4} a^{2}fa​(x)=(lnx)2+a⋅lnx−43​a2.

Problem 32704

Problem 32705

Find the limits of the piecewise function f(x)f(x)f(x) as x→4x \to 4x→4 and x→2+x \to 2^+x→2+. What are the correct choices?

Problem 32706

Find lim⁡f(x)\lim f(x)limf(x) as x→4−x \to 4^{-}x→4− and x→4+x \to 4^{+}x→4+. Also, determine f(4)f(4)f(4).

Problem 32707

Given the function f(x)={6−x,x<43,x=4x2,x>4f(x)=\left\{\begin{array}{ll}6-x, & x<4 \\ 3, & x=4 \\ \frac{x}{2}, & x>4\end{array}\right.f(x)=⎩⎨⎧​6−x,3,2x​,​x<4x=4x>4​, find f(4)f(4)f(4) and determine if lim⁡x→4f(x)\lim _{x \rightarrow 4} f(x)limx→4​f(x) exists.

Problem 32708

Problem 32709

Evaluate the limits of the piecewise function f(x)f(x)f(x) as xxx approaches 4 from the right and left.

Problem 32710

Problem 32711

Problem 32712

Evaluate the piecewise function f(x)f(x)f(x) defined as: f(x)={6−x,x<43,x=4x2,x>4f(x)=\begin{cases} 6-x, & x<4 \\ 3, & x=4 \\ \frac{x}{2}, & x>4 \end{cases}f(x)=⎩⎨⎧​6−x,3,2x​,​x<4x=4x>4​. Find the limit as xxx approaches 4 from the right.

Problem 32713

Find the xxx where f(x)=(x−1)2f(x)=(x-1)^{2}f(x)=(x−1)2 has a gradient of -8.

Problem 32714

Find the work done by the force F=3x2+1F=3 x^{2}+1F=3x2+1 moving a particle from x=3x=3x=3 to x=10 mx=10 \mathrm{~m}x=10 m. Options: a. 785 J, b. 1758 J, c. 1466 J, d. 980 J, e. 1175 J.

Problem 32715

Find the largest δ>0\delta>0δ>0 so that ∣f(x)−11∣<1.25|f(x)-11|<1.25∣f(x)−11∣<1.25 when 0<∣x−3∣<δ0<|x-3|<\delta0<∣x−3∣<δ, where f(x)=5x−4f(x)=5x-4f(x)=5x−4. The largest value of δ\deltaδ is □\square□.

Problem 32716

Find the largest interval around c=4c=4c=4 where ∣f(x)−28∣<0.35|f(x)-28|<0.35∣f(x)−28∣<0.35 for f(x)=5x+8f(x)=5x+8f(x)=5x+8. What is the largest δ>0\delta>0δ>0?

Problem 32717

Problem 32718

Problem 32719

Prove that lim⁡x→6f(x)=36\lim_{x \rightarrow 6} f(x)=36limx→6​f(x)=36 for f(x)={x2x≠65x=6f(x)=\begin{cases} x^2 & x \neq 6 \\ 5 & x=6 \end{cases}f(x)={x25​x=6x=6​.

Problem 32720

Find the work done in stretching a rubber band from x=0x=0x=0 to x=Lx=Lx=L with force F=ax+bx2F=a x+b x^{2}F=ax+bx2. Choices: a. aL+2bL2a L+2 b L^{2}aL+2bL2 b. 12aL2+13bL3\frac{1}{2} a L^{2}+\frac{1}{3} b L^{3}21​aL2+31​bL3 c. aL2+bL3a L^{2}+b L^{3}aL2+bL3 d. bLb LbL e. a+2bLa+2 b La+2bL

Problem 32721

Prove that lim⁡x→0(9x−6)=−6\lim _{x \rightarrow 0}(9 x-6)=-6limx→0​(9x−6)=−6 using an ε−δ\varepsilon-\deltaε−δ argument. Choose the correct δ\deltaδ.

Problem 32722

Define lim⁡x→0g(x)=k\lim _{x \rightarrow 0} g(x)=klimx→0​g(x)=k and choose the true statement about it from the options A, B, C, or D.

Problem 32723

Find the largest interval around c=5c=5c=5 where ∣f(x)−11∣<0.07|f(x)-11|<0.07∣f(x)−11∣<0.07 for f(x)=7x+86f(x)=\sqrt{7x+86}f(x)=7x+86​. Determine δ>0\delta>0δ>0 for 0<∣x−5∣<δ0<|x-5|<\delta0<∣x−5∣<δ.

Problem 32724

Find L=lim⁡x→−7x2+8x+7x+7L=\lim_{x \rightarrow -7} \frac{x^{2}+8x+7}{x+7}L=limx→−7​x+7x2+8x+7​ and the largest δ>0\delta>0δ>0 for ∣f(x)−L∣<0.3|f(x)-L|<0.3∣f(x)−L∣<0.3.

Problem 32725

Find the largest interval around c=−8c=-8c=−8 where ∣f(x)−64∣<0.1|f(x)-64|<0.1∣f(x)−64∣<0.1 for f(x)=x2f(x)=x^{2}f(x)=x2 and the largest δ>0\delta>0δ>0.

Problem 32726

Find the largest interval around c=−9c=-9c=−9 where ∣f(x)−81∣<0.3|f(x)-81|<0.3∣f(x)−81∣<0.3 for f(x)=x2f(x)=x^{2}f(x)=x2. Also, determine δ>0\delta>0δ>0 for 0<∣x+9∣<δ0<|x+9|<\delta0<∣x+9∣<δ.

Problem 32727

Prove that lim⁡x→52x=25\lim _{x \rightarrow 5} \frac{2}{x}=\frac{2}{5}limx→5​x2​=52​ using the ε−δ\varepsilon-\deltaε−δ definition of limits.

Problem 32728

Calculate f(x)=x+4(x−1)2f(x)=\frac{x+4}{(x-1)^{2}}f(x)=(x−1)2x+4​ for values near x=1x=1x=1 and find lim⁡x→1f(x)\lim _{x \rightarrow 1} f(x)limx→1​f(x).

Problem 32729

Find the largest interval around c=−7c=-7c=−7 where ∣f(x)−49∣<0.35|f(x)-49|<0.35∣f(x)−49∣<0.35 for f(x)=x2f(x)=x^2f(x)=x2. Also, find δ\deltaδ such that ∣f(x)−49∣<0.35|f(x)-49|<0.35∣f(x)−49∣<0.35 for 0<∣x+7∣<δ0<|x+7|<\delta0<∣x+7∣<δ.

Problem 32730

Find the largest interval around c=−4c=-4c=−4 where ∣f(x)−16∣<0.35|f(x)-16|<0.35∣f(x)−16∣<0.35 for f(x)=x2f(x)=x^{2}f(x)=x2. Then, find δ\deltaδ such that ∣f(x)−16∣<0.35|f(x)-16|<0.35∣f(x)−16∣<0.35 for 0<∣x+4∣<δ0<|x+4|<\delta0<∣x+4∣<δ.

Problem 32731

Find the limit L=lim⁡x→1(7−2x)L=\lim _{x \rightarrow 1} (7-2x)L=limx→1​(7−2x) and the largest δ>0\delta>0δ>0 for ∣f(x)−L∣<0.02|f(x)-L|<0.02∣f(x)−L∣<0.02 when 0<∣x−1∣<δ0<|x-1|<\delta0<∣x−1∣<δ. L=□L=\squareL=□

Problem 32732

Find the largest interval around c=−8c=-8c=−8 where ∣f(x)−64∣<0.35|f(x)-64|<0.35∣f(x)−64∣<0.35 for f(x)=x2f(x)=x^{2}f(x)=x2. Also, find δ>0\delta>0δ>0 for 0<∣x+8∣<δ0<|x+8|<\delta0<∣x+8∣<δ.

Problem 32733

Problem 32734

Prove that lim⁡x→6f(x)=36\lim_{x \rightarrow 6} f(x) = 36limx→6​f(x)=36 for f(x)=x2f(x) = x^2f(x)=x2 if x≠6x \neq 6x=6 and f(6)=5f(6) = 5f(6)=5.

Problem 32735

Find the largest interval around c=−7c=-7c=−7 where ∣f(x)−49∣<0.25|f(x)-49|<0.25∣f(x)−49∣<0.25 for f(x)=x2f(x)=x^{2}f(x)=x2 and the largest δ>0\delta>0δ>0.

Problem 32736

Find the largest open interval around c=5c=5c=5 where ∣f(x)−11∣<0.06|f(x)-11|<0.06∣f(x)−11∣<0.06 for f(x)=6x+91f(x)=\sqrt{6x+91}f(x)=6x+91​. Round to four decimal places.

Problem 32737

Find the largest interval around c=3c=3c=3 where ∣f(x)−L∣<0.07|f(x)-L|<0.07∣f(x)−L∣<0.07 holds for f(x)=mxf(x)=mxf(x)=mx, L=3mL=3mL=3m, m>0m>0m>0. What is δ\deltaδ?

Problem 32738

Find the limit as xxx approaches 5 from the right for f(x)=(3xx+1)(5x+5x2+x)f(x)=\left(\frac{3 x}{x+1}\right)\left(\frac{5 x+5}{x^{2}+x}\right)f(x)=(x+13x​)(x2+x5x+5​).

Problem 32739

Problem 32740

Find the limit as xxx approaches 2 from the left: lim⁡x→2−(2x+1)(x+5x)(4−x9)=□\lim _{x \rightarrow 2^{-}}\left(\frac{2}{x+1}\right)\left(\frac{x+5}{x}\right)\left(\frac{4-x}{9}\right)=\squarelimx→2−​(x+12​)(xx+5​)(94−x​)=□

Problem 32741

Find the limit as hhh approaches 0 from the positive side: lim⁡h→0+h2+2h+19−19h\lim _{h \rightarrow 0^{+}} \frac{\sqrt{h^{2}+2 h+19}-\sqrt{19}}{h}limh→0+​hh2+2h+19​−19​​.

Problem 32742

Find the limit: lim⁡t→0sin⁡ktt\lim _{t \rightarrow 0} \frac{\sin k t}{t}limt→0​tsinkt​ using lim⁡θ→0sin⁡θθ=1\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1limθ→0​θsinθ​=1.

Problem 32743

Find the limit: lim⁡θ→07sin⁡5θ5θ\lim _{\theta \rightarrow 0} \frac{7 \sin \sqrt{5} \theta}{\sqrt{5} \theta}limθ→0​5​θ7sin5​θ​. A. □\square□ B. Does not exist.

Problem 32744

Find the position of the particle when its speed is maximum, given $x=6.0 t^{2}-1.0 t^{3}$ . Choose A, B, C, D, or E.

Find the time for a mailbag to hit the ground if a helicopter's height is given by $h=3.50 t^{3}$ at $t=1.95$ s.

Bestimme die Koordinaten des lokalen Extrempunkts der Funktion $f_{a}(x)=(\ln x)^{2}+a \cdot \ln x-\frac{3}{4} a^{2}$ .

Find the limits of the piecewise function $f(x)$ as $x \to 4$ and $x \to 2^+$ . What are the correct choices?

Find $\lim f(x)$ as $x \to 4^{-}$ and $x \to 4^{+}$ . Also, determine $f(4)$ .

Given the function $f(x)=\left\{\begin{array}{ll}6-x, & x<4 \\ 3, & x=4 \\ \frac{x}{2}, & x>4\end{array}\right.$ , find $f(4)$ and determine if $\lim _{x \rightarrow 4} f(x)$ exists.

Evaluate the limits of the piecewise function $f(x)$ as $x$ approaches 4 from the right and left.

Evaluate the piecewise function $f(x)$ defined as: $f(x)=\begin{cases} 6-x, & x<4 \\ 3, & x=4 \\ \frac{x}{2}, & x>4 \end{cases}$ . Find the limit as $x$ approaches 4 from the right.

Find the $x$ where $f(x)=(x-1)^{2}$ has a gradient of -8.

Find the work done by the force $F=3 x^{2}+1$ moving a particle from $x=3$ to $x=10 \mathrm{~m}$ . Options: a. 785 J, b. 1758 J, c. 1466 J, d. 980 J, e. 1175 J.

Find the largest $\delta>0$ so that $|f(x)-11|<1.25$ when $0<|x-3|<\delta$ , where $f(x)=5x-4$ . The largest value of $\delta$ is $\square$ .

Find the largest interval around $c=4$ where $|f(x)-28|<0.35$ for $f(x)=5x+8$ . What is the largest $\delta>0$ ?

Prove that $\lim_{x \rightarrow 6} f(x)=36$ for $f(x)=\begin{cases} x^2 & x \neq 6 \\ 5 & x=6 \end{cases}$ .

Find the work done in stretching a rubber band from $x=0$ to $x=L$ with force $F=a x+b x^{2}$ . Choices: a. $a L+2 b L^{2}$ b. $\frac{1}{2} a L^{2}+\frac{1}{3} b L^{3}$ c. $a L^{2}+b L^{3}$ d. $b L$ e. $a+2 b L$

Prove that $\lim _{x \rightarrow 0}(9 x-6)=-6$ using an $\varepsilon-\delta$ argument. Choose the correct $\delta$ .

Define $\lim _{x \rightarrow 0} g(x)=k$ and choose the true statement about it from the options A, B, C, or D.

Find the largest interval around $c=5$ where $|f(x)-11|<0.07$ for $f(x)=\sqrt{7x+86}$ . Determine $\delta>0$ for $0<|x-5|<\delta$ .

Find $L=\lim_{x \rightarrow -7} \frac{x^{2}+8x+7}{x+7}$ and the largest $\delta>0$ for $|f(x)-L|<0.3$ .

Find the largest interval around $c=-8$ where $|f(x)-64|<0.1$ for $f(x)=x^{2}$ and the largest $\delta>0$ .

Find the largest interval around $c=-9$ where $|f(x)-81|<0.3$ for $f(x)=x^{2}$ . Also, determine $\delta>0$ for $0<|x+9|<\delta$ .

Prove that $\lim _{x \rightarrow 5} \frac{2}{x}=\frac{2}{5}$ using the $\varepsilon-\delta$ definition of limits.

Calculate $f(x)=\frac{x+4}{(x-1)^{2}}$ for values near $x=1$ and find $\lim _{x \rightarrow 1} f(x)$ .

Find the largest interval around $c=-7$ where $|f(x)-49|<0.35$ for $f(x)=x^2$ . Also, find $\delta$ such that $|f(x)-49|<0.35$ for $0<|x+7|<\delta$ .

Find the largest interval around $c=-4$ where $|f(x)-16|<0.35$ for $f(x)=x^{2}$ . Then, find $\delta$ such that $|f(x)-16|<0.35$ for $0<|x+4|<\delta$ .

Find the limit $L=\lim _{x \rightarrow 1} (7-2x)$ and the largest $\delta>0$ for $|f(x)-L|<0.02$ when $0<|x-1|<\delta$ . $L=\square$

Find the largest interval around $c=-8$ where $|f(x)-64|<0.35$ for $f(x)=x^{2}$ . Also, find $\delta>0$ for $0<|x+8|<\delta$ .

Prove that $\lim_{x \rightarrow 6} f(x) = 36$ for $f(x) = x^2$ if $x \neq 6$ and $f(6) = 5$ .

Find the largest interval around $c=-7$ where $|f(x)-49|<0.25$ for $f(x)=x^{2}$ and the largest $\delta>0$ .

Find the largest open interval around $c=5$ where $|f(x)-11|<0.06$ for $f(x)=\sqrt{6x+91}$ . Round to four decimal places.

Find the largest interval around $c=3$ where $|f(x)-L|<0.07$ holds for $f(x)=mx$ , $L=3m$ , $m>0$ . What is $\delta$ ?

Find the limit as $x$ approaches 5 from the right for $f(x)=\left(\frac{3 x}{x+1}\right)\left(\frac{5 x+5}{x^{2}+x}\right)$ .

Find the limit as $x$ approaches 2 from the left: $\lim _{x \rightarrow 2^{-}}\left(\frac{2}{x+1}\right)\left(\frac{x+5}{x}\right)\left(\frac{4-x}{9}\right)=\square$

Find the limit as $h$ approaches 0 from the positive side: $\lim _{h \rightarrow 0^{+}} \frac{\sqrt{h^{2}+2 h+19}-\sqrt{19}}{h}$ .

Find the limit: $\lim _{t \rightarrow 0} \frac{\sin k t}{t}$ using $\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1$ .

Find the limit: $\lim _{\theta \rightarrow 0} \frac{7 \sin \sqrt{5} \theta}{\sqrt{5} \theta}$ . A. $\square$ B. Does not exist.

Find the integral $\int \sec (5 x) d x$ . Which is the correct answer from the options given?

Find the limit: $\lim _{y \rightarrow 0} 49 y^{2}(\cot y)(\csc 7 y)$ . A. $\lim _{y \rightarrow 0} 49 y^{2}(\cot y)(\csc 7 y)=-\quad$ B. Limit does not exist.

Find the relative maximum point of the function $f(x)=-x^{3}+3x^{2}-4$ using $f'(x)$ and $f''(x)$ .

Bestimmen Sie $p$ für $f(x)=3x^{2}+p^{2}$ , sodass die Fläche im Intervall $[-1; 2]$ gleich $21 \mathrm{FE}$ ist.

Find the end behavior of $f(x) = 6 - \ln x$ . Choose A or B for the limits as $x \to \infty$ and $x \to 0^{+}$ .

Find the limit as $x$ approaches $8$ from the right for $f(x) = \left(\frac{2 x}{x+1}\right)\left(\frac{4 x+8}{x^{2}+x}\right)$ .

Find the limit as $x$ approaches 4 from the right for $f(x)=\left(\frac{8 x}{x+1}\right)\left(\frac{2 x+4}{x^{2}+x}\right)$ .

Finde die Extremstellen der Funktion $f(x)=x^{4}$ . Bestimme $f'(x)$ und $f''(x)$ , um das Extremum zu analysieren.

Find the limit: $\lim _{y \rightarrow 0} \frac{\sin 11 y}{5 y}$ . What to multiply by to get $\frac{\sin \theta}{\theta}$ ? Multiply by $\frac{11}{11}$ .

Find the tangent line equation to the curve at $(0,2)$ where $x=\ln(t)$ and $y=1+t^{2}$ .

Calculate the integral $\int_{1}^{2} \frac{\ln x^{3}}{x} d x$ . What is the result?

مطلوب إيجاد معادلة الخط المماس لمنحنى الدالة $f(x)=x^{4}+2 x-1$ عند النقطة $x=1$ .

Find the limit: $\lim _{n \rightarrow \infty} \frac{4 \cdot 6 + 9 \cdot 4 + 14 \cdot 5 + \cdots + (5n-1)(n+2)}{n^{3}}$

Finde die Extremstellen der Funktion $f(x) = (x+2)^3 \cdot x$ .

If $y=x^{2} e^{x}$ , find $\frac{d y}{d x}$ and show $\frac{d y}{d x}=0$ for $x=0$ and $x=-2$ .

Berechne die erste und zweite Ableitung von $h(t) = -\frac{2}{75} t^{4} + \frac{22}{15} t^{3} - \frac{80}{3} t^{2} + 170 t$ .

Solve the differential equation: $\frac{d y}{d x}=3 x \sqrt{y}$ for $y \neq 0$ .

Evaluate the limit: $\lim _{x \rightarrow 2}\left(\frac{3}{2 x^{4}+4 x^{2}-44}\right)^{2}$ .

Find a solution to the equation $\frac{d y}{d x}=(y-3)(2 x+1)$ that passes through $(-1,5)$ .

Find the limit as $b$ approaches 7: $\lim _{b \rightarrow 7} \frac{8 b}{\sqrt{5 b+29}-1}$ .

Solve the differential equation $\frac{d u}{d t}=\frac{2 t+\sec ^{2}(t)}{2 u}$ with initial condition $u(0)=-2$ .

Find $g^{\prime}(4)$ if $g(x)=2 f(x)+3$ and $f^{\prime}(4)=5$ .

Find the Taylor polynomial of degree 2 for $f(x) = \sqrt{x}$ at $a$ where $\sqrt{a} = 16$ to approximate $\sqrt{18}$ . Show true value is in $\left(\eta - \frac{1}{2048}, \eta + \frac{1}{2048}\right)$ .

Find the limit: $\lim _{x \rightarrow 3 \pi / 2} \frac{\sin ^{2} x+7 \sin x+6}{\sin x+1}$

Find where the function $h(x)=\frac{2+2 \sin x}{\cos x}$ is continuous and analyze the limits $\lim _{x \rightarrow \pi / 2^{-}} h(x)$ and $\lim _{x \rightarrow 4 \pi / 3} h(x)$ .

Find the limit as $x$ approaches 5 for $\sqrt{x^{2}+11}$ .

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

Find the intervals where the function $f(x)=(8 x+9)^{\frac{2}{7}}$ is continuous, considering endpoints.

Berechne das Integral von $f(x)=2-x^{2}$ von 0 bis 1 mit 5 und 500 Rechtecken, links für Lisa und rechts für Theo. Vergleiche die Ergebnisse.