Calculus

Problem 29801

Analyze the graph of $f(x)=e^{x}$ to solve: a. Find $\lim _{x \rightarrow-\infty} e^{x}$ . b. Identify the horizontal asymptote.

See Solution

Problem 29802

Find $\lim _{x \rightarrow 4} \frac{f(x)}{g(x)}$ given $\lim _{x \rightarrow 4} f(x)=4$ and $\lim _{x \rightarrow 4} g(x)=20$ . Answer: $\square$

See Solution

Problem 29803

Find the limits for $f(x)=\frac{x^{2}-9 x+8}{x+8}$ as $x \to -8^{-}$ , $x \to -8^{+}$ , and $x \to -8$ .

See Solution

Problem 29804

Find the limit: $\lim _{x \rightarrow 3} \frac{f(x)+g(x)}{2 g(x)}$ given $\lim _{x \rightarrow 3} f(x)=7$ and $\lim _{x \rightarrow 3} g(x)=5$ .

See Solution

Problem 29805

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{4 x^{2}+2 x}{5 x^{2}-2 x+1}$ and find its value.

See Solution

Problem 29806

Find the limit: $\lim _{x \rightarrow 2} 3^{f(x)}$ given that $\lim f(x)=5$ . Simplify your answer.

See Solution

Problem 29807

Find the limit: $\lim _{x \rightarrow 121} \frac{\sqrt{x}-11}{x-121}$ . Choose A or B for the answer.

See Solution

Problem 29808

Find and interpret $\lim_{s \rightarrow \infty} P(s)$ where $P(s)=\frac{70 s}{s+6}$ . What does the limit indicate?

See Solution

Problem 29809

Supply function: $q=8^{p}+6$ . Find inverse function $p=h(q)$ , sketch graph, and evaluate $\lim _{q \rightarrow 6^{+}} h(q)$ .

See Solution

Problem 29810

Supply function: $q=8^{P}+6$ a) Find the inverse supply function. b) Graph the inverse function $p=h(q)$ . c) Explain $\lim _{q \rightarrow 0^{+}} h(q)$ .

See Solution

Problem 29811

Estimate $\lim _{x \rightarrow 6} f(x)$ using values: $f(5.9)=8.9$ , $f(6)=9$ , $f(6.1)=9.1$ . Choices: A. $\lim _{x \rightarrow 6} f(x)=$ or B. limit does not exist.

See Solution

Problem 29812

Calculate $F=19,620 \times \int_{0}^{b}\left[11 y^{\frac{1}{2}}-y^{\frac{3}{2}}\right] d y$ .

See Solution

Problem 29813

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{2 x^{3}-5 x-4}{3 x^{2}-3 x-4}$ . If yes, find its value.

See Solution

Problem 29814

Given the piecewise function $f(x)$ , find the limits as $x$ approaches 2 and 8. Choose the correct options for each limit.

See Solution

Problem 29815

Calculate $F=19,620 \times \int_{0}^{6}\left[11 y^{\frac{1}{2}}-y^{\frac{3}{2}}\right] dy$ .

See Solution

Problem 29816

Evaluate the function and limit for $f(x)=\frac{7 x^{4}-14 x^{2}}{18 x^{5}+9}$ : (A) $f(-6)$ , (B) $f(-12)$ , (C) $\lim_{x \to -\infty} f(x)$ .

See Solution

Problem 29817

Determine if the limit exists:
$\lim _{x \rightarrow-3} \frac{x^{2}-9}{x+3}$
A. Find the limit value. B. The limit does not exist.

See Solution

Problem 29818

Find horizontal and vertical asymptotes for $f(x)=\frac{3 x+5}{4 x^{6}+3}$ .

See Solution

Problem 29819

Find the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}=\square$ (Type an integer or simplified fraction.)

See Solution

Problem 29820

Determine if the limit exists: $\lim _{h \rightarrow 0} \frac{\frac{4}{9+h}-\frac{4}{9}}{h}$ . If so, find its value.

See Solution

Problem 29821

Find $\lim _{x \rightarrow 2} \frac{7 x^{2}-1}{(x-2)^{2}}$ using a graph. Options: Does not exist, $-\infty$ , 7, $\infty$ .

See Solution

Problem 29822

Find the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ . Choose A (value) or B (does not exist).

See Solution

Problem 29823

Find horizontal and vertical asymptotes for $f(x)=\frac{3 x+5}{4 x^{6}+3}$ .

See Solution

Problem 29824

Find the gradient of these functions using differentiation rules: a) $y=5 x^{2}-3 \sqrt[4]{x}+2$ b) $y=\left(x^{2}-8\right)^{3}$ c) $y=\left(2 x^{2}+6 x+3\right)\left(9-x^{3}+8 x^{5}\right)$ d) $y=\frac{x^{2}-4}{2+x}$ and verify part (d) with chain and product rules.

See Solution

Problem 29825

Find the limits and value of $f(x)=\frac{|x-15|}{x-15}$ as $x \to 15^{+}$ , $x \to 15^{-}$ , and at $x=15$ .

See Solution

Problem 29826

Determine if the limit exists and find its value:
$\lim _{x \rightarrow \infty} \frac{7 x}{6 x-8}$
Choose A for the limit value or B if it doesn't exist.

See Solution

Problem 29827

Find the limit for the left end behavior of the function $f(x)=\frac{40 x^{6}+3 x^{2}}{17 x^{5}-4 x}$ . What is $\lim _{x \rightarrow-\infty} f(x)=\square$ ?

See Solution

Problem 29828

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{3 x^{2}+2 x}{4 x^{2}-3 x+1}$ . If so, find its value.

See Solution

Problem 29829

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{2 x^{2}-1}}{x+8}$ .

See Solution

Problem 29830

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{6 x^{3}+6 x-1}{6 x^{4}-8 x^{3}-9}$ . If so, find its value.

See Solution

Problem 29831

Find local max/min of $f(x)=x^{3}-4 x^{2}+12$ using the second derivative test.

See Solution

Problem 29832

Check if Rolle's theorem applies for $f(x)=\frac{5}{x^{2}}$ on $[-3,3]$ and find $c$ if conditions are met.

See Solution

Problem 29833

Given the piecewise function $f(x)=\left\{\begin{array}{ll}x-2 & \text { if } x<4 \\ 2 & \text { if } 4 \leq x \leq 6 \\ x+3 & \text { if } x>6\end{array}\right.$ , find the limits: a. $\lim _{x \rightarrow 4} f(x)$ b. $\lim _{x \rightarrow 6} f(x)$ Choose A or B for your answer.

See Solution

Problem 29834

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{6 x^{3}-2 x-6}{7 x^{2}-5 x-6}$ . What is its value?

See Solution

Problem 29835

Find the domain, asymptotes, intercepts, intervals of increase/decrease, concavity, relative extrema, inflection points, and graph for $f(x)=\frac{1}{4} x^{4}+x^{3}$ .

See Solution

Problem 29836

Find the limit: $\lim _{x \rightarrow 0^{+}} x^{-1}\left(e^{4 x}-1\right)$ . What is its value?

See Solution

Problem 29837

Find the limit as $x$ approaches 3 for the expression $\frac{x^{2}-4 x+3}{2 x^{2}-7 x+3}$ .

See Solution

Problem 29838

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{6 x^{3}-9 x-2}{4 x^{2}-2 x-9}$ . If yes, find its value.

See Solution

Problem 29839

Estimate $\lim _{x \rightarrow 3} f(x)$ using the provided values. Choose A or B for your answer.

See Solution

Problem 29840

Determine for which $x \in \mathbb{R}$ the function $f(x) = \sum_{n=0}^{\infty} e^{-2 n x}$ is defined.

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Problem 29841

A city's population is given by $P(t)=1.2(1.05)^{t}$ .
a) What is the average rate of change in population per year? b) Estimate the instantaneous rate of change in 2010.

See Solution

Problem 29842

Explain the Intermediate Value Theorem and its plausibility. Choose the correct answer about continuous functions on $[a, b]$ .

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Problem 29843

Evaluate the following integrals:
a) $\int_{0}^{2} x \, dx$ b) $\int_{-1}^{1} 3 x^{2} \, dx$ c) $\int_{1}^{3} (4x + 1) \, dx$ d) $\int_{2}^{5} 4 \, dx$ e) $\int_{0}^{2} 4x \, dx$ f) $\int_{0}^{3} 6x^{2} \, dx$ g) $\int_{-1}^{1} 2 \, dx$ h) $\int_{1}^{5} (3x^{2} + 2x + 2) \, dx$ i) $\int_{-3}^{1} (7x^{6} + 6x^{5} - 2) \, dx$ j) $\int_{0.5\pi}^{\pi} \sin x \, dx$ k) $\int_{1}^{3} (x^{2} + 2x) \, dx$ l) $\int_{1}^{e} \frac{1}{x} \, dx$ m) $\int_{-2}^{2} (4x^{7} - 3x^{2} + 4) \, dx$ n) $\int_{\pi}^{2\pi} \cos x \, dx$ o) $\int_{0}^{2} (x^{3} - x) \, dx$ p) $\int_{0}^{4} 6x \, dx$ q) $\int_{0}^{3} (\cos x - x^{2}) \, dx$

See Solution

Problem 29844

Find the values of $x$ where $r(x)=\ln \left|\frac{x}{x-7}\right|$ is discontinuous and state the limits.

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Problem 29845

Find the limits of the postage function $C(x)$ as $x$ approaches 3 from the left, right, and both sides.

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Problem 29846

Find local min/max points of $y=-x^{\frac{2}{3}}-4$ using the first derivative test.

See Solution

Problem 29847

Estimate the glacier's movement rate after 20 days using the model $d(t)=0.01 t^{2}+0.5 t$ .

See Solution

Problem 29848

Find the intervals where the graph of $y=-x^{4}-8 x^{3}+5 x-7$ is concave up or down.

See Solution

Problem 29849

Find the limit: $\lim _{x \rightarrow 0^{+}}(7 x+1)^{3 \cot x}$ . What does it equal?

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Problem 29850

Bestimmen Sie eine Stammfunktion $F$ für die Funktionen $f(x)$ in den Aufgaben a) bis 1) und prüfen Sie Ihre Ergebnisse.

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Problem 29851

Check if the mean value theorem applies to $f(x)=x^{3}-2x$ on $[1,3]$ and find $c$ in $(1,3)$ .

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Problem 29852

Find critical numbers of $f(x)=\frac{3 x^{2}}{x+4}$ . Options: $x=-8$ , $x=-4$ , $x=0$ , or combinations.

See Solution

Problem 29853

Evaluate the limit as $x$ approaches infinity for $\frac{7 x^{2}+5 x-8}{9 x^{3}-2 x^{2}+1}$ .

See Solution

Problem 29854

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

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Problem 29855

Find critical numbers of $f(x)=\frac{3}{2} x^{4}-4 x^{3}+3 x^{2}+2$ and identify local min/max points using a graph.

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Problem 29856

Untersuchen Sie Wendepunkte der Funktion $f(x)=2 \cdot e^{x}-e^{-x}$ und skizzieren Sie den Graphen.

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Problem 29857

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{5 x^{2}-8}}{x+3}$ .

See Solution

Problem 29858

Find the limit of postage cost $C(x)$ as $x$ approaches 3 oz from the left: $\lim_{x \rightarrow 3^{-}} C(x)=\$ \square$ .

See Solution

Problem 29859

Find the inflection point(s) for the function $f(x)=-2 x^{6}+3 x^{5}+12 x-2$ . Options: $(0,-2)$ , $(1,11)$ , $(0,12)$ , $(1,15)$ .

See Solution

Problem 29860

Find the inflection point(s) for $f(x)=-x^{5}+5 x^{4}+12 x-2$ . Options: (3,0), (0,-2), (3,196), (3,147).

See Solution

Problem 29861

Postage costs \ $0.34 for the first ounce and \$0.24 for each additional ounce. Find$ \lim C(x) $as$ x \rightarrow 3^{-} $. A.$ \lim C(x)=\ $\quad$ (integer or decimal) B. Limit does not exist.

See Solution

Problem 29862

Is the function $f(x)=\sqrt[3]{x^{2}}$ differentiable at $x=0$ ? If not, explain why.

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Problem 29863

Evaluate the integral: $\int_{0}^{1}\left(2 e^{5 x+1}-x^{3}\right) d x$ .

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Problem 29864

Use Newton's formula to approximate $\sqrt[3]{7.5}$ from $x^{3}-7.5=0$ and find the third iteration value.

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Problem 29865

Find the limit of postage cost $C(x)$ as $x$ approaches 3 ounces: $\lim_{x \rightarrow 3^{-}} C(x) = \$.

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Problem 29866

Berechne die Fläche $A$ zwischen den Funktionen $f(x)=0,5 x^{2}+2$ und $g(x)=-0,5 x+1$ im Intervall $[-1, 1,5]$ . Zeige, dass $A=\int_{-1}^{1,5} (f(x)-g(x)) \, dx$ .

See Solution

Problem 29867

Estimate $\int_{2}^{4} \frac{15}{x^{2}+1} d x$ using Simpson's rule with $n=4$ . Round to five decimal places.

See Solution

Problem 29868

Berechnen Sie die Fläche A zwischen den Funktionen $f(x)=2-x^{2}$ und $g(x)=0,5 x^{2}+0,5$ im Intervall $[-1, 1]$ .

See Solution

Problem 29869

Calculate the integral $\int x^{3} \sqrt{x^{2}+1} \, dx$ .

See Solution

Problem 29870

Find local minima and maxima of the function $y=x^{3}-6 x^{2}+2$ using the first derivative test.

See Solution

Problem 29871

Calculate the average value of the function $f(x)=3 x^{2}-2 x$ on the interval $[-2,3]$ .

See Solution

Problem 29872

Find the limit: $\lim _{h \rightarrow 0} \frac{(8+h)^{2}-64}{h}$ or state if it doesn't exist.

See Solution

Problem 29873

Find $\bar{o}$ for the limit as $x$ approaches 7 in $\lim _{x \rightarrow 7}(2 x-6)=8$ with $\varepsilon=0.001$ .

See Solution

Problem 29874

Find the tangent line equation to $f(x)=x^{5}$ at $x=-2$ . Choices: $y=-80 x+128$ , $y=-80 x-192$ , $y=80 x-192$ , $y=80 x+128$ .

See Solution

Problem 29875

Calculate the limit: $\lim _{x \rightarrow-4} \frac{2(2 x+3)^{2}-50}{x+4}$ or state if it doesn't exist.

See Solution

Problem 29876

Find the rate of change of paycheck with respect to hours worked given hours: 40, 80, 125, 160 and paychecks: 828, 1788, 2868, 3708.

See Solution

Problem 29877

Evaluate $\lim _{x \rightarrow 6^{+}} \sqrt{x-6}$ and explain why this limit does not exist.

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Problem 29878

Find the limits of the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}+10, & x<-10 \\ \sqrt{x+10}, & x \geq-10\end{array}\right.$ at $x=-10$ .
a. $\lim _{x \rightarrow-10^{-}} f(x)$ b. $\lim _{x \rightarrow-10^{+}} f(x)$ c. $\lim _{x \rightarrow-10} f(x)$

See Solution

Problem 29879

Find the limit or state if it doesn't exist: $\lim _{h \rightarrow 0} \frac{\sqrt{4+h}-2}{h}$

See Solution

Problem 29880

Find the differential $dy$ for $y=\cos(7x^2+1)$ .

See Solution

Problem 29881

Find the limit $\lim _{x \rightarrow 1}[4 f(x)]$ given $\lim _{x \rightarrow 1} f(x)=5$ and state the limit laws used.

See Solution

Problem 29882

Find the limit: $\lim _{x \rightarrow 144} \frac{\sqrt{x}-12}{x-144}$ or state if it doesn't exist.

See Solution

Problem 29883

Find the limit or state if it doesn't exist: $\lim _{x \rightarrow 196} \frac{\sqrt{x}-14}{x-196}$

See Solution

Problem 29884

Find $\delta$ for $\varepsilon=0.5, 0.1, 0.05$ given $\lim _{x \rightarrow 4}(4 x-2)=14$ .

See Solution

Problem 29885

Given $g(x)=f(9-x)$ , with $\lim _{x \rightarrow 9^{+}} f(x)=6$ and $\lim _{x \rightarrow 9^{-}} f(x)=5$ , find $\lim _{x \rightarrow 0^{+}} g(x)$ and $\lim _{x \rightarrow 0^{-}} g(x)$ . $\lim _{x \rightarrow 0^{+}} g(x)=\square$ (Type an integer or a decimal.)

See Solution

Problem 29886

Show that $\lim _{x \rightarrow 0}|x|=0$ by finding $\lim _{x \rightarrow 0^{-}}|x|$ and $\lim _{x \rightarrow 0^{+}}|x|$ .

See Solution

Problem 29887

Show that $\lim _{x \rightarrow 0}|x|=0$ by finding $\lim _{x \rightarrow 0^{-}}|x|$ and $\lim _{x \rightarrow 0^{+}}|x|$ .

See Solution

Problem 29888

Find $\lim _{x \rightarrow 7} \frac{f(x)}{g(x)-h(x)}$ given $\lim _{x \rightarrow 7} f(x)=28$ , $\lim _{x \rightarrow 7} g(x)=6$ , $\lim _{x \rightarrow 7} h(x)=4$ . State the limit laws used.

See Solution

Problem 29889

Find where the tangent line to $y=f(x)=x^{3} \ln (x)$ is horizontal and if it intersects at $(1,0)$ .

See Solution

Problem 29890

Calculate the limit: $\lim _{x \rightarrow 0} \frac{x^{3}}{\sin \left(3 x^{2}\right)}$ . Does it exist?

See Solution

Problem 29891

Find the limit as $x$ approaches $-\infty$ for $e^{x} \sin(x^{2})$ . Does it exist?

See Solution

Problem 29892

Calculate the limit: $\lim _{x \rightarrow-1^{-}} e^{\frac{-1}{1-x^{2}}}$ . Does it exist?

See Solution

Problem 29893

Find the radius of convergence for the series $\sum_{n=1}^{\infty} 2^{n}\left(n+\frac{1}{n}\right) x^{n}$ .

See Solution

Problem 29894

Analyze the function $f(x)=\frac{1}{x^{2}-1}$ for max/min and if it's increasing: (a) max? (b) min? (c) increasing?

See Solution

Problem 29895

Find the value of the series $\sum_{n=0}^{\infty} \frac{n^{50}}{3^{n}}$ .

See Solution

Problem 29896

Find where $f(x)=\sum_{n=0}^{\infty} e^{-2 n x}$ is defined and calculate $f^{\prime}(x)$ .

See Solution

Problem 29897

Find where the tangent line to $f(x)=x^{3} \ln (x)$ is horizontal for $x \in (0, \infty)$ .

See Solution

Problem 29898

Is the function $f(x)=\left\{\begin{array}{cc}(x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ 1 & x=1\end{array}\right.$ continuous on $\mathbb{R}$ ?

See Solution

Problem 29899

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{3}}{\sin \left(x^{2}\right)}$ or state if it doesn't exist.

See Solution

Problem 29900

Find the derivative of $f(x) = (1 + \cos^2 x)^6$ .

See Solution

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Calculus

Problem 29801

Analyze the graph of f(x)=exf(x)=e^{x}f(x)=ex to solve: a. Find lim⁡x→−∞ex\lim _{x \rightarrow-\infty} e^{x}limx→−∞​ex. b. Identify the horizontal asymptote.

Problem 29802

Find lim⁡x→4f(x)g(x)\lim _{x \rightarrow 4} \frac{f(x)}{g(x)}limx→4​g(x)f(x)​ given lim⁡x→4f(x)=4\lim _{x \rightarrow 4} f(x)=4limx→4​f(x)=4 and lim⁡x→4g(x)=20\lim _{x \rightarrow 4} g(x)=20limx→4​g(x)=20. Answer: □\square□

Problem 29803

Find the limits for f(x)=x2−9x+8x+8f(x)=\frac{x^{2}-9 x+8}{x+8}f(x)=x+8x2−9x+8​ as x→−8−x \to -8^{-}x→−8−, x→−8+x \to -8^{+}x→−8+, and x→−8x \to -8x→−8.

Problem 29804

Find the limit: lim⁡x→3f(x)+g(x)2g(x)\lim _{x \rightarrow 3} \frac{f(x)+g(x)}{2 g(x)}limx→3​2g(x)f(x)+g(x)​ given lim⁡x→3f(x)=7\lim _{x \rightarrow 3} f(x)=7limx→3​f(x)=7 and lim⁡x→3g(x)=5\lim _{x \rightarrow 3} g(x)=5limx→3​g(x)=5.

Problem 29805

Determine if the limit exists: lim⁡x→∞4x2+2x5x2−2x+1\lim _{x \rightarrow \infty} \frac{4 x^{2}+2 x}{5 x^{2}-2 x+1}x→∞lim​5x2−2x+14x2+2x​ and find its value.

Problem 29806

Find the limit: lim⁡x→23f(x)\lim _{x \rightarrow 2} 3^{f(x)}limx→2​3f(x) given that lim⁡f(x)=5\lim f(x)=5limf(x)=5. Simplify your answer.

Problem 29807

Find the limit: lim⁡x→121x−11x−121\lim _{x \rightarrow 121} \frac{\sqrt{x}-11}{x-121}x→121lim​x−121x​−11​. Choose A or B for the answer.

Problem 29808

Find and interpret lim⁡s→∞P(s)\lim_{s \rightarrow \infty} P(s)lims→∞​P(s) where P(s)=70ss+6P(s)=\frac{70 s}{s+6}P(s)=s+670s​. What does the limit indicate?

Problem 29809

Supply function: q=8p+6q=8^{p}+6q=8p+6. Find inverse function p=h(q)p=h(q)p=h(q), sketch graph, and evaluate lim⁡q→6+h(q)\lim _{q \rightarrow 6^{+}} h(q)limq→6+​h(q).

Problem 29810

Supply function: q=8P+6q=8^{P}+6q=8P+6 a) Find the inverse supply function. b) Graph the inverse function p=h(q)p=h(q)p=h(q). c) Explain lim⁡q→0+h(q)\lim _{q \rightarrow 0^{+}} h(q)limq→0+​h(q).

Problem 29811

Estimate lim⁡x→6f(x)\lim _{x \rightarrow 6} f(x)limx→6​f(x) using values: f(5.9)=8.9f(5.9)=8.9f(5.9)=8.9, f(6)=9f(6)=9f(6)=9, f(6.1)=9.1f(6.1)=9.1f(6.1)=9.1. Choices: A. lim⁡x→6f(x)=\lim _{x \rightarrow 6} f(x)=limx→6​f(x)= or B. limit does not exist.

Problem 29812

Calculate F=19,620×∫0b[11y12−y32]dyF=19,620 \times \int_{0}^{b}\left[11 y^{\frac{1}{2}}-y^{\frac{3}{2}}\right] d yF=19,620×∫0b​[11y21​−y23​]dy.

Problem 29813

Determine if the limit exists: lim⁡x→∞2x3−5x−43x2−3x−4\lim _{x \rightarrow \infty} \frac{2 x^{3}-5 x-4}{3 x^{2}-3 x-4}x→∞lim​3x2−3x−42x3−5x−4​. If yes, find its value.

Problem 29814

Given the piecewise function f(x)f(x)f(x), find the limits as xxx approaches 2 and 8. Choose the correct options for each limit.

Problem 29815

Calculate F=19,620×∫06[11y12−y32]dyF=19,620 \times \int_{0}^{6}\left[11 y^{\frac{1}{2}}-y^{\frac{3}{2}}\right] dyF=19,620×∫06​[11y21​−y23​]dy.

Problem 29816

Evaluate the function and limit for f(x)=7x4−14x218x5+9f(x)=\frac{7 x^{4}-14 x^{2}}{18 x^{5}+9}f(x)=18x5+97x4−14x2​: (A) f(−6)f(-6)f(−6), (B) f(−12)f(-12)f(−12), (C) lim⁡x→−∞f(x)\lim_{x \to -\infty} f(x)limx→−∞​f(x).

Problem 29817

Determine if the limit exists: lim⁡x→−3x2−9x+3\lim _{x \rightarrow-3} \frac{x^{2}-9}{x+3}x→−3lim​x+3x2−9​ A. Find the limit value. B. The limit does not exist.

Problem 29818

Find horizontal and vertical asymptotes for f(x)=3x+54x6+3f(x)=\frac{3 x+5}{4 x^{6}+3}f(x)=4x6+33x+5​.

Problem 29819

Find the limit: lim⁡x→64x−8x−64=□\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}=\squarex→64lim​x−64x​−8​=□ (Type an integer or simplified fraction.)

Problem 29820

Determine if the limit exists: lim⁡h→049+h−49h\lim _{h \rightarrow 0} \frac{\frac{4}{9+h}-\frac{4}{9}}{h}h→0lim​h9+h4​−94​​. If so, find its value.

Problem 29821

Find lim⁡x→27x2−1(x−2)2\lim _{x \rightarrow 2} \frac{7 x^{2}-1}{(x-2)^{2}}limx→2​(x−2)27x2−1​ using a graph. Options: Does not exist, −∞-\infty−∞, 7, ∞\infty∞.

Problem 29822

Find the limit: lim⁡x→64x−8x−64\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}x→64lim​x−64x​−8​. Choose A (value) or B (does not exist).

Problem 29823

Find horizontal and vertical asymptotes for f(x)=3x+54x6+3f(x)=\frac{3 x+5}{4 x^{6}+3}f(x)=4x6+33x+5​.

Problem 29824

Problem 29825

Find the limits and value of f(x)=∣x−15∣x−15f(x)=\frac{|x-15|}{x-15}f(x)=x−15∣x−15∣​ as x→15+x \to 15^{+}x→15+, x→15−x \to 15^{-}x→15−, and at x=15x=15x=15.

Problem 29826

Determine if the limit exists and find its value: lim⁡x→∞7x6x−8 \lim _{x \rightarrow \infty} \frac{7 x}{6 x-8} x→∞lim​6x−87x​ Choose A for the limit value or B if it doesn't exist.

Problem 29827

Find the limit for the left end behavior of the function f(x)=40x6+3x217x5−4xf(x)=\frac{40 x^{6}+3 x^{2}}{17 x^{5}-4 x}f(x)=17x5−4x40x6+3x2​. What is lim⁡x→−∞f(x)=□\lim _{x \rightarrow-\infty} f(x)=\squarelimx→−∞​f(x)=□?

Problem 29828

Determine if the limit exists: lim⁡x→∞3x2+2x4x2−3x+1\lim _{x \rightarrow \infty} \frac{3 x^{2}+2 x}{4 x^{2}-3 x+1}x→∞lim​4x2−3x+13x2+2x​. If so, find its value.

Problem 29829

Find the limit: lim⁡x→−∞2x2−1x+8\lim _{x \rightarrow-\infty} \frac{\sqrt{2 x^{2}-1}}{x+8}limx→−∞​x+82x2−1​​.

Problem 29830

Determine if the limit exists: lim⁡x→∞6x3+6x−16x4−8x3−9\lim _{x \rightarrow \infty} \frac{6 x^{3}+6 x-1}{6 x^{4}-8 x^{3}-9}x→∞lim​6x4−8x3−96x3+6x−1​. If so, find its value.

Problem 29831

Find local max/min of f(x)=x3−4x2+12f(x)=x^{3}-4 x^{2}+12f(x)=x3−4x2+12 using the second derivative test.

Problem 29832

Check if Rolle's theorem applies for f(x)=5x2f(x)=\frac{5}{x^{2}}f(x)=x25​ on [−3,3][-3,3][−3,3] and find ccc if conditions are met.

Problem 29833

Problem 29834

Determine if the limit exists: lim⁡x→∞6x3−2x−67x2−5x−6\lim _{x \rightarrow \infty} \frac{6 x^{3}-2 x-6}{7 x^{2}-5 x-6}x→∞lim​7x2−5x−66x3−2x−6​. What is its value?

Problem 29835

Find the domain, asymptotes, intercepts, intervals of increase/decrease, concavity, relative extrema, inflection points, and graph for f(x)=14x4+x3f(x)=\frac{1}{4} x^{4}+x^{3}f(x)=41​x4+x3.

Problem 29836

Find the limit: lim⁡x→0+x−1(e4x−1)\lim _{x \rightarrow 0^{+}} x^{-1}\left(e^{4 x}-1\right)limx→0+​x−1(e4x−1). What is its value?

Problem 29837

Find the limit as xxx approaches 3 for the expression x2−4x+32x2−7x+3\frac{x^{2}-4 x+3}{2 x^{2}-7 x+3}2x2−7x+3x2−4x+3​.

Problem 29838

Determine if the limit exists: lim⁡x→∞6x3−9x−24x2−2x−9\lim _{x \rightarrow \infty} \frac{6 x^{3}-9 x-2}{4 x^{2}-2 x-9}x→∞lim​4x2−2x−96x3−9x−2​. If yes, find its value.

Problem 29839

Estimate lim⁡x→3f(x)\lim _{x \rightarrow 3} f(x)limx→3​f(x) using the provided values. Choose A or B for your answer.

Problem 29840

Determine for which x∈Rx \in \mathbb{R}x∈R the function f(x)=∑n=0∞e−2nxf(x) = \sum_{n=0}^{\infty} e^{-2 n x}f(x)=∑n=0∞​e−2nx is defined.

Problem 29841

Analyze the graph of $f(x)=e^{x}$ to solve: a. Find $\lim _{x \rightarrow-\infty} e^{x}$ . b. Identify the horizontal asymptote.

Find $\lim _{x \rightarrow 4} \frac{f(x)}{g(x)}$ given $\lim _{x \rightarrow 4} f(x)=4$ and $\lim _{x \rightarrow 4} g(x)=20$ . Answer: $\square$

Find the limits for $f(x)=\frac{x^{2}-9 x+8}{x+8}$ as $x \to -8^{-}$ , $x \to -8^{+}$ , and $x \to -8$ .

Find the limit: $\lim _{x \rightarrow 3} \frac{f(x)+g(x)}{2 g(x)}$ given $\lim _{x \rightarrow 3} f(x)=7$ and $\lim _{x \rightarrow 3} g(x)=5$ .

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{4 x^{2}+2 x}{5 x^{2}-2 x+1}$ and find its value.

Find the limit: $\lim _{x \rightarrow 2} 3^{f(x)}$ given that $\lim f(x)=5$ . Simplify your answer.

Find the limit: $\lim _{x \rightarrow 121} \frac{\sqrt{x}-11}{x-121}$ . Choose A or B for the answer.

Find and interpret $\lim_{s \rightarrow \infty} P(s)$ where $P(s)=\frac{70 s}{s+6}$ . What does the limit indicate?

Supply function: $q=8^{p}+6$ . Find inverse function $p=h(q)$ , sketch graph, and evaluate $\lim _{q \rightarrow 6^{+}} h(q)$ .

Supply function: $q=8^{P}+6$ a) Find the inverse supply function. b) Graph the inverse function $p=h(q)$ . c) Explain $\lim _{q \rightarrow 0^{+}} h(q)$ .

Estimate $\lim _{x \rightarrow 6} f(x)$ using values: $f(5.9)=8.9$ , $f(6)=9$ , $f(6.1)=9.1$ . Choices: A. $\lim _{x \rightarrow 6} f(x)=$ or B. limit does not exist.

Calculate $F=19,620 \times \int_{0}^{b}\left[11 y^{\frac{1}{2}}-y^{\frac{3}{2}}\right] d y$ .

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{2 x^{3}-5 x-4}{3 x^{2}-3 x-4}$ . If yes, find its value.

Given the piecewise function $f(x)$ , find the limits as $x$ approaches 2 and 8. Choose the correct options for each limit.

Calculate $F=19,620 \times \int_{0}^{6}\left[11 y^{\frac{1}{2}}-y^{\frac{3}{2}}\right] dy$ .

Evaluate the function and limit for $f(x)=\frac{7 x^{4}-14 x^{2}}{18 x^{5}+9}$ : (A) $f(-6)$ , (B) $f(-12)$ , (C) $\lim_{x \to -\infty} f(x)$ .

Determine if the limit exists:
$\lim _{x \rightarrow-3} \frac{x^{2}-9}{x+3}$
A. Find the limit value. B. The limit does not exist.

Find horizontal and vertical asymptotes for $f(x)=\frac{3 x+5}{4 x^{6}+3}$ .

Find the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}=\square$ (Type an integer or simplified fraction.)

Determine if the limit exists: $\lim _{h \rightarrow 0} \frac{\frac{4}{9+h}-\frac{4}{9}}{h}$ . If so, find its value.

Find $\lim _{x \rightarrow 2} \frac{7 x^{2}-1}{(x-2)^{2}}$ using a graph. Options: Does not exist, $-\infty$ , 7, $\infty$ .

Find the limit: $\lim _{x \rightarrow 64} \frac{\sqrt{x}-8}{x-64}$ . Choose A (value) or B (does not exist).

Find horizontal and vertical asymptotes for $f(x)=\frac{3 x+5}{4 x^{6}+3}$ .

Find the limits and value of $f(x)=\frac{|x-15|}{x-15}$ as $x \to 15^{+}$ , $x \to 15^{-}$ , and at $x=15$ .

Determine if the limit exists and find its value:
$\lim _{x \rightarrow \infty} \frac{7 x}{6 x-8}$
Choose A for the limit value or B if it doesn't exist.

Find the limit for the left end behavior of the function $f(x)=\frac{40 x^{6}+3 x^{2}}{17 x^{5}-4 x}$ . What is $\lim _{x \rightarrow-\infty} f(x)=\square$ ?

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{3 x^{2}+2 x}{4 x^{2}-3 x+1}$ . If so, find its value.

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{2 x^{2}-1}}{x+8}$ .

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{6 x^{3}+6 x-1}{6 x^{4}-8 x^{3}-9}$ . If so, find its value.

Find local max/min of $f(x)=x^{3}-4 x^{2}+12$ using the second derivative test.

Check if Rolle's theorem applies for $f(x)=\frac{5}{x^{2}}$ on $[-3,3]$ and find $c$ if conditions are met.

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{6 x^{3}-2 x-6}{7 x^{2}-5 x-6}$ . What is its value?

Find the domain, asymptotes, intercepts, intervals of increase/decrease, concavity, relative extrema, inflection points, and graph for $f(x)=\frac{1}{4} x^{4}+x^{3}$ .

Find the limit: $\lim _{x \rightarrow 0^{+}} x^{-1}\left(e^{4 x}-1\right)$ . What is its value?

Find the limit as $x$ approaches 3 for the expression $\frac{x^{2}-4 x+3}{2 x^{2}-7 x+3}$ .

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{6 x^{3}-9 x-2}{4 x^{2}-2 x-9}$ . If yes, find its value.

Estimate $\lim _{x \rightarrow 3} f(x)$ using the provided values. Choose A or B for your answer.

Determine for which $x \in \mathbb{R}$ the function $f(x) = \sum_{n=0}^{\infty} e^{-2 n x}$ is defined.

A city's population is given by $P(t)=1.2(1.05)^{t}$ .
a) What is the average rate of change in population per year? b) Estimate the instantaneous rate of change in 2010.

Explain the Intermediate Value Theorem and its plausibility. Choose the correct answer about continuous functions on $[a, b]$ .

Find the values of $x$ where $r(x)=\ln \left|\frac{x}{x-7}\right|$ is discontinuous and state the limits.

Find the limits of the postage function $C(x)$ as $x$ approaches 3 from the left, right, and both sides.

Find local min/max points of $y=-x^{\frac{2}{3}}-4$ using the first derivative test.

Estimate the glacier's movement rate after 20 days using the model $d(t)=0.01 t^{2}+0.5 t$ .

Find the intervals where the graph of $y=-x^{4}-8 x^{3}+5 x-7$ is concave up or down.

Find the limit: $\lim _{x \rightarrow 0^{+}}(7 x+1)^{3 \cot x}$ . What does it equal?

Bestimmen Sie eine Stammfunktion $F$ für die Funktionen $f(x)$ in den Aufgaben a) bis 1) und prüfen Sie Ihre Ergebnisse.

Check if the mean value theorem applies to $f(x)=x^{3}-2x$ on $[1,3]$ and find $c$ in $(1,3)$ .

Find critical numbers of $f(x)=\frac{3 x^{2}}{x+4}$ . Options: $x=-8$ , $x=-4$ , $x=0$ , or combinations.

Evaluate the limit as $x$ approaches infinity for $\frac{7 x^{2}+5 x-8}{9 x^{3}-2 x^{2}+1}$ .

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

Find critical numbers of $f(x)=\frac{3}{2} x^{4}-4 x^{3}+3 x^{2}+2$ and identify local min/max points using a graph.

Untersuchen Sie Wendepunkte der Funktion $f(x)=2 \cdot e^{x}-e^{-x}$ und skizzieren Sie den Graphen.

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt{5 x^{2}-8}}{x+3}$ .

Find the limit of postage cost $C(x)$ as $x$ approaches 3 oz from the left: $\lim_{x \rightarrow 3^{-}} C(x)=\$ \square$ .

Find the inflection point(s) for the function $f(x)=-2 x^{6}+3 x^{5}+12 x-2$ . Options: $(0,-2)$ , $(1,11)$ , $(0,12)$ , $(1,15)$ .

Find the inflection point(s) for $f(x)=-x^{5}+5 x^{4}+12 x-2$ . Options: (3,0), (0,-2), (3,196), (3,147).

Is the function $f(x)=\sqrt[3]{x^{2}}$ differentiable at $x=0$ ? If not, explain why.

Evaluate the integral: $\int_{0}^{1}\left(2 e^{5 x+1}-x^{3}\right) d x$ .

Use Newton's formula to approximate $\sqrt[3]{7.5}$ from $x^{3}-7.5=0$ and find the third iteration value.

Find the limit of postage cost $C(x)$ as $x$ approaches 3 ounces: $\lim_{x \rightarrow 3^{-}} C(x) = \$.

Berechne die Fläche $A$ zwischen den Funktionen $f(x)=0,5 x^{2}+2$ und $g(x)=-0,5 x+1$ im Intervall $[-1, 1,5]$ . Zeige, dass $A=\int_{-1}^{1,5} (f(x)-g(x)) \, dx$ .

Estimate $\int_{2}^{4} \frac{15}{x^{2}+1} d x$ using Simpson's rule with $n=4$ . Round to five decimal places.

Berechnen Sie die Fläche A zwischen den Funktionen $f(x)=2-x^{2}$ und $g(x)=0,5 x^{2}+0,5$ im Intervall $[-1, 1]$ .

Calculate the integral $\int x^{3} \sqrt{x^{2}+1} \, dx$ .

Find local minima and maxima of the function $y=x^{3}-6 x^{2}+2$ using the first derivative test.

Calculate the average value of the function $f(x)=3 x^{2}-2 x$ on the interval $[-2,3]$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{(8+h)^{2}-64}{h}$ or state if it doesn't exist.

Find $\bar{o}$ for the limit as $x$ approaches 7 in $\lim _{x \rightarrow 7}(2 x-6)=8$ with $\varepsilon=0.001$ .

Find the tangent line equation to $f(x)=x^{5}$ at $x=-2$ . Choices: $y=-80 x+128$ , $y=-80 x-192$ , $y=80 x-192$ , $y=80 x+128$ .