Calculus

Problem 28901

Find the equation of the tangent line $y=\frac{-3}{5 x+2}$ at $x=3$ .

See Solution

Problem 28902

Find the value(s) of $x$ where the function $g(x)=\frac{x+1}{\sqrt{x}}$ has a horizontal tangent.

See Solution

Problem 28903

Differentiate the function $f(x)=\sqrt{\cos x \sin x}$ .

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Problem 28904

How long until $80 \mathrm{~g}$ of carbon-14 remains from $100 \mathrm{~g}$ ? Also, find the rate of change at 100 years.

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Problem 28905

Carbon-14 has a half-life of 5730 years. A plant had $100 \mathrm{~g}$ at death. Find: a) remaining after 5730 years, b) equation for remaining after $t$ years, c) years for $80 \mathrm{~g}$ left, d) rate of change at 100 years.

See Solution

Problem 28906

Find the absolute extrema of $f(x)=(x-3)e^{x}$ for $-1 \leq x \leq 3$ .

See Solution

Problem 28907

A cone-shaped tank is 18 ft tall and 12 ft wide. If drained at $4 \mathrm{ft}^3/\mathrm{min}$ , how fast drops water level at 6 ft?

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Problem 28908

Find $b$ such that the slope of the tangent to $f(x)=(3 x^{2}+1)(2 x^{2}+b)$ at $x=-1$ is -16.

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Problem 28909

How fast is the diameter of a balloon increasing at a radius of 5 cm if air is pumped in at $100 \mathrm{~cm}^{3} / \mathrm{sec}$ ?

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Problem 28910

Find the limit: $\lim _{x \rightarrow 0} \frac{6 x+2 \ln x}{x+\sin x}$ .

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Problem 28911

Find the instantaneous rate of change of $w$ for $w=\frac{1}{z}+\frac{z}{2}$ . Choices: (A) $\frac{3}{2}$ (B) -2 (C) $\frac{z^{2}-2}{2 z^{2}}$ (D) $-\frac{1}{z^{2}}$ (E) None.

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Problem 28912

Sia $f$ una funzione integrabile con $\int_{0}^{1} f(x) d x = 3$ . Quale affermazione è vera tra (a), (b), (c), (d) o (e)?

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Problem 28913

Sia $f$ una funzione integrabile con $\int_{0}^{1} f(x) d x = 3$ . Quale delle seguenti affermazioni è vera? (a), (b), (c), (d) o (e)?

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Problem 28914

Trova l'espressione esplicita di $f^{\prime}(x)$ per la funzione $f(x) = \int_{3 x^{2}}^{7 x} e^{-t^{2}} \mathrm{d} t$ , con $x \in [0,1]$ .

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Problem 28915

Find the limit: $\lim _{x \rightarrow 0^{-}}\left(1+3^{x}\right)^{1 / x}$ .

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Problem 28916

Approximate the rate of change for $f(x)=x^{3}-5$ at $x=1$ using $h=0.1, h=0.01, h=0.001$ .

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Problem 28917

Find the normal line equation to $y=x-\frac{1}{x}$ at point $(1,0)$ in the form $y=mx+b$ .

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Problem 28918

Find a real number between $n = -5$ and $b = -4$ for the function $f(x) = x^{3} + 3x^{2} - 9x - 13$ using the Intermediate Value Theorem.

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Problem 28919

Calculate the volume of the solid using the formula $\pi\left[\frac{x^{2}}{2}-\frac{x^{9}}{9}\right]_{0}^{1}$ .

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Problem 28920

Find the rate of change of revenue when producing 4000 products, given $R(x)=x(-0.25 x^{2}+7.5 x+18.75)$ .

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Problem 28921

The cost to produce $x$ boxes of cookies is $C=0.0001 x^{3}-0.03 x^{2}+4 x+300$ . Production after $t$ weeks is $x=1400+100 t$ .
(a) Find the marginal cost $C^{\prime}(x)$ . $C^{\prime}(x)=$ (b) Use $\frac{d C}{d t}=\frac{d C}{d x} \cdot \frac{d x}{d t}$ to find the rate of cost change over time. $\frac{d C}{d t}=$ (c) Find the cost increase rate (in \ $per week) when$ t=3$ weeks.

See Solution

Problem 28922

Find the tangent line equation for $y=\sin \left(2 x+\frac{\pi}{3}\right)$ at $x=\frac{\pi}{3}$ .

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Problem 28923

Find the melting rate of a snowball with volume $S=\frac{4}{3} \pi r^{3}$ , where $r=\frac{1}{(t+1)^{2}}-\frac{1}{17}$ .

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Problem 28924

Find the average velocity of a ball rolling down a ramp given $s(t)=10 t^{2}$ from $t_{1}=3$ to $t_{2}=4$ .

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Problem 28925

Calculate the limit: $\lim _{x \rightarrow 1^{+}} \arctan \left(\frac{x^{2}}{1-x^{2}}\right)$ . Does it exist?

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Problem 28926

1. Find the limit: $\lim _{n \rightarrow \infty} \frac{(e-1) \sum_{k=0}^{\infty} \frac{1}{e^{k}}}{(1+\frac{2}{n})^{n}}$ .
2. Determine if the series $\sum_{n=1}^{\infty} \frac{n ! \sin (1 / n)}{n^{n}}$ converges or diverges.
3. Calculate the radius of convergence for $\sum_{n=1}^{\infty} 2^{n}\left(n+\frac{1}{n}\right) x^{n}$ .
4. Does $f(x)=x-\sqrt{x}$ for $x \geq 1$ have an inverse? (YES / NO)
5. For $f(x)=\frac{1}{x^{2}-1}$ , determine: (a) has a maximum? (YES / NO) (b) has a minimum? (YES / NO) (c) is increasing? (YES / NO)

See Solution

Problem 28927

Find the rate of change of depth $D(t)=7 \sin \left(\frac{\pi}{6} t-\frac{7 \pi}{6}\right)+4$ at $t=6$ hours. Round to one decimal place.

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Problem 28928

Find the general antiderivative of $f(x)=\frac{3 x^{2}-4 x+8}{x^{2}}$ for $x>0$ .

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Problem 28929

Given the function $y=\cot(\csc(x))$ , decompose it as $y=f(u)$ and $u=g(x)$ . Then find $\frac{d y}{d x}$ .

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Problem 28930

Estimate the area under the curve $y=f(x)$ from $x=0$ to $x=10$ using 5 rectangles. Round to 1 decimal place.

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Problem 28931

Which properties are needed to evaluate $\int_{1}^{4}(2 x+3) d x$ ? Options: a) P only b) Q only c) Both P and Q d) Neither P nor Q

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Problem 28932

Berechnen Sie die Ableitung von $f(x)=\frac{(x+3)^{2}}{(x+2)^{3}}$ .

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Problem 28933

Evaluate the limit as $x$ approaches 4 for the expression $x^{4}-5x+6$ . What is the limit?

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Problem 28934

Find the value of $\int_{0}^{2/3}(4-x^{2})dx$ given that $\int_{0}^{1}(4-x^{2})dx=\frac{11}{3}$ . Options: a) $\frac{16}{9}$ b) $\frac{22}{9}$ c) $\frac{10}{3}$ d) less than $\frac{11}{3}$ .

See Solution

Problem 28935

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

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Problem 28936

Which properties are needed to evaluate $\int_{0}^{3}|x-1| d x$ ? a) P only b) Q only c) Both P and Q d) Neither P nor Q

See Solution

Problem 28937

Estimate the limit: $\lim _{\theta \rightarrow 0} \frac{\sin (7 \theta)}{\theta}=\square$ , with $\theta$ in radians.

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Problem 28938

Find bounds for $I = \int_{-3}^{0} (x^{2} + 2x) \, dx$ . Options: a) $1 \leq I \leq 3$ b) $\bigcirc$ c) $-3 \leq I \leq -5$ d) $-3 \leq I \leq 9$ .

See Solution

Problem 28939

Fill in the table to estimate $\lim_{x \to 3} f(x)$ for $f(x) = \frac{x^3 - 27}{x^2 - 9}$ .

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Problem 28940

Which integral equals $\int_{3}^{4} f(x) dx + \int_{5}^{8} f(x) dx + \int_{4}^{3} f(x) dx$ ? a) $0 \int_{1}^{3} f(x) dx$ b) $0 \int_{3}^{5} f(x) dx$ c) $0 \int_{5}^{8} f(x) dx$ d) $O \int_{12}^{15} f(x) dx$

See Solution

Problem 28941

Which statement is true? P: For non-negative integer $k$ , $\left|\int_{0}^{1} x^{k} f(x) dx\right| \geq \int_{0}^{1}|f(x)| dx$ . Q: $\int_{0}^{1} x dx \leq \int_{0}^{1} x^{2} dx$ . a) P Only b) Q Only c) Both P and Q d) Neither P nor Q

See Solution

Problem 28942

6. Find the limit: $\lim _{x \rightarrow 1^{-}} \frac{x^{2}+2 x-3}{|x-1|}$ . Does it exist?
7. Find the limit: $\lim _{x \rightarrow 0} \frac{x^{3}}{\sin \left(x^{2}\right)}$ . Does it exist?
8. Find the limit: $\lim _{x \rightarrow 1^{-}} \arctan \left(\frac{x^{2}}{1-x^{2}}\right)$ . Does it exist?
9. Is the function $\int(x)=\left\{\begin{array}{cc} (x-1) \sin \left(\frac{1}{x-1}\right) & x \neq 1 \\ 1 & x=1 \end{array}\right.$ continuous on $\mathbb{R}$ ?
10. For the function $f(x)=x^{3} \ln (x)$ : (a) Where is the tangent line horizontal in $(0, \infty)$ ? (b) Does the tangent line at $(1,0)$ intersect $y=f(x)$ for some $x \in(1, \infty)$ ? (YES / NO)

See Solution

Problem 28943

Evaluate $\lim _{x \rightarrow 6}(f(x)+g(x))$ given $\lim _{x \rightarrow 6} f(x)=4$ and $\lim _{x \rightarrow 6} g(x)=3$ . Limit =

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Problem 28944

Find the limit: $\lim _{x \rightarrow a} h(x)+g(x)$ given $\lim _{x \rightarrow a} h(x)=4$ and $\lim _{x \rightarrow a} g(x)=0$ .

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Problem 28945

Evaluate the integral from 1 to 5 of $x^{2}$ with respect to $x$ .

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Problem 28946

Evaluate the integral from 0 to 1 of the function $x^{2}+4x+3$ .

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Problem 28947

Find the sum of the series: $\sum_{n=0}^{\infty} e^{\frac{1}{n}}$ .

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Problem 28948

Find the limits: 1. $\lim _{x \rightarrow a} h(x)+g(x)$ , 2. $\lim _{x \rightarrow a} h(x)-g(x)$ given limits are 4, 0, and 8.

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Problem 28949

Given limits:
1. $\lim _{x \rightarrow \infty} h(x)+g(x)$
2. $\lim _{x \rightarrow \infty} h(x)-g(x)$
3. $\lim _{x \rightarrow a} h(x) \cdot f(x)$
4. $\lim _{x \rightarrow \infty} \frac{h(x)}{g(x)}$
5. $\lim _{x \rightarrow \infty} \frac{h(x)}{f(x)}$
6. $\lim _{x \rightarrow a} \frac{f(x)}{h(x)}$
7. $\lim _{x \rightarrow a}(g(x))^{2}$
8. $\lim _{x \rightarrow a} \frac{1}{g(x)}$
9. $\lim _{x \rightarrow a} \frac{1}{g(x)-f(x)}$
Find these limits or state DNE if they do not exist.

See Solution

Problem 28950

Find the limit: $\lim _{x \rightarrow 3} \frac{\sqrt{x+6}-3}{x-3}=\square$

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Problem 28951

Given limits: $\lim _{x \rightarrow a} h(x)=4$ , $\lim _{x \rightarrow a} g(x)=0$ , $\lim _{x \rightarrow a} f(x)=8$ . Find the limits: 1. $\lim _{x \rightarrow a} (h(x)+g(x))$ , 2. $\lim _{x \rightarrow a} (h(x)-g(x))$ , 3. $\lim _{x \rightarrow a} (h(x) \cdot f(x))$ , 4. $\lim _{x \rightarrow a} \frac{h(x)}{g(x)}$ , 5. $\lim _{x \rightarrow a} \frac{h(x)}{f(x)}$ .

See Solution

Problem 28952

Find the limit as $x$ approaches 1 for $\frac{x^{3}-x}{x^{2}-1}$ . If it doesn't exist, write "DNE".

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Problem 28953

Find the limit as $x$ approaches 6 for $\frac{x^{2}-3x-18}{x-6}$ . Answer with I, -I, or DNE.

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Problem 28954

Calculate $m=\frac{1}{3}\left[0.5 x^{2}\right]_{-1}^{2}$ .

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Problem 28955

1. Find the limit: $\lim _{x \rightarrow 3^{+}} \frac{x^{2}+2 x-3}{|x+3|}$
2. Find the limit: $\lim _{x \rightarrow 0} \frac{x^{3}}{\sin \left(3 x^{2}\right)}$
3. Find the limit: $\lim _{x \rightarrow 1^{+}} \arctan \left(\frac{x^{2}}{1-x^{2}}\right)$
4. 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 $\mathbb{R}$ ?
5. For $f(x)=x^{3} \ln (x)$ : (a) Where is the tangent line horizontal in $(0, \infty)$ ? (b) Does the tangent line at $(1,0)$ intersect the graph in $(1, \infty)$ ? (YES / NO)

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Problem 28956

Calculate the integral: $\int_{1}^{4}\left(\frac{8}{\sqrt{t}}-12 \sqrt{t^{3}}\right) d t$

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Problem 28957

1. Find function $h$ such that $h \circ f(x) = \frac{1}{1-\cos^{2}(x)} + \sin(x)$ for $f(x)=\sin(x)$ .
2. Does $f(x)=\frac{8}{(x-2)^{4}+1}$ have an inverse for $x \in [2, \infty)$ ?
3. Find the radius of convergence for $\sum_{n=1}^{\infty} 6^{n}(n+\frac{1}{n})x^{n}$ .
4. Does $f(x)=2x-\sqrt{2x}$ for $x \geq \frac{1}{2}$ have an inverse? (YES / NO)
5. For $f(x)=\frac{1}{1-x^{2}}$ , determine if it has a max (YES/NO), min (YES/NO), and if it is increasing (YES/NO).

See Solution

Problem 28958

Evaluate $\lim _{x \rightarrow 8} f(x) \cdot g(x)$ given $\lim _{x \rightarrow 8} f(x)=8$ and $\lim _{x \rightarrow 8} g(x)=3$ . Limit $=$

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Problem 28959

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{2}+2 x}{x}=\square$

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Problem 28960

Evaluate $\lim _{x \rightarrow 6} \frac{f(x)}{g(x)}$ given $\lim _{x \rightarrow 6} f(x)=8$ and $\lim _{x \rightarrow 6} g(x)=7$ . Limit =

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Problem 28961

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}=\square$

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Problem 28962

Find the limit: $\lim _{x \rightarrow 5} \frac{x^{2}-3 x-10}{x^{2}-8 x+15}.$

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Problem 28963

Find the limit: $\lim _{x \rightarrow 0} \frac{\cos 2 x \sin 2 x}{7 x}=\square$

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Problem 28964

Find the limit: $\lim _{x \rightarrow 5} \frac{x^{2}-7 x+10}{x^{2}-25}=\square$

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Problem 28965

Find the limit: $\lim _{x \rightarrow 2} \frac{\sqrt{x+2}-2}{x-2}=\square$

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Problem 28966

Evaluate the limit as $\theta$ approaches 0: $\lim _{\theta \rightarrow 0} \frac{\sin 2 \theta \sin 5 \theta}{\theta^{2}}$

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Problem 28967

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

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Problem 28968

Determine the critical points of $f(x)=\frac{4}{3} x-\tan x$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ . Options include: 1 point at $x=\frac{2}{\sqrt{3}}$ , 1 point at $x=\frac{\pi}{6}$ , 2 points at $x=-\frac{2}{\sqrt{3}}$ and $x=\frac{2}{\sqrt{3}}$ , none, or 2 points at $x=-\frac{\pi}{6}$ and $x=\frac{\pi}{6}$ .

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Problem 28969

Determine if the function $f(x)$ has a jump discontinuity at $x=8$ by finding $\lim _{x \rightarrow 8^{-}} f(x)$ and $\lim _{x \rightarrow 8^{+}} f(x)$ .

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Problem 28970

Berechne die Wendepunkte der Funktion $f(x) = \frac{x}{x^{2}+4}$ .

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Problem 28971

Find the derivative of $h(t)=(t^{2}+9)(t+7)$ at $t=1$ using the Product Rule. Provide a whole number answer. $\left.\frac{d h}{d t}\right|_{t=1}=$

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Problem 28972

Find the second derivative at 1 for the function $f(t)=\frac{15 t}{t+1}$ .

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Problem 28973

Find the first five derivatives of $f(x)=\sin(x)$ , then calculate $f^{(110)}(x)$ and $f^{(9)}(x)$ .

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Problem 28974

Find the limit as $x$ approaches infinity for $\frac{5x + 2 \ln x}{x + 3 \ln x}$ .

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Problem 28975

Find $U_{1}$ in the DE $y^{\prime \prime}-2 y^{\prime}+y=\frac{e^{x}}{x^{2}}$ with $y_{p}=U_{1} e^{x}+U_{2} x e^{x}$ . Options: I. $\frac{e^{x}}{6 x^{2}}$ , II. $-\ln x$ , III. $e^{x}(x+1)$ , IV. $-\frac{1}{x}$ , V. $\frac{x+1}{x}$ .

See Solution

Problem 28976

Find $y^{\prime}$ and $y^{\prime \prime}$ for $y^{5}-\frac{5}{2} x^{2}=3$ . Express $y^{\prime}$ as a function of $x$ and $y$ .

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Problem 28977

Bestimme die Temperaturen von $f(t)$ für $t=0$ und $t=2$ , berechne die prozentuale Abweichung. Zeige, dass $f$ einen Extrempunkt hat.

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Problem 28978

Find the value of $k$ for the point $(4.2, k)$ on the curve $-8x^2 + 5xy + y^3 = -149$ using the tangent line $y = 3x - 13$ .

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Problem 28979

Zwei Läusepopulationen: Eine wächst täglich um 500, die andere um $5\%$ . a) Funktionsgleichungen aufstellen. b) $f^{\prime}(t)=k \cdot f(t)$ prüfen und k bestimmen. c) Funktionstyp für b) untersuchen.

See Solution

Problem 28980

Berechnen Sie $f(0)$ und $f(2)$ für $f(t)=23+20 \cdot t \cdot e^{-\frac{1}{10} t}$ und die prozentuale Abweichung. Zeigen Sie, dass $f$ einen Extrempunkt hat.

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Problem 28981

Berechne die Steigung von $f(x)=4 x^{5}+6 x^{3}+\frac{1}{4} x^{2}$ bei $x_{0}=1$ .

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Problem 28982

Find where the function $f(x)=4 x^{4}-3 x^{3}+5 x-10$ is concave up. Options: A, B, C, D, E.

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Problem 28983

Find the relative extreme point of $f(x)=-3+6x-x^{2}$ and determine if it's a max or min at $(3,6)$ . What is $f''$ ?

See Solution

Problem 28984

Find the derivative: $\frac{d y}{d x} = (2 x^{3} - x^{2} + 3 x + 2)^{5}$ .

See Solution

Problem 28985

Find the $x$ -coordinates of relative extrema for $f(x)=\frac{2}{3} x^{3}-7 x^{2}+24 x-72$ .

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Problem 28986

3 Berechne $f(-1)$ für $f(x)=-x^{4}+2 x^{2}+2 x+\sqrt{7}$ 4 Berechne die Steigung bei $x=2$ für $f(x)=0.25 x^{4}+2 x^{3}+3 x^{2}+7$ 5 Berechne $f'(-3)$ für $f(x)=-\frac{2}{3} x^{3}+\frac{1}{3} x^{3}+2 \sqrt{3}$ 6 Finde die fehlende Koordinate für $f(x)=-\frac{3}{4} x^{8}+2.5 x^{4}-2 x+4$ , $P(1.5 / \quad)$ 7 Berechne $f'(0.2)$ für $f(x)=5 x^{4}-0.2 x^{3}+0.4 x^{2}-7$ 8 Finde die fehlende Koordinate für $f(x)=0.3 x^{5}+\frac{1}{7} x^{3}+2 x$ , $P(-3 / 4 I,)$ 9 Berechne die Steigung bei $x=1.5$ für $f(x)=-\frac{3}{4} x^{4}+\frac{1}{2} x^{4}+x-3$ 10 Berechne $f(-3)$ für $f(x)=-\frac{1}{3} x^{4}-2 x^{3}+\frac{3}{4} x^{2}$ 11 Berechne die Geschwindigkeit bei $t=5$ für $s(t)=t^{4}-0.8 t^{2}-7 t$ 12 Berechne die Strecke bei $t=-2$ für $s(t)=0.5 t^{3}-1.8 t^{2}-7$

See Solution

Problem 28987

Finde die Stammfunktion von $f(x) = \frac{1}{2}x^3 + \frac{3}{2}x^2$ ohne Konstante $C$ .

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Problem 28988

Berechne die Stammfunktion von $g(x)=\frac{1}{3} x^{2}$ .

See Solution

Problem 28989

Zeige, dass $F(x) = e^{x}(x-1)$ die Stammfunktion von $f(x) = x \cdot e^{x}$ ist.

See Solution

Problem 28990

Zeige, dass $F(x)=e^{x}(x-1)$ eine Stammfunktion von $f(x)=x \cdot e^{x}$ ist.

See Solution

Problem 28991

Find the curve where $\frac{\mathrm{d} y}{\mathrm{~d} x}=\sqrt{2 x+1}$ and it passes through $(4,11)$ .

See Solution

Problem 28992

Find the vertical asymptotes of $f(x)=\frac{3x}{(x-3)(x+2)}$ and evaluate $\lim_{x \to -\infty} f(x)$ and $\lim_{x \to \infty} f(x)$ .

See Solution

Problem 28993

Find the area between the curves $y=\frac{1}{x^{2}}$ , $y=-x^{2}$ from $x=1$ to $x=2$ .

See Solution

Problem 28994

What fits in the blank? "If $\lim _{x \rightarrow a} f(x)=L$ , then $f(x)$ is ____ to L." Options: slightly larger, always equal, slightly smaller, never equal.

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Problem 28995

Find $\delta$ values that satisfy $\lim _{x \rightarrow 2} 5 x+2=12$ for $\varepsilon=0.3$ . Options: $0.0036$ , $0.06$ , $0.12$ , $0.02$ .

See Solution

Problem 28996

Complete the sentence: "If $\lim _{x \rightarrow a} f(x) \neq L$ , then there exists $\varepsilon>0$ such that $f(x)$ is at least $\varepsilon$ away from $L$ ."

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Problem 28997

Find $\delta$ values for $\lim _{x \rightarrow 1} 5 x+4=9$ with $\varepsilon=0.3$ . Options: $\delta=0.0036$ , $\delta=0.02$ , $\delta=0.06$ , $\delta=0.12$ .

See Solution

Problem 28998

Find a suitable $\delta$ value for $\lim _{x \rightarrow 1} 5 x+2=7$ with $\varepsilon=0.4$ . Options: $0.16$ , $0.0064$ , $0.08$ , $0.0266666666666667$ .

See Solution

Problem 28999

Find the definite integral for the area between $y=x$ and $y=5x^2-x^3$ . Options are A, B, C, or D.

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Problem 29000

Find a suitable $\delta$ for $\lim _{x \rightarrow 2} 4 x+5=13$ with $\varepsilon=0.4$ : $\delta=0.1$ , $\delta=0.0333$ , $\delta=0.2$ , $\delta=0.01$ .

See Solution

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Calculus

Problem 28901

Find the equation of the tangent line y=−35x+2y=\frac{-3}{5 x+2}y=5x+2−3​ at x=3x=3x=3.

Problem 28902

Find the value(s) of xxx where the function g(x)=x+1xg(x)=\frac{x+1}{\sqrt{x}}g(x)=x​x+1​ has a horizontal tangent.

Problem 28903

Differentiate the function f(x)=cos⁡xsin⁡xf(x)=\sqrt{\cos x \sin x}f(x)=cosxsinx​.

Problem 28904

How long until 80 g80 \mathrm{~g}80 g of carbon-14 remains from 100 g100 \mathrm{~g}100 g? Also, find the rate of change at 100 years.

Problem 28905

Carbon-14 has a half-life of 5730 years. A plant had 100 g100 \mathrm{~g}100 g at death. Find: a) remaining after 5730 years, b) equation for remaining after ttt years, c) years for 80 g80 \mathrm{~g}80 g left, d) rate of change at 100 years.

Problem 28906

Find the absolute extrema of f(x)=(x−3)exf(x)=(x-3)e^{x}f(x)=(x−3)ex for −1≤x≤3-1 \leq x \leq 3−1≤x≤3.

Problem 28907

A cone-shaped tank is 18 ft tall and 12 ft wide. If drained at 4ft3/min4 \mathrm{ft}^3/\mathrm{min}4ft3/min, how fast drops water level at 6 ft?

Problem 28908

Find bbb such that the slope of the tangent to f(x)=(3x2+1)(2x2+b)f(x)=(3 x^{2}+1)(2 x^{2}+b)f(x)=(3x2+1)(2x2+b) at x=−1x=-1x=−1 is -16.

Problem 28909

How fast is the diameter of a balloon increasing at a radius of 5 cm if air is pumped in at 100 cm3/sec100 \mathrm{~cm}^{3} / \mathrm{sec}100 cm3/sec?

Problem 28910

Find the limit: lim⁡x→06x+2ln⁡xx+sin⁡x\lim _{x \rightarrow 0} \frac{6 x+2 \ln x}{x+\sin x}limx→0​x+sinx6x+2lnx​.

Problem 28911

Find the instantaneous rate of change of www for w=1z+z2w=\frac{1}{z}+\frac{z}{2}w=z1​+2z​. Choices: (A) 32\frac{3}{2}23​ (B) -2 (C) z2−22z2\frac{z^{2}-2}{2 z^{2}}2z2z2−2​ (D) −1z2-\frac{1}{z^{2}}−z21​ (E) None.

Problem 28912

Sia fff una funzione integrabile con ∫01f(x)dx=3\int_{0}^{1} f(x) d x = 3∫01​f(x)dx=3. Quale affermazione è vera tra (a), (b), (c), (d) o (e)?

Problem 28913

Sia fff una funzione integrabile con ∫01f(x)dx=3\int_{0}^{1} f(x) d x = 3∫01​f(x)dx=3. Quale delle seguenti affermazioni è vera? (a), (b), (c), (d) o (e)?

Problem 28914

Trova l'espressione esplicita di f′(x)f^{\prime}(x)f′(x) per la funzione f(x)=∫3x27xe−t2dtf(x) = \int_{3 x^{2}}^{7 x} e^{-t^{2}} \mathrm{d} tf(x)=∫3x27x​e−t2dt, con x∈[0,1]x \in [0,1]x∈[0,1].

Problem 28915

Find the limit: lim⁡x→0−(1+3x)1/x\lim _{x \rightarrow 0^{-}}\left(1+3^{x}\right)^{1 / x}limx→0−​(1+3x)1/x.

Problem 28916

Approximate the rate of change for f(x)=x3−5f(x)=x^{3}-5f(x)=x3−5 at x=1x=1x=1 using h=0.1,h=0.01,h=0.001h=0.1, h=0.01, h=0.001h=0.1,h=0.01,h=0.001.

Problem 28917

Find the normal line equation to y=x−1xy=x-\frac{1}{x}y=x−x1​ at point (1,0)(1,0)(1,0) in the form y=mx+by=mx+by=mx+b.

Problem 28918

Find a real number between n=−5n = -5n=−5 and b=−4b = -4b=−4 for the function f(x)=x3+3x2−9x−13f(x) = x^{3} + 3x^{2} - 9x - 13f(x)=x3+3x2−9x−13 using the Intermediate Value Theorem.

Problem 28919

Calculate the volume of the solid using the formula π[x22−x99]01\pi\left[\frac{x^{2}}{2}-\frac{x^{9}}{9}\right]_{0}^{1}π[2x2​−9x9​]01​.

Problem 28920

Find the rate of change of revenue when producing 4000 products, given R(x)=x(−0.25x2+7.5x+18.75)R(x)=x(-0.25 x^{2}+7.5 x+18.75)R(x)=x(−0.25x2+7.5x+18.75).

Problem 28921

Problem 28922

Find the tangent line equation for y=sin⁡(2x+π3)y=\sin \left(2 x+\frac{\pi}{3}\right)y=sin(2x+3π​) at x=π3x=\frac{\pi}{3}x=3π​.

Problem 28923

Find the melting rate of a snowball with volume S=43πr3S=\frac{4}{3} \pi r^{3}S=34​πr3, where r=1(t+1)2−117r=\frac{1}{(t+1)^{2}}-\frac{1}{17}r=(t+1)21​−171​.

Problem 28924

Find the average velocity of a ball rolling down a ramp given s(t)=10t2s(t)=10 t^{2}s(t)=10t2 from t1=3t_{1}=3t1​=3 to t2=4t_{2}=4t2​=4.

Problem 28925

Calculate the limit: lim⁡x→1+arctan⁡(x21−x2)\lim _{x \rightarrow 1^{+}} \arctan \left(\frac{x^{2}}{1-x^{2}}\right)limx→1+​arctan(1−x2x2​). Does it exist?

Problem 28926

Problem 28927

Find the rate of change of depth D(t)=7sin⁡(π6t−7π6)+4D(t)=7 \sin \left(\frac{\pi}{6} t-\frac{7 \pi}{6}\right)+4D(t)=7sin(6π​t−67π​)+4 at t=6t=6t=6 hours. Round to one decimal place.

Problem 28928

Find the general antiderivative of f(x)=3x2−4x+8x2f(x)=\frac{3 x^{2}-4 x+8}{x^{2}}f(x)=x23x2−4x+8​ for x>0x>0x>0.

Problem 28929

Given the function y=cot⁡(csc⁡(x))y=\cot(\csc(x))y=cot(csc(x)), decompose it as y=f(u)y=f(u)y=f(u) and u=g(x)u=g(x)u=g(x). Then find dydx\frac{d y}{d x}dxdy​.

Problem 28930

Estimate the area under the curve y=f(x)y=f(x)y=f(x) from x=0x=0x=0 to x=10x=10x=10 using 5 rectangles. Round to 1 decimal place.

Problem 28931

Which properties are needed to evaluate ∫14(2x+3)dx\int_{1}^{4}(2 x+3) d x∫14​(2x+3)dx? Options: a) P only b) Q only c) Both P and Q d) Neither P nor Q

Problem 28932

Berechnen Sie die Ableitung von f(x)=(x+3)2(x+2)3f(x)=\frac{(x+3)^{2}}{(x+2)^{3}}f(x)=(x+2)3(x+3)2​.

Problem 28933

Evaluate the limit as xxx approaches 4 for the expression x4−5x+6x^{4}-5x+6x4−5x+6. What is the limit?

Problem 28934

Find the value of ∫02/3(4−x2)dx\int_{0}^{2/3}(4-x^{2})dx∫02/3​(4−x2)dx given that ∫01(4−x2)dx=113\int_{0}^{1}(4-x^{2})dx=\frac{11}{3}∫01​(4−x2)dx=311​. Options: a) 169\frac{16}{9}916​ b) 229\frac{22}{9}922​ c) 103\frac{10}{3}310​ d) less than 113\frac{11}{3}311​.

Problem 28935

Find the limit as x approaches 2 from the right: lim⁡x→2+−1x−2=□\lim _{x \rightarrow 2^{+}}-\frac{1}{x-2}=\squarex→2+lim​−x−21​=□

Problem 28936

Which properties are needed to evaluate ∫03∣x−1∣dx\int_{0}^{3}|x-1| d x∫03​∣x−1∣dx? a) P only b) Q only c) Both P and Q d) Neither P nor Q

Problem 28937

Estimate the limit: lim⁡θ→0sin⁡(7θ)θ=□\lim _{\theta \rightarrow 0} \frac{\sin (7 \theta)}{\theta}=\squareθ→0lim​θsin(7θ)​=□, with θ\thetaθ in radians.

Problem 28938

Find bounds for I=∫−30(x2+2x) dxI = \int_{-3}^{0} (x^{2} + 2x) \, dxI=∫−30​(x2+2x)dx. Options: a) 1≤I≤31 \leq I \leq 31≤I≤3 b) ◯\bigcirc◯ c) −3≤I≤−5-3 \leq I \leq -5−3≤I≤−5 d) −3≤I≤9-3 \leq I \leq 9−3≤I≤9.

Problem 28939

Fill in the table to estimate lim⁡x→3f(x)\lim_{x \to 3} f(x)limx→3​f(x) for f(x)=x3−27x2−9f(x) = \frac{x^3 - 27}{x^2 - 9}f(x)=x2−9x3−27​.

Problem 28940

Problem 28941

Problem 28942

Find the equation of the tangent line $y=\frac{-3}{5 x+2}$ at $x=3$ .

Find the value(s) of $x$ where the function $g(x)=\frac{x+1}{\sqrt{x}}$ has a horizontal tangent.

Differentiate the function $f(x)=\sqrt{\cos x \sin x}$ .

How long until $80 \mathrm{~g}$ of carbon-14 remains from $100 \mathrm{~g}$ ? Also, find the rate of change at 100 years.

Carbon-14 has a half-life of 5730 years. A plant had $100 \mathrm{~g}$ at death. Find: a) remaining after 5730 years, b) equation for remaining after $t$ years, c) years for $80 \mathrm{~g}$ left, d) rate of change at 100 years.

Find the absolute extrema of $f(x)=(x-3)e^{x}$ for $-1 \leq x \leq 3$ .

A cone-shaped tank is 18 ft tall and 12 ft wide. If drained at $4 \mathrm{ft}^3/\mathrm{min}$ , how fast drops water level at 6 ft?

Find $b$ such that the slope of the tangent to $f(x)=(3 x^{2}+1)(2 x^{2}+b)$ at $x=-1$ is -16.

How fast is the diameter of a balloon increasing at a radius of 5 cm if air is pumped in at $100 \mathrm{~cm}^{3} / \mathrm{sec}$ ?

Find the limit: $\lim _{x \rightarrow 0} \frac{6 x+2 \ln x}{x+\sin x}$ .

Find the instantaneous rate of change of $w$ for $w=\frac{1}{z}+\frac{z}{2}$ . Choices: (A) $\frac{3}{2}$ (B) -2 (C) $\frac{z^{2}-2}{2 z^{2}}$ (D) $-\frac{1}{z^{2}}$ (E) None.

Sia $f$ una funzione integrabile con $\int_{0}^{1} f(x) d x = 3$ . Quale affermazione è vera tra (a), (b), (c), (d) o (e)?

Sia $f$ una funzione integrabile con $\int_{0}^{1} f(x) d x = 3$ . Quale delle seguenti affermazioni è vera? (a), (b), (c), (d) o (e)?

Trova l'espressione esplicita di $f^{\prime}(x)$ per la funzione $f(x) = \int_{3 x^{2}}^{7 x} e^{-t^{2}} \mathrm{d} t$ , con $x \in [0,1]$ .

Find the limit: $\lim _{x \rightarrow 0^{-}}\left(1+3^{x}\right)^{1 / x}$ .

Approximate the rate of change for $f(x)=x^{3}-5$ at $x=1$ using $h=0.1, h=0.01, h=0.001$ .

Find the normal line equation to $y=x-\frac{1}{x}$ at point $(1,0)$ in the form $y=mx+b$ .

Find a real number between $n = -5$ and $b = -4$ for the function $f(x) = x^{3} + 3x^{2} - 9x - 13$ using the Intermediate Value Theorem.

Calculate the volume of the solid using the formula $\pi\left[\frac{x^{2}}{2}-\frac{x^{9}}{9}\right]_{0}^{1}$ .

Find the rate of change of revenue when producing 4000 products, given $R(x)=x(-0.25 x^{2}+7.5 x+18.75)$ .

Find the tangent line equation for $y=\sin \left(2 x+\frac{\pi}{3}\right)$ at $x=\frac{\pi}{3}$ .

Find the melting rate of a snowball with volume $S=\frac{4}{3} \pi r^{3}$ , where $r=\frac{1}{(t+1)^{2}}-\frac{1}{17}$ .

Find the average velocity of a ball rolling down a ramp given $s(t)=10 t^{2}$ from $t_{1}=3$ to $t_{2}=4$ .

Calculate the limit: $\lim _{x \rightarrow 1^{+}} \arctan \left(\frac{x^{2}}{1-x^{2}}\right)$ . Does it exist?

Find the rate of change of depth $D(t)=7 \sin \left(\frac{\pi}{6} t-\frac{7 \pi}{6}\right)+4$ at $t=6$ hours. Round to one decimal place.

Find the general antiderivative of $f(x)=\frac{3 x^{2}-4 x+8}{x^{2}}$ for $x>0$ .

Given the function $y=\cot(\csc(x))$ , decompose it as $y=f(u)$ and $u=g(x)$ . Then find $\frac{d y}{d x}$ .

Estimate the area under the curve $y=f(x)$ from $x=0$ to $x=10$ using 5 rectangles. Round to 1 decimal place.

Which properties are needed to evaluate $\int_{1}^{4}(2 x+3) d x$ ? Options: a) P only b) Q only c) Both P and Q d) Neither P nor Q

Berechnen Sie die Ableitung von $f(x)=\frac{(x+3)^{2}}{(x+2)^{3}}$ .

Evaluate the limit as $x$ approaches 4 for the expression $x^{4}-5x+6$ . What is the limit?

Find the value of $\int_{0}^{2/3}(4-x^{2})dx$ given that $\int_{0}^{1}(4-x^{2})dx=\frac{11}{3}$ . Options: a) $\frac{16}{9}$ b) $\frac{22}{9}$ c) $\frac{10}{3}$ d) less than $\frac{11}{3}$ .

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

Which properties are needed to evaluate $\int_{0}^{3}|x-1| d x$ ? a) P only b) Q only c) Both P and Q d) Neither P nor Q

Estimate the limit: $\lim _{\theta \rightarrow 0} \frac{\sin (7 \theta)}{\theta}=\square$ , with $\theta$ in radians.

Find bounds for $I = \int_{-3}^{0} (x^{2} + 2x) \, dx$ . Options: a) $1 \leq I \leq 3$ b) $\bigcirc$ c) $-3 \leq I \leq -5$ d) $-3 \leq I \leq 9$ .

Fill in the table to estimate $\lim_{x \to 3} f(x)$ for $f(x) = \frac{x^3 - 27}{x^2 - 9}$ .

Evaluate $\lim _{x \rightarrow 6}(f(x)+g(x))$ given $\lim _{x \rightarrow 6} f(x)=4$ and $\lim _{x \rightarrow 6} g(x)=3$ . Limit =

Find the limit: $\lim _{x \rightarrow a} h(x)+g(x)$ given $\lim _{x \rightarrow a} h(x)=4$ and $\lim _{x \rightarrow a} g(x)=0$ .

Evaluate the integral from 1 to 5 of $x^{2}$ with respect to $x$ .

Evaluate the integral from 0 to 1 of the function $x^{2}+4x+3$ .

Find the sum of the series: $\sum_{n=0}^{\infty} e^{\frac{1}{n}}$ .

Find the limits: 1. $\lim _{x \rightarrow a} h(x)+g(x)$ , 2. $\lim _{x \rightarrow a} h(x)-g(x)$ given limits are 4, 0, and 8.

Find the limit: $\lim _{x \rightarrow 3} \frac{\sqrt{x+6}-3}{x-3}=\square$

Find the limit as $x$ approaches 1 for $\frac{x^{3}-x}{x^{2}-1}$ . If it doesn't exist, write "DNE".

Find the limit as $x$ approaches 6 for $\frac{x^{2}-3x-18}{x-6}$ . Answer with I, -I, or DNE.

Calculate $m=\frac{1}{3}\left[0.5 x^{2}\right]_{-1}^{2}$ .

Calculate the integral: $\int_{1}^{4}\left(\frac{8}{\sqrt{t}}-12 \sqrt{t^{3}}\right) d t$

Evaluate $\lim _{x \rightarrow 8} f(x) \cdot g(x)$ given $\lim _{x \rightarrow 8} f(x)=8$ and $\lim _{x \rightarrow 8} g(x)=3$ . Limit $=$

Find the limit: $\lim _{x \rightarrow 0} \frac{x^{2}+2 x}{x}=\square$

Evaluate $\lim _{x \rightarrow 6} \frac{f(x)}{g(x)}$ given $\lim _{x \rightarrow 6} f(x)=8$ and $\lim _{x \rightarrow 6} g(x)=7$ . Limit =

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}=\square$

Find the limit: $\lim _{x \rightarrow 5} \frac{x^{2}-3 x-10}{x^{2}-8 x+15}.$

Find the limit: $\lim _{x \rightarrow 0} \frac{\cos 2 x \sin 2 x}{7 x}=\square$

Find the limit: $\lim _{x \rightarrow 5} \frac{x^{2}-7 x+10}{x^{2}-25}=\square$

Find the limit: $\lim _{x \rightarrow 2} \frac{\sqrt{x+2}-2}{x-2}=\square$

Evaluate the limit as $\theta$ approaches 0: $\lim _{\theta \rightarrow 0} \frac{\sin 2 \theta \sin 5 \theta}{\theta^{2}}$

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

Determine if the function $f(x)$ has a jump discontinuity at $x=8$ by finding $\lim _{x \rightarrow 8^{-}} f(x)$ and $\lim _{x \rightarrow 8^{+}} f(x)$ .

Berechne die Wendepunkte der Funktion $f(x) = \frac{x}{x^{2}+4}$ .

Find the derivative of $h(t)=(t^{2}+9)(t+7)$ at $t=1$ using the Product Rule. Provide a whole number answer. $\left.\frac{d h}{d t}\right|_{t=1}=$

Find the second derivative at 1 for the function $f(t)=\frac{15 t}{t+1}$ .

Find the first five derivatives of $f(x)=\sin(x)$ , then calculate $f^{(110)}(x)$ and $f^{(9)}(x)$ .

Find the limit as $x$ approaches infinity for $\frac{5x + 2 \ln x}{x + 3 \ln x}$ .