Browse Calculus problems Studdy Solved!

Problem 2301

Find the slope of the tangent line to $f(x)=3 x^{2}$ at $x=4$ using the limit definition of the derivative.

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Problem 2302

For the function $F(x)=-x^{4}+4 x^{2}+192$ , find if $F$ is even/odd, a second local max, and area from $x=-4$ to $x=0$ .

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Problem 2303

A ball is dropped from a 94 ft building. Find the time to fall half the distance and to ground level: (a) $t=$ sec, (b) $t=$ sec.

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Problem 2304

A ball drops from a 94 ft building. (Round to 3 decimal places.) (a) Time to fall half distance? $t=1.8$ (b) Time to reach ground? $t=2.4$

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Problem 2305

Find all numbers $x$ in $(-1,2)$ where $f'(x)=3x^2+4x-1$ equals the average rate of change of $f(x)=x^3+2x^2-x+6$ .

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Problem 2306

Show that if $y=\frac{5+\sqrt{x}}{\sqrt{x}}$ , then $6 \frac{d y}{d x}+4 x \frac{d^{2} y}{d x^{2}}=0$ .

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Problem 2307

Differentiate the following functions:
1. Constants and powers: (a) $\frac{d}{d x}(7)$ (b) $\frac{d}{d x}(\ln 3)$ (c) $\frac{d}{d x}(x^{3})$ (d) $\frac{d}{d x}(\frac{1}{x^{8}})$ (e) $\frac{d}{d x}(4x^{5})$
2. Find $\frac{d y}{d x}$ for: (a) $y=(x+3)(2x^{2}+4)$ (b) $y=\frac{3x^{2}-7x-4}{x^{3}}$ (c) $y=(4x-1)^{2}(x^{2}-3)$ (product rule) (d) $y=\frac{x-2}{x+4}$ (quotient rule) (e) $y=\frac{e^{3x}}{x}$ (f) $y=\log_{2} x$ (g) $y=\ln(4+9x^{2})$
3. For $y=\sqrt{3x-7}$ , find: (a) $\frac{d y}{d x}$ (b) $\frac{d^{2} y}{d x^{2}}$

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Problem 2308

Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for these functions: (a) $f(x, y)=(x^{2}-1)(y+2)$ (b) $f(x, y)=e^{x+y+1}$ (c) $f(x, y)=\ln (x+y)$

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Problem 2309

Find the limit as $x$ approaches infinity for $\frac{x^{2}}{2^{x}}$ . Evaluate $f(x)=\frac{x^{2}}{2^{x}}$ for $x=0,1,\ldots,100$ .

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Problem 2310

Find the limit as $x$ approaches infinity for $\sqrt{x^{2}+6}$ .

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Problem 2311

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ .

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Problem 2312

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)$ . If it doesn't exist, enter DNE.

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Problem 2313

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[S]}{0.021+[S]}$ . What does it indicate?

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Problem 2314

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ . What does it signify?

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Problem 2315

Bertalanffy growth function:
1. Find $\lim_{t \rightarrow \infty} L(t)$ and interpret.
2. With $L_{0}=1 \mathrm{~cm}$ , $L_{T}=38 \mathrm{~cm}$ , graph $L(t)$ for various $k$ .

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Problem 2316

Find $\lim_{t \rightarrow \infty} B(t)$ for the biomass model $B(t)=\frac{9 \times 10^{7}}{1+3 e^{-0.72 t}}$ . What does this limit mean?

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Problem 2317

Evaluate $f(x)=\frac{x^{2}-3 x}{x^{2}-x-6}$ at $x=3.5,3.1,...,2.999$ and find $\lim_{x \to 3} f(x) \approx 0.600000$ .

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Problem 2318

Evaluate $f(x)=\frac{x^{2}-8 x}{x^{2}-7 x-8}$ at $x=0,-0.5,-0.9,-0.95,-0.99,-0.999,-2,-1.5,-1.1,-1.01,-1.001$ . Find $\lim_{x \to -1} f(x)$ to six decimal places.

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Problem 2319

Find the tangent line equation for the polar curve $r=3 \cos \theta$ at $\theta=\frac{\pi}{3}$ .

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Problem 2320

Find the slope of the tangent line for $r=3 \cos \theta$ at $\theta=\frac{\pi}{3}$ .

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Problem 2321

Find the average rate of change of $f(x)=-2 x^{3}+4 x+7$ on $[-1,3]$ . Is it even, odd, or neither?

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Problem 2322

Find $a$ so that $\lim_{{x \rightarrow \infty}} \frac{ax^{3}+4x^{2}-x-1}{3x^{3}+x^{2}-9} = -5$ .

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Problem 2323

An automobile accelerated from rest to $345.6 \mathrm{~m/s}$ in $3.07 \mathrm{~s}$ . What distance did it travel?

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Problem 2324

How long does it take for a football to fall $1719 \mathrm{~m}$ on Mars? $(\mathrm{g}=3.72 \mathrm{~m}/\mathrm{s}^2)$ ?

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Problem 2325

11. Calculate $\int_{0}^{10}(8-x) d x$ in two ways: a. As the limit $\lim _{N \rightarrow \infty} R_{N}$ b. By sketching the relevant signed area and using geometry

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Problem 2326

Find the volume $V$ of the solid obtained by rotating the region bounded by the given curves about the specified line., $y=e^{x}, \quad y=0, \quad x=-2, x=2 ;$ about the $x$ -axis $V=$

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Problem 2327

7 What is the minimum period of oscillation of a uniform rod of length 0.960 m with a moveable pivot point? (That is, you are free to place the pivot wherever you like such that the period is a minimum.)

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Problem 2328

aluate. $\int_{a}^{b} e^{4 t} d t$

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Problem 2329

Evaluate. $\int_{1}^{e^{3}}\left(3 x+\frac{1}{x}\right) d x$

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Problem 2330

A company finds that the marginal profit, in dollars per foot, from drilling a well that is $x$ feet deep is given by $P^{\prime}(x)=2 \sqrt[3]{x}$ . Find the profit when a well 200 ft deep is drilled.
Set up the integral for the total profit for a well that is 200 feet deep. $P(200)=\int_{0}^{\square}(\square) \mathrm{dx}$

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Problem 2331

Find the general antiderivative of $f(x)=-6 x^{5}+\frac{5}{x}+\frac{5}{x^{4}}+5 \sqrt{x}$

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Problem 2332

Find the volume $V$ of the solid obtained by rotating the region bounded by the given curves about the specified line. $y=x^{3}, \quad y=1, \quad x=2 ;$ about $y=-4$ $V=$

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Problem 2333

Problem \#2 (30 points) In this circuit, the switch has been open for a long time and at $\mathrm{t}=0$ it closes. It is given that at $\mathrm{t}=0$ - there is no energy stored in the inductor or the capacitor. a) Show that when the switch is closed, the current $I(\mathrm{~s})$ in the s-domain is given by: b) Perform an inverse Laplace transform and determine the current $i(\mathrm{t})$ in the time-domain for $\mathrm{t} \geq 0$ . $I(s)=\frac{5}{s^{2}+s 200+10^{5}}$ c) Plot the current $i(\mathrm{t})$ from $\mathrm{t}=0$ until a time at which the current reaches a steady state value. d) On the plot of part (c), indicate the transient and steady state periods of time. e) Using the known behavior of the inductor and capacitor explain the steady state value of the current. a)

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Problem 2334

7. Let $E=\{(x, y, z) \mid x+y+z \leq 1, x \geq 0, y \geq 0, z \geq 0\}$ . Show that $\iiint_{E} e^{z} d V=e-\frac{5}{2}$

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Problem 2335

a. Find the slant asymptote of the graph of the rational function. b. Follow the seven-step strategy and use the slant asymptote to graph the rational function. $f(x)=\frac{x^{3}+64}{x^{2}+5 x}$ a. Select the correct choice below and, if necessary, fill in the answer box to complete the choice. A. The equation of the slant asymptote is $\square$ (Type an equation.) B. There is no slant asymptote.

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Problem 2336

Find the area under the curve $y=\frac{1}{x}$ over the interval $[1,6]$ . $\square$ Find the average value of $y=\frac{1}{x}$ over the interval $[1,6]$ . $\square$

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Problem 2337

Determine where $f^{\prime}(x)=0$ . Use the Second Derivative Test to determine the local maxima and local minima of each function. Give the coordinates of the points. $f(x)=x+\frac{1}{x}$
Sorry, that's incorrect. Try again?
Local Maximum = $\square$
Local Minimum = $\square$

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Problem 2338

Question
Logistic Curve The sales of a new stereo system over a period of time are expected to follow the logistic curve $f(x)=\frac{7000}{1+25 e^{-x}}$ where $x$ is measured in years. Determine the year in which the sales rate is a maximum. Truncate the answer to the integer.

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Problem 2339

Question Watch Video
If $-2=x y+5 y+3 x^{3}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ .
Answer Attempt 1 out of 2

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Problem 2340

14. Radius The area $A$ of a circle is decreasing at a rate of 2 square feet per minute. Find the rate of change of the radius when $A=24$ square feet.

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Problem 2341

Question Find the absolute maximum and absolute minimum of the function on the given interval. $f(x)=x^{4 / 3}-16 x^{1 / 3} \text { on }[-1,8]$
Round answers to 3 decimal places.

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Problem 2342

18. Surface Area All edges of a cube are expanding at rate of 6 centimeters per second. How fast is the surface area changing when each edge is (a) 2 centimeters and (b) 10 centimeters?

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Problem 2343

In Exercises $1,2,3,4,5,6,7,8,9$ , and 10, (a) find the intervals where the function $f$ is increasing and where it is decreasing, (b) find the relative extrema of $f$ , (c) find the intervals where the graph of $f$ is concave upward and wnere it is concave downward, and (d) find the inflection points, if any, of $f$ .
1. $f(x)=\frac{1}{3} x^{3}-x^{2}+x-6$

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Problem 2344

20,100 is invester at $3.50^{\circ}$ intrest compounded continously. how lony will it tuke for the investment to double.

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Problem 2345

ii. $\mathcal{L}\left\{\cosh 2 t e^{4 t}\right\} \quad$ ; First Shift Theorem ; Teorem Anjakan Pertama

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Problem 2346

$\sin(x+y) = 3x$ Find $\frac{dy}{dx}$ .

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Problem 2347

Question The twice-differentiable function $f$ is shown below on the domain $(-9,9)$ . The function $f$ has points of inflection at $x=-7.6$ , $x=-1.3, x=2.4$ , shown with small green circles on the graph. Determine what could be said about the values of $f(3), f^{\prime}(3)$ , and $f^{\prime \prime}(3)$ .
Answer Attempt 1 out of 2 f(3) $\qquad$ and $f^{\prime}(3)$ $\square$ and $f^{\prime \prime}(3)$ $\square$ Submit Answer

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Problem 2348

$\sqrt{y} - \ln y = x + 8$
Find $\frac{dy}{du}$ .

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Problem 2349

$\begin{array}{l}\text { Find } \frac{d y}{d t} \quad y=(t \sin t)^{6} \\ \frac{d y}{d t}=\text { ? }\end{array}$

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Problem 2350

Sketch the graph of the following function. Indicate where the function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. $f(x)=\frac{8 x-9}{x}$
On what interval(s) is fincreasing and on what interval(s) is decreasing? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function is increasing on $\square$ The function is never decreasing. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. The function is decreasing on $\square$ - The function is never increasing. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) C. The function is increasing on $\square$ and decreasing on $\square$ 1. (Simplify your answers. Type your answers in interval notation. Type exact answers, using radicals as needed. Use a comma to separate answers as needed.) D. The function is never increasing or decreasing.

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Problem 2351

The radioactive isotope ${ }^{226} \mathrm{Ra}$ has a half-life of approximately 1599 years. Consider a lab that currently has 45 g of ${ }^{226} \mathrm{Ra}$ . (a.) How much of the isotope remains after 1200 years? Round your answer to three decimal places.

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Problem 2352

1. $\left(6402 \# 3\right.$ p. 303) A particle is moving on the $x$ -axis with velocity given by $v(l)=-3 \iota^{2}+18 \iota-15$ . Find both the displacement (net change of position) and the total distance travelled for this particle during the time interval $0 \leq \iota \leq 8$ .

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Problem 2353

9. (6405 \#3 p. 320) Let $R$ denote the region bounded by the curve $y=\frac{1}{x^{2}+1}$ , the $y$ -axis, the $x$ -axis, and the line $x=4$ . Set $S$ denote the solid generated when the region $R$ is revolved about the $y$ -axis. Find the volume of $S$ .

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Problem 2354

For the following function $f$ , find the antiderivative $F$ that satisfies the given condition. $f(x)=6 \sqrt{x}+4 ; F(1)=4$
The antiderivative that satisfies the given condition is $F(x)=4 x \sqrt{x}+4 x+c$ .

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Problem 2355

Compute the integral $\int \frac{e^{x}}{\sin ^{2}\left(e^{x}\right)} d x=$ $\square$ $\text { 마온 }+C$ where $C$ is the constant of integration. Do not include the constant of integration in yo

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Problem 2356

uences and Series ee Dimensions tor Functions Ex 16.8.15 Find the maximum and minimum values of tial Differentiation A tions of Several Variables f(x, y) = = xy + 9 - x² - y² ts and Continuity al Differentiation when x² + y² ≤ 9. (answer) Chain Rule

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Problem 2357

Question 8 0/1 pt 5 8 Details
Find the inflection point(s) for the function shown below. If there is more than one, be sure to separate them by using a comma. If there is not an inflection point, type DNE in the answer box. If necessary, round all numbers to two decimal places. $f(x)=-4 x^{3}+36 x^{2}+324 x-8$ $\square$ Question Help: Message instructor Submit Question Jump to Answer

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Problem 2358

Find the arc length of the following curve on the given interval. $y=\frac{1}{3} x^{3 / 2} \text { on }[0,54]$

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Problem 2359

MATH102 (Calculus II) First exam - Page 2 of 4 April 25, 2024
1. $\left(21 / 2\right.$ points) The sum of the series $\sum_{k=0}^{\infty} \frac{5^{k} 3^{-k}}{2^{k}}$ is A. 6 B. 5 C. -6 D. -5
2. ( $21 / 2$ points) One of the following values of $p$ makes the series $\sum_{k=0}^{\infty} \frac{p^{k} k^{2}}{k^{3}+1}$ conditionally convergent. A. -1 B. 1 C. 2 D. 3
3. (2 $1 / 2$ points) The series $\sum_{k=0}^{\infty}(-1)^{k} \frac{3}{k^{2}+1}$ is A. conditionally convergent. B. divergent. C. absolutely convergent.
4. ( $21 / 2$ points) Which of the following series is convergent? A. $\sum_{k=1}^{\infty} \frac{k}{k^{2}+4}$ B. $\sum_{k=1}^{\infty} \frac{1}{k+3}$ $\begin{array}{l} \frac{k}{k^{2}+4}<\frac{n}{n^{2}}=\frac{1}{n} \times \frac{n}{n^{2}+4}>\frac{n}{2 n^{1}}=\frac{1}{2 n} \mathrm{dim} \\ \frac{1}{k+3}<\frac{1}{k+3}>\frac{1}{2 k} \text { Jir } \end{array}$ C. $\sum_{k=1}^{\infty} \frac{1}{k\left(1+\ln ^{2}(k)\right)}$ D. $\sum_{k=1}^{\infty}(-1)^{k} \frac{4 k+3}{3 k-5}$
5. ( $21 / 2$ points) One of the following values of $p$ makes the series $\sum_{n=1}^{\infty} \frac{n!}{p^{n}((n+1)!+1)}$ convergent A. 2 B. 0.2 C. 0.5 D. 1 $\begin{aligned} \lim & \frac{(n+1)!}{P^{n+1}((n+1+1)!+1)} \cdot \frac{P^{n}((n+1)!+1)}{n!} \\ & =\frac{(n+1) n!}{P^{2} \cdot P((n+2)(n+1)!+1)} P^{D^{2}((n+1)!+1)}=\ln \frac{(n+1)((n+1)!+1}{P((n+2)(n+1)!} \end{aligned}$ $\begin{array}{l} \lim \frac{(n+1) \cdot((n+1)!+1)}{P((n+2)!+1)}=\lim \frac{(n+1)((n+1)!+1)(n+1)!+(n}{P(n+2)!+P} \\ \left|\frac{1}{P}\right| \lim \frac{(n+1)((n+1)!+1)}{(n+2)!+1)} \quad P((n+2)!+1) \quad P(n+2)!+P \end{array}$

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Problem 2360

Find the area of the surface generated when the given curve is revolved about the given axis. $y=22 \sqrt{x}$ , for $9 \leq x \leq 17$ ; about the $x$ -axis

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Problem 2361

Find the area of the surface generated when the given curve is revolved about the given axis. $y=(8 x)^{\frac{1}{3}}$ , for $0 \leq x \leq 8$ ; about the $y$ -axis

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Problem 2362

Consider the function $f(x)=2 x^{3}+21 x^{2}-48 x+11, \quad-8 \leq x \leq 2$ . Use the derivatives to algebraically answer the question:
This function has an absolutelminimum value equal to $\square$ and an absolute maximum value equal to $\square$ Question Help: Message instructor Submit Question Jump to Answer

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Problem 2363

17. The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied.

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Problem 2364

You go to the doctor and he gives you 17 milligrams of radioactive dye which decays exponentially. After 24 minutes, 5.75 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm if more than 2 milligrams of the dye are in your system, how long will you have to remain at the doctor's office, assuming you were given the dye as soon as you arrived? Give your answer to the nearest minute.
You will spend $\square$ minutes at the doctor's office.

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Problem 2365

$\int_{0}^{1} \frac{3 x}{4 x^{2}-1} d x$

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Problem 2366

Find the derivative of the function $y = 2^x - \sqrt[3]{x^2} + 4 \arccos x$ .

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Problem 2367

10) Find approximate expressions for $1-\cos 4 \theta$ and $\tan ^{2} 2 \theta$ when $\theta$ is small enough for both $4 \theta$ and $2 \theta$ to be considered small and hence find $\lim _{\theta \rightarrow 0} \frac{1-\cos 4 \theta}{\tan ^{2} 2 \theta}$ .

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Problem 2368

$\int \frac{1}{16-x^{2}} d x$

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Problem 2369

7. Given that $y=\frac{x^{3}}{(1-2 x)}$ , then $\frac{1}{y} \frac{\mathrm{~d} y}{\mathrm{~d} x}=$ (Hint: take logarithms of both sides and then (a) $\frac{3}{x}-\frac{1}{1-2 x}$ (b) $\frac{3}{x}+\frac{2}{1-2 x}$ (c) $\frac{3}{x}-\frac{2}{1-2 x}$ (d) $\frac{3}{x}+\frac{1}{1-2 x}$

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Problem 2370

Exercies find the second derivative of $y=f(x)$ at $x=0$ where $y$ is determined by $y^{5}+2 y-x-3 x^{7}=0$

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Problem 2371

Укажіть похідну функції $f(x)=-\operatorname{ctg} x+2 x^{5}$

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Problem 2372

Давайте теперь подробно решим задачу с использованием формулы Пуассона для волнового уравнения. Мы будем решать задачу Коши для уравнения: $u_{t t}=a^{2}\left(u_{x x}+u_{y y}\right)$
с начальными условиями: $u(x, y, 0)=0, \quad u_{t}(x, y, 0)=(2 x+3 y)^{2}$

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Problem 2373

Укажіть значення похідної функції $y=f(x)$ у точці з абсцисою $x_{0}$ $\sqrt{3}$ 0 $-1$ 1 $-\frac{1}{\sqrt{3}}$

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Problem 2374

Установіть відповідність між функцією (1-3) та числовим значенням тангенса кута (А-Д), який утворює з додатним напрямком осі Ох дотична до графіка цієї функції в точці з абсцисою $x_{0}$
Функція $\begin{array}{l} y=1-2 x-\frac{1}{4} x^{2}, x_{0}=-3 \\ y=\operatorname{tg} x, x_{0}=\frac{\pi}{4} \\ y=\frac{1}{3} x^{3}, x_{0}=1 \end{array}$ $\square$ $\square$ $\square$
Значення похідної функції $1 \quad-2$ $2 \quad-\frac{1}{2}$ $3 \quad \frac{1}{2}$ 41 52

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Problem 2375

$\int \frac{x}{2 x^{2}+x+1} d x$

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Problem 2376

Данная задача представляет собой задачу Коши для волнового уравнения в трёхмерном пространстве. Уравнение: $u_{t t}=9\left(u_{x x}+u_{y y}+u_{z z}\right)$
где $u(t, x, y, z)$ - функция, зависящая от времени $t$ и пространственных координат $x, y, z$ . Также даны начальные условия:
1. $u(t=0, x, y, z)=0$ ,
2. $\quad u_{t}(t=0, x, y, z)=(2 x+3 y+4 z)^{2}$ .

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Problem 2377

Oblicz: $\int\left(4-x^{2}\right) \sqrt{2-x} d x$

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Problem 2378

ZADANIE 2 Wyznacz punkty przegięcia funkcji: $f(x)=\frac{x}{x^{2}+1}$ $\int(x)=\frac{x}{x^{2}+1}$

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Problem 2379

2. [-/3 Points] DETAILS MY NOTES SCALCET9 11.10.062.
Evaluate the indefinite integral as an infinite series. $\begin{array}{r} \int \arctan \left(x^{6}\right) d x \\ \sum_{n=0}^{\infty}(\square)+c \end{array}$ Need Help? Readif Submit Answer

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Problem 2380

(7 pts) Let $f(x)$ be a twice differentiable function and the graph of the derivative $f^{\prime}(x)$ is given below. i) $f(x)$ is increasing on the interval $(1,2)$ . ii) $f(x)$ is decreasing on the interval $(2,5)$ . iii) $(5, f(5))$ is a critical point of $f(x)$ . iv) $(3.5, f(3.5))$ is an inflection point of $f(x)$ .
Choose all the correct statements

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Problem 2381

1. Solve this DE using Exact Techrique $\frac{d y}{d x}=\frac{y-y^{2}}{2 x y+1}, \quad y>1$

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Problem 2382

Evaluate $f^{\prime}(x)$ at $x=1$ . $\begin{array}{l} \text { Evaluate } f^{\prime}(x) \text { at } x=1 . \\ f^{\prime}(1)=\square \end{array}$

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Problem 2383

b) $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n}}$

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Problem 2384

$\int e^{\sin (x)} \times \cos (x) d x$

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Problem 2385

i) $\lim _{x \rightarrow 3} \frac{x^{3}+x^{2}-5 x-21}{x-3}=$

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Problem 2386

Question Pat 1 of 2 Completed: 6 of 11 My score: $6 / 11$ pts (54.55\%) Save
Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n. $f(x)=\frac{2}{x} \text { on }[1,5] ; n=4$
The left Riemann sum for f is $\square$ . (Simplify your answer.)

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Problem 2387

Evaluate the integral. $\frac{\int \frac{3}{x^{3 / 2}-\sqrt{x}} d x}{\int \frac{3}{x^{3 / 2}-\sqrt{x}} d x=\square}$ $\square$

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Problem 2388

Question Compl
Use graphs to evaluate $\int_{0}^{2 \pi} \sin x d x$ and $\int_{0}^{2 \pi} \cos x d x$ . $\int_{0}^{2 \pi} \sin x d x=$ $\square$ $\int_{0}^{2 \pi} \cos x d x=$ $\square$

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Problem 2389

The function $f(x)=x^{3}+14 x^{2}+56 x+74$ is graphed below. Plot a line segment connecting the points on $f$ where $x=-9$ and $x=-4$ . Afterwards, determine all values of $c$ which satisfy the conclusion of the Mean Value Theore for $f$ on the closed interval $-9 \leq x \leq-4$ .
Plot a line by clicking in two locations. Click the line to delete it.

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Problem 2390

1. Solve this DE using Exact Technique $\frac{d y}{d x}=\frac{y-y^{2}}{2 x y+1}, \quad y>1$

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Problem 2391

2. Flächeninhalt
Die Funktion $f(x)=(x-1) \cdot e^{x}(s$ . Bild oben) beschreibt den Verlauf eines Flusses, der von zwei Straßen überbrückt wird, die längs der Koordinatenachsen laufen. (1 LE = 1 km ) Die beiden Straßen und der Fluss schließen im 4. Quadranten ein Grundstück A ein, welches für $80 €$ pro $\mathrm{m}^{2}$ zum Kauf angeboten wird. a) Zeigen Sie, dass $F(x)=(x-2) \cdot e^{x}$ eine Stammfunktion von $f$ ist.

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Problem 2392

$y=\cos (2 \theta), x=\tan (\theta)$ find $\frac{d y}{d x}$ .

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Problem 2393

The intensity of sunlight below the ocean's surface decreased exponentially with depth below the surface. When the intensity at the surface is 100 units, then intensity at a depth of 3 m is 6 units. A particular plant cannot grow if the intensity of the sunlight is less than 0.001 unit, What is the maximum depth, to the nearest centimetre, at which this plant can grow?

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Problem 2394

$\begin{array}{c}f(x, y)=x^{9} e^{y^{2}} \\ f_{x}=\square\end{array}$

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Problem 2395

$\int_{0}^{t} 3 S\left(\frac{3 S^{2}}{2}\right) d s$

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Problem 2396

89. If $f(x)$ is a differentiable function for all $x$ and has a relative minimum at $x=a$ , which of the following must be true about $f(x)$ ? A. $f^{\prime}(a)=0$ B. $f^{\prime}(x)$ changes from positive to negative at $x=a$ . C. $f^{\prime}(x)$ changes from decreasing to increasing at $x=a$ . D. $f^{\prime}(x)$ changes from concave down to concave up at $x=a$ .

See Solution

Problem 2397

The half-life of a drug in the bloodstream is 9 hours. What fraction of the original drug dose remains in 12 hours? in 24 hours?
What fraction of the original drug dose remains in 12 hours? $\square$ (Do not round until the final answer. Then round to the nearest hundredth as needed.)

See Solution

Problem 2398

Use a Maclaurin series in this table to obtain the Maclaurin series for the given function. $\begin{array}{r} f(x)=x^{2} \ln \left(1+x^{3}\right) \\ \sum_{n=1}^{\infty}(\square) \end{array}$

See Solution

Problem 2399

Question 2: Find the indefinite integral of each of the following functions. Check you results by differentiating. (a) $f(x)=\frac{x}{2} \cos (5 x)$ (d) $f(x)=x 3^{x}$

See Solution

Problem 2400

Find the Taylor series for $f(x)$ centered at the given value of $a$ . [Assume that $f$ has a power series expansion. Do not show that $R_{n}(x) \rightarrow 0$ .] $\begin{array}{r} f(x)=\sin (x), \quad a=\pi \\ f(x)=\sum_{n=0}^{\infty}(\square) \end{array}$
Find the associated radius of convergence, $R$ . $\square$ $R=$

See Solution

Calculus

Problem 2301

Find the slope of the tangent line to f(x)=3x2f(x)=3 x^{2}f(x)=3x2 at x=4x=4x=4 using the limit definition of the derivative.

Problem 2302

For the function F(x)=−x4+4x2+192F(x)=-x^{4}+4 x^{2}+192F(x)=−x4+4x2+192, find if FFF is even/odd, a second local max, and area from x=−4x=-4x=−4 to x=0x=0x=0.

Problem 2303

A ball is dropped from a 94 ft building. Find the time to fall half the distance and to ground level: (a) t=t=t= sec, (b) t=t=t= sec.

Problem 2304

A ball drops from a 94 ft building. (Round to 3 decimal places.) (a) Time to fall half distance? t=1.8 t=1.8 t=1.8 (b) Time to reach ground? t=2.4 t=2.4 t=2.4

Problem 2305

Find all numbers xxx in (−1,2)(-1,2)(−1,2) where f′(x)=3x2+4x−1f'(x)=3x^2+4x-1f′(x)=3x2+4x−1 equals the average rate of change of f(x)=x3+2x2−x+6f(x)=x^3+2x^2-x+6f(x)=x3+2x2−x+6.

Problem 2306

Show that if y=5+xxy=\frac{5+\sqrt{x}}{\sqrt{x}}y=x​5+x​​, then 6dydx+4xd2ydx2=06 \frac{d y}{d x}+4 x \frac{d^{2} y}{d x^{2}}=06dxdy​+4xdx2d2y​=0.

Problem 2307

Problem 2308

Problem 2309

Find the limit as xxx approaches infinity for x22x\frac{x^{2}}{2^{x}}2xx2​. Evaluate f(x)=x22xf(x)=\frac{x^{2}}{2^{x}}f(x)=2xx2​ for x=0,1,…,100x=0,1,\ldots,100x=0,1,…,100.

Problem 2310

Find the limit as xxx approaches infinity for x2+6\sqrt{x^{2}+6}x2+6​.

Problem 2311

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: v=0.17[ S]0.021+[S]v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}v=0.021+[S]0.17[ S]​.

Problem 2312

Find the limit: lim⁡x→∞(25x2+x−5x)\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)x→∞lim​(25x2+x​−5x). If it doesn't exist, enter DNE.

Problem 2313

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: v=0.17[S]0.021+[S]v=\frac{0.17[S]}{0.021+[S]}v=0.021+[S]0.17[S]​. What does it indicate?

Problem 2314

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: v=0.17[ S]0.021+[S]v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}v=0.021+[S]0.17[ S]​. What does it signify?

Problem 2315

Bertalanffy growth function: 1. Find lim⁡t→∞L(t)\lim_{t \rightarrow \infty} L(t)limt→∞​L(t) and interpret.2. With L0=1 cmL_{0}=1 \mathrm{~cm}L0​=1 cm, LT=38 cmL_{T}=38 \mathrm{~cm}LT​=38 cm, graph L(t)L(t)L(t) for various kkk.

Problem 2316

Find lim⁡t→∞B(t)\lim_{t \rightarrow \infty} B(t)limt→∞​B(t) for the biomass model B(t)=9×1071+3e−0.72tB(t)=\frac{9 \times 10^{7}}{1+3 e^{-0.72 t}}B(t)=1+3e−0.72t9×107​. What does this limit mean?

Problem 2317

Evaluate f(x)=x2−3xx2−x−6f(x)=\frac{x^{2}-3 x}{x^{2}-x-6}f(x)=x2−x−6x2−3x​ at x=3.5,3.1,...,2.999x=3.5,3.1,...,2.999x=3.5,3.1,...,2.999 and find lim⁡x→3f(x)≈0.600000\lim_{x \to 3} f(x) \approx 0.600000limx→3​f(x)≈0.600000.

Problem 2318

Problem 2319

Find the tangent line equation for the polar curve r=3cos⁡θr=3 \cos \thetar=3cosθ at θ=π3\theta=\frac{\pi}{3}θ=3π​.

Problem 2320

Find the slope of the tangent line for r=3cos⁡θr=3 \cos \thetar=3cosθ at θ=π3\theta=\frac{\pi}{3}θ=3π​.

Problem 2321

Find the average rate of change of f(x)=−2x3+4x+7f(x)=-2 x^{3}+4 x+7f(x)=−2x3+4x+7 on [−1,3][-1,3][−1,3]. Is it even, odd, or neither?

Problem 2322

Find aaa so that lim⁡x→∞ax3+4x2−x−13x3+x2−9=−5\lim_{{x \rightarrow \infty}} \frac{ax^{3}+4x^{2}-x-1}{3x^{3}+x^{2}-9} = -5limx→∞​3x3+x2−9ax3+4x2−x−1​=−5.

Problem 2323

An automobile accelerated from rest to 345.6 m/s345.6 \mathrm{~m/s}345.6 m/s in 3.07 s3.07 \mathrm{~s}3.07 s. What distance did it travel?

Problem 2324

How long does it take for a football to fall 1719 m1719 \mathrm{~m}1719 m on Mars? (g=3.72 m/s2)(\mathrm{g}=3.72 \mathrm{~m}/\mathrm{s}^2)(g=3.72 m/s2)?

Problem 2325

11. Calculate ∫010(8−x)dx\int_{0}^{10}(8-x) d x∫010​(8−x)dx in two ways: a. As the limit lim⁡N→∞RN\lim _{N \rightarrow \infty} R_{N}limN→∞​RN​ b. By sketching the relevant signed area and using geometry

Problem 2326

Find the volume VVV of the solid obtained by rotating the region bounded by the given curves about the specified line., y=ex,y=0,x=−2,x=2;y=e^{x}, \quad y=0, \quad x=-2, x=2 ;y=ex,y=0,x=−2,x=2; about the xxx-axis V=V=V=

Problem 2327

7 What is the minimum period of oscillation of a uniform rod of length 0.960 m with a moveable pivot point? (That is, you are free to place the pivot wherever you like such that the period is a minimum.)

Problem 2328

aluate. ∫abe4tdt\int_{a}^{b} e^{4 t} d t∫ab​e4tdt

Problem 2329

Evaluate. ∫1e3(3x+1x)dx\int_{1}^{e^{3}}\left(3 x+\frac{1}{x}\right) d x∫1e3​(3x+x1​)dx

Problem 2330

Problem 2331

Find the general antiderivative of f(x)=−6x5+5x+5x4+5xf(x)=-6 x^{5}+\frac{5}{x}+\frac{5}{x^{4}}+5 \sqrt{x}f(x)=−6x5+x5​+x45​+5x​

Problem 2332

Find the volume VVV of the solid obtained by rotating the region bounded by the given curves about the specified line. y=x3,y=1,x=2;y=x^{3}, \quad y=1, \quad x=2 ;y=x3,y=1,x=2; about y=−4y=-4y=−4 V=V=V=

Problem 2333

Problem 2334

7. Let E={(x,y,z)∣x+y+z≤1,x≥0,y≥0,z≥0}E=\{(x, y, z) \mid x+y+z \leq 1, x \geq 0, y \geq 0, z \geq 0\}E={(x,y,z)∣x+y+z≤1,x≥0,y≥0,z≥0}. Show that ∭EezdV=e−52\iiint_{E} e^{z} d V=e-\frac{5}{2}∭E​ezdV=e−25​

Problem 2335

Problem 2336

Find the area under the curve y=1xy=\frac{1}{x}y=x1​ over the interval [1,6][1,6][1,6]. □\square□ Find the average value of y=1xy=\frac{1}{x}y=x1​ over the interval [1,6][1,6][1,6]. □\square□

Problem 2337

Problem 2338

Problem 2339

Question Watch VideoIf −2=xy+5y+3x3-2=x y+5 y+3 x^{3}−2=xy+5y+3x3 then find dydx\frac{d y}{d x}dxdy​ in terms of xxx and yyy.Answer Attempt 1 out of 2

Problem 2340

14. Radius The area AAA of a circle is decreasing at a rate of 2 square feet per minute. Find the rate of change of the radius when A=24A=24A=24 square feet.

Problem 2341

Question Find the absolute maximum and absolute minimum of the function on the given interval. f(x)=x4/3−16x1/3 on [−1,8]f(x)=x^{4 / 3}-16 x^{1 / 3} \text { on }[-1,8]f(x)=x4/3−16x1/3 on [−1,8]Round answers to 3 decimal places.

Problem 2342

18. Surface Area All edges of a cube are expanding at rate of 6 centimeters per second. How fast is the surface area changing when each edge is (a) 2 centimeters and (b) 10 centimeters?

Problem 2343

Problem 2344

20,100 is invester at 3.50∘3.50^{\circ}3.50∘ intrest compounded continously. how lony will it tuke for the investment to double.

Find the slope of the tangent line to $f(x)=3 x^{2}$ at $x=4$ using the limit definition of the derivative.

For the function $F(x)=-x^{4}+4 x^{2}+192$ , find if $F$ is even/odd, a second local max, and area from $x=-4$ to $x=0$ .

A ball is dropped from a 94 ft building. Find the time to fall half the distance and to ground level: (a) $t=$ sec, (b) $t=$ sec.

A ball drops from a 94 ft building. (Round to 3 decimal places.) (a) Time to fall half distance? $t=1.8$ (b) Time to reach ground? $t=2.4$

Find all numbers $x$ in $(-1,2)$ where $f'(x)=3x^2+4x-1$ equals the average rate of change of $f(x)=x^3+2x^2-x+6$ .

Show that if $y=\frac{5+\sqrt{x}}{\sqrt{x}}$ , then $6 \frac{d y}{d x}+4 x \frac{d^{2} y}{d x^{2}}=0$ .

Find the limit as $x$ approaches infinity for $\frac{x^{2}}{2^{x}}$ . Evaluate $f(x)=\frac{x^{2}}{2^{x}}$ for $x=0,1,\ldots,100$ .

Find the limit as $x$ approaches infinity for $\sqrt{x^{2}+6}$ .

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{25 x^{2}+x}-5 x\right)$ . If it doesn't exist, enter DNE.

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[S]}{0.021+[S]}$ . What does it indicate?

Find the horizontal asymptote of the Michaelis-Menten equation for chymotrypsin: $v=\frac{0.17[\mathrm{~S}]}{0.021+[\mathrm{S}]}$ . What does it signify?

Bertalanffy growth function:
1. Find $\lim_{t \rightarrow \infty} L(t)$ and interpret.
2. With $L_{0}=1 \mathrm{~cm}$ , $L_{T}=38 \mathrm{~cm}$ , graph $L(t)$ for various $k$ .

Find $\lim_{t \rightarrow \infty} B(t)$ for the biomass model $B(t)=\frac{9 \times 10^{7}}{1+3 e^{-0.72 t}}$ . What does this limit mean?

Evaluate $f(x)=\frac{x^{2}-3 x}{x^{2}-x-6}$ at $x=3.5,3.1,...,2.999$ and find $\lim_{x \to 3} f(x) \approx 0.600000$ .

Find the tangent line equation for the polar curve $r=3 \cos \theta$ at $\theta=\frac{\pi}{3}$ .

Find the slope of the tangent line for $r=3 \cos \theta$ at $\theta=\frac{\pi}{3}$ .

Find the average rate of change of $f(x)=-2 x^{3}+4 x+7$ on $[-1,3]$ . Is it even, odd, or neither?

Find $a$ so that $\lim_{{x \rightarrow \infty}} \frac{ax^{3}+4x^{2}-x-1}{3x^{3}+x^{2}-9} = -5$ .

An automobile accelerated from rest to $345.6 \mathrm{~m/s}$ in $3.07 \mathrm{~s}$ . What distance did it travel?

How long does it take for a football to fall $1719 \mathrm{~m}$ on Mars? $(\mathrm{g}=3.72 \mathrm{~m}/\mathrm{s}^2)$ ?

11. Calculate $\int_{0}^{10}(8-x) d x$ in two ways: a. As the limit $\lim _{N \rightarrow \infty} R_{N}$ b. By sketching the relevant signed area and using geometry

Find the volume $V$ of the solid obtained by rotating the region bounded by the given curves about the specified line., $y=e^{x}, \quad y=0, \quad x=-2, x=2 ;$ about the $x$ -axis $V=$

aluate. $\int_{a}^{b} e^{4 t} d t$

Evaluate. $\int_{1}^{e^{3}}\left(3 x+\frac{1}{x}\right) d x$

Find the general antiderivative of $f(x)=-6 x^{5}+\frac{5}{x}+\frac{5}{x^{4}}+5 \sqrt{x}$

Find the volume $V$ of the solid obtained by rotating the region bounded by the given curves about the specified line. $y=x^{3}, \quad y=1, \quad x=2 ;$ about $y=-4$ $V=$

7. Let $E=\{(x, y, z) \mid x+y+z \leq 1, x \geq 0, y \geq 0, z \geq 0\}$ . Show that $\iiint_{E} e^{z} d V=e-\frac{5}{2}$

Find the area under the curve $y=\frac{1}{x}$ over the interval $[1,6]$ . $\square$ Find the average value of $y=\frac{1}{x}$ over the interval $[1,6]$ . $\square$

Question Watch Video
If $-2=x y+5 y+3 x^{3}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ .
Answer Attempt 1 out of 2

14. Radius The area $A$ of a circle is decreasing at a rate of 2 square feet per minute. Find the rate of change of the radius when $A=24$ square feet.

Question Find the absolute maximum and absolute minimum of the function on the given interval. $f(x)=x^{4 / 3}-16 x^{1 / 3} \text { on }[-1,8]$
Round answers to 3 decimal places.

20,100 is invester at $3.50^{\circ}$ intrest compounded continously. how lony will it tuke for the investment to double.

ii. $\mathcal{L}\left\{\cosh 2 t e^{4 t}\right\} \quad$ ; First Shift Theorem ; Teorem Anjakan Pertama

$\sin(x+y) = 3x$ Find $\frac{dy}{dx}$ .

$\sqrt{y} - \ln y = x + 8$
Find $\frac{dy}{du}$ .

$\begin{array}{l}\text { Find } \frac{d y}{d t} \quad y=(t \sin t)^{6} \\ \frac{d y}{d t}=\text { ? }\end{array}$

The radioactive isotope ${ }^{226} \mathrm{Ra}$ has a half-life of approximately 1599 years. Consider a lab that currently has 45 g of ${ }^{226} \mathrm{Ra}$ . (a.) How much of the isotope remains after 1200 years? Round your answer to three decimal places.

9. (6405 \#3 p. 320) Let $R$ denote the region bounded by the curve $y=\frac{1}{x^{2}+1}$ , the $y$ -axis, the $x$ -axis, and the line $x=4$ . Set $S$ denote the solid generated when the region $R$ is revolved about the $y$ -axis. Find the volume of $S$ .

For the following function $f$ , find the antiderivative $F$ that satisfies the given condition. $f(x)=6 \sqrt{x}+4 ; F(1)=4$
The antiderivative that satisfies the given condition is $F(x)=4 x \sqrt{x}+4 x+c$ .

Compute the integral $\int \frac{e^{x}}{\sin ^{2}\left(e^{x}\right)} d x=$ $\square$ $\text { 마온 }+C$ where $C$ is the constant of integration. Do not include the constant of integration in yo

Find the arc length of the following curve on the given interval. $y=\frac{1}{3} x^{3 / 2} \text { on }[0,54]$

Find the area of the surface generated when the given curve is revolved about the given axis. $y=22 \sqrt{x}$ , for $9 \leq x \leq 17$ ; about the $x$ -axis

Find the area of the surface generated when the given curve is revolved about the given axis. $y=(8 x)^{\frac{1}{3}}$ , for $0 \leq x \leq 8$ ; about the $y$ -axis

$\int_{0}^{1} \frac{3 x}{4 x^{2}-1} d x$

Find the derivative of the function $y = 2^x - \sqrt[3]{x^2} + 4 \arccos x$ .

$\int \frac{1}{16-x^{2}} d x$

Exercies find the second derivative of $y=f(x)$ at $x=0$ where $y$ is determined by $y^{5}+2 y-x-3 x^{7}=0$

Укажіть похідну функції $f(x)=-\operatorname{ctg} x+2 x^{5}$

Укажіть значення похідної функції $y=f(x)$ у точці з абсцисою $x_{0}$ $\sqrt{3}$ 0 $-1$ 1 $-\frac{1}{\sqrt{3}}$