Calculus

Problem 1901

A sample of 9 grams of radioactive material is placed in a vault. Let $P(t)$ be the amount remaining after $t$ years, and let $P(t)$ satisfy the differential equation $P^{\prime}(t)=-0.032 P(t)$ . Answer parts $(a)$ through $(g)$ . (Type an expression using t as the variable.) (b) What is $P(0)$ ? $P(0)=9$ (c) What is the decay constant? $\square$

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Problem 1902

A sample of 9 grams of radioactive material is placed in a vault. Let $P(t)$ be the amount remaining after $t$ years, and let $P(t)$ satisfy the differential equation $P^{\prime}(t)=-0.032 P(t)$ . Answer parts $(a)$ through $(g)$ . (c) What is the decay constant? 0.032 (d) How much of the material will remain after 10 years?
(Type an integer or decimal rounded to one decimal place as needed.)

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Problem 1903

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Evaluate the integral $\begin{array}{l} I=\int_{0}^{1} \frac{x^{3}}{\sqrt{1-x^{8}}} \mathrm{~d} x \\ I= \end{array}$
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Problem 1904

A sample of 9 grams of radioactive material is placed in a vault. Let $P(t)$ be the amount remaining after $t$ years, and let $P(t)$ satisfy the differential equation $P^{\prime}(t)=-0.032 P(t)$ . Answer parts $(a)$ through $(g)$ . (e) How fast is the sample disintegrating when just 1 gram remains? Use the differential equation.
Choose the correct process to find how fast the sample disintegrating when just 1 gram remains. A. Evaluate $P^{\prime}(t)=-0.032(1)$ . B. Solve $1=-0.032 P(t)$ for $P(t)$ . C. Evaluate $P^{\prime}(t)=-0.032 P(1)$ . D. Solve $P^{\prime}(1)=-0.032 P(t)$ for $P(t)$ .

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Problem 1905

In practice, we take $-M$ to be the maximum of $\left|f^{(n+1)}(x)\right|$ on the interval $I$ .
Question: (a) Approximate $\ln (1.5)$ using the 3-rd degree Taylor polynomial of $f(x)=\ln (x)$ centred at $x=1$ (b) Use Taylor's Remainder Theorem to approximate the error between $\ln (1.5)$ and the 3 -rd degree Taylor polynomial approximation on the interval $I=\left[\frac{1}{2}, \frac{3}{2}\right]$ . (c) Let $T_{n}(x)$ be the $n$ -th degree Taylor polynomial for $f(x)=\ln (x)$ centred at $x=1$ . Using Taylor's Remainder Theorem, find the smallest integer $n$ so that $\left|f(x)-T_{n}(x)\right| \leq 0.0001$ for all $x \in\left[\frac{1}{2}, \frac{3}{2}\right]$ .

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Problem 1906

Points: 0 of 1 Save
A sample of 9 grams of radioactive material is placed in a vault. Let $P(t)$ be the amount remaining after $t$ years, and let $P(t)$ satisfy the differential equation $P^{\prime}(t)=-0.032 P(t)$ . Answer parts (a) through $(\mathrm{g})$ . B. Solve $-0.119=-0.032 P(t)$ for $P(t)$ . C. Evaluate $P^{\prime}(t)=-0.032(-0.119)$ . D. Evaluate $P^{\prime}(t)=-0.032 P(-0.119)$ .
There will be $\square$ of radioactive material left when it is disintegrating at a rate of 0.119 gram per year. (Type an integer or decimal rounded to one decimal place as needed.)

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Problem 1907

Evaluate the following indefinite integral. If you use any trigonometric substitution, y should assume that $|\sin (\theta)|=\sin (\theta)$ and $|\cos (\theta)|=\cos (\theta)$ in the region we are interested in. $4 \int \frac{x}{\sqrt{9-x^{2}}} \mathrm{~d} x=$ $\square$ Check

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Problem 1908

Evaluate the following definite integral: To input the inverse triginometric functions $\cos ^{-1}$ and $\sin ^{-1}$ , type acos or asin respectively (short for arccos and arcsin). $\int_{2}^{5} \frac{\sqrt{x^{2}-4}}{x} d x=$ $\square$ Check

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Problem 1909

Archimedes (287 - 212 B.C.E), inventor, military engineer, physicist, and the greatest mathematician..of classical times in the Western world, discovered that the area under a parabolic arch is two-thirds the base times the height. Sketch the parabolic arch $y=h-\frac{4 h}{b^{2}} x^{2},-\frac{b}{2} \leq x \leq \frac{b}{2}$ , assuming that $h$ and $b$ are positive. Then use calculus to find the are of the region enclosed between the arch and the $x$ -axis.

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Problem 1910

Question: (a) Approximate $\ln (1.5)$ using the 3-rd degree Taylor polynomial of $f(x)=\ln (x)$ centred at $x=1$ . (b) Use Taylor's Remainder Theorem to approximate the error between $\ln (1.5)$ and the 3 -rd degree Taylor polynomial approximation on the interval $I=\left[\frac{1}{2}, \frac{3}{2}\right]$ . (c) Let $T_{n}(x)$ be the $n$ -th degree Taylor polynomial for $f(x)=\ln (x)$ centred at $x=1$ . Using Taylor's Remainder Theorem, find the smallest integer $n$ so that $\left|f(x)-T_{n}(x)\right| \leq 0.0001$ for all $x \in\left[\frac{1}{2}, \frac{3}{2}\right]$ .

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Problem 1911

Calculate the integral: $\int t^{2}(t^{3}-3)^{3} dt$

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Problem 1912

Find the integral of the inverse cosine function: $\int \cos^{-1} x \, dx$ .

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Problem 1913

Evaluate the integral $\int e^{2 \theta} \sin 3 \theta d \theta$ .

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Problem 1914

Find the tangent line equation for $f(x)=\sqrt{3-x}$ at a given point.

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Problem 1915

Find the derivative of the function $y=x^{4}+12x$ . What is $\frac{dy}{dx}$ ?

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Problem 1916

Find $\lim _{x \rightarrow \infty} \cos \left(\frac{1+\pi x}{x}\right)$ .

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Problem 1917

Find the limit as $x$ approaches 0 for $\cos \left(\frac{1+\pi x}{x}\right)$ .

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Problem 1918

Calculate the integral $\int_{0}^{2}\left(x^{2}+1\right) e^{-x} dx$ .

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Problem 1919

Calculate the integral from 0 to 7 of $(x^{2}+1) e^{-x}$ .

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Problem 1920

Find the limit as $x$ approaches 4 for the piecewise function $f(x)$ defined as: $f(x) = -7x + 46$ if $x < 4$ , $12$ if $x = 4$ , $2x^2 - 14$ if $x > 4$ . What is $\lim_{x \rightarrow 4} f(x)$ ?

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Problem 1921

A ball is thrown with a velocity of $41 \mathrm{ft} / \mathrm{s}$ . Its height is $y=41 t-22 t^{2}$ . Find average velocity from $t=2$ and instantaneous velocity at $t=2$ .

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Problem 1922

Find the rate of change of the city's population modeled by $P(t)=22 e^{0.08 t}$ from 2025 to 2034.

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Problem 1923

Calculate the integral from 1 to 2 of $w^{2} \ln w \, dw$ .

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Problem 1924

Calculate the integral $\int_{0}^{2 \pi} t^{2} \sin 2 t \, dt$ .

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Problem 1925

Find the average rate of change of COVID-19 infections from 0 to 8 days, assuming 0 at day 0. Round to two decimals.

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Problem 1926

Find the integral of the function $e^{\sqrt{x}}$ with respect to $x$ : $\int e^{\sqrt{x}} d x$ .

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Problem 1927

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Height after $t$ seconds is $y=41t-22t^2$ . Find average velocity from $t=2$ for 0.01, 0.005, and 0.002 seconds. What is the instantaneous velocity at $t=2$ ?

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Problem 1928

Encuentra la derivada de $f(x) = 3x^3 - x$ .

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Problem 1929

Approximate the average rate of change of COVID-19 infections in Minnesota from 0 to 8 days after April 1, 2020.

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Problem 1930

Find the integral: $\int x^{2} \ln (x) \, dx$ .

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Problem 1931

Find the rate of change of the city's population, modeled by $P(t)=10 e^{0.04 t}$ , from 2021 to 2030 in thousand persons/year.

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Problem 1932

Calculate the average rates of change for COVID-19 infections in Minnesota over these intervals using $\frac{\Delta y}{\Delta x}$ :
1. 0 to 8 days: (calculate)
2. 8 to 16 days: 0.59 thousand/day
3. 0 to 16 days: 0.46 thousand/day

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Problem 1933

Find the average velocity of a ball thrown with $y=41t-22t^{2}$ at $t=2$ for intervals of 0.01, 0.005, 0.002, and 0.001 seconds.

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Problem 1934

Calculate the average rate of change of the function $k(x)=-16 \sqrt{x}$ from $x=12$ to $x=15$ .

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Problem 1935

Two engines start at the origin and meet at $t=e-1$ . Thomas' equation is $x=300 \ln(t+1)$ and Henry's is $x=kt$ .
a. Sketch the graphs. b. Show $k=\frac{300}{e-1}$ . c. Find the max distance between them in the first $e-1$ minutes and when it occurs.

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Problem 1936

A particle $P$ has a velocity of $v=24-4 t^{2}$ m/s. Find: (a) distance in the first second, (b) when it changes direction, (c) when it returns to start.

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Problem 1937

A particle $P$ moves along a line with displacement $x = \frac{1}{3} t^{3} - 4 t^{2} + 15 t$ . Find: (a) when $P$ is at rest, (b) distance in 5 seconds.

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Problem 1938

Find the distance from the origin when the particle stops, given $v=9t-3t^{2}$ m/s and starts at $t=0$ .

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Problem 1939

Does the function $f(x)=x$ have a limit as $x$ approaches 3 from all real numbers except 3?

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Problem 1940

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Its height after $t$ seconds is $y=41t-22t^2$ . Find average velocity for $t=2$ over 0.01, 0.005, 0.002, and 0.001 seconds, then determine instantaneous velocity at $t=2$ .

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Problem 1941

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Its height is $y=41t-22t^2$ . Find average velocity from $t=2$ for given intervals and the instantaneous velocity at $t=2$ .

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Problem 1942

Calculate the integral $\int_{0}^{5} 4 x^{2} 8 x \, dx$ .

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Problem 1943

Find $\lim _{x \rightarrow 2} f(x)$ given that $\lim _{x \rightarrow 2} \sqrt{\frac{[f(x)]^{2}-8 x+3}{x+1}}=9$ and $f(x) \geq 0$ .

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Problem 1944

Find the limit as $x$ approaches 3 for the expression $x^{-1/2}(5x-7)^{1/3}$ .

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Problem 1945

Find the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$ .

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Problem 1946

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

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Problem 1947

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}$ .

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Problem 1948

Find the limit as $x$ approaches 0 for the expression $\frac{\sin x}{x}$ .

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Problem 1949

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{5+x}-\frac{1}{5}}{x}$ .

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Problem 1950

Find the limit as $x$ approaches -1 for the expression $-2x^3 - x + 5$ .

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Problem 1951

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

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Problem 1952

Find the limit: $\lim _{x \rightarrow 4} \sqrt{1+2 x}$ .

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Problem 1953

Find the limit as $x$ approaches 0 for the expression $e^{-x} \cos x$ .

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Problem 1954

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

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Problem 1955

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

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Problem 1956

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

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Problem 1957

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3}{x^{2}}$ , where $h \neq 0$ .

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Problem 1958

The function $f$ is shown below. If $g$ is the function defined by $g(x)=\int_{-2}^{x} f(t) d t$ , what is the value of $g(9)$ ?

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Problem 1959

Given $\frac{d y}{d t}=7 y$ and $y(0)=450$ , determine $y(t)$ $y(t)=$ $\square$

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Problem 1960

Problem 1. (1 point) $\int_{0}^{2} \int_{0}^{5} e^{5 x+2 y} d x d y=\square$ (Give the exact value, not a decimal approximation)

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Problem 1961

pago 34
Solution. (a) On a $f_{X, Y}(x, y)=\left\{\begin{array}{c} 10 x y^{2}, 0 \leq x \leq y \leq 1 \\ 0, \text { ailleurs } \end{array}\right.$ $\begin{array}{l} \text { Pour } 0<x<y<1, \text { on a } \\ \begin{aligned} F_{X, Y}(x, y) & =\int_{0}^{x} \int_{u}^{y} 10 u v^{2} d v d u=10 \int_{0}^{x} u \int_{u}^{y} v^{2} \mathrm{~d} v \mathrm{~d} u \\ & =\left.10 \int_{0}^{x} u\left(\frac{v^{3}}{3}\right)\right|_{u} ^{y} \mathrm{~d} u=10 \int_{0}^{x} u\left(\frac{y^{3}-u^{3}}{3}\right) \mathrm{d} u \\ & =\frac{10}{3} \int_{0}^{x} y^{3} u-u^{4} \mathrm{~d} u=\left.\frac{10}{3}\left(y^{3} \frac{u^{2}}{2}-\frac{u^{5}}{5}\right)\right|_{0} ^{x} \\ & =\frac{10}{3}\left(y^{3} \frac{x^{2}}{2}-\frac{x^{5}}{5}\right) \\ & =\frac{5}{3} x^{2} y^{3}-\frac{2}{3} x^{5} \end{aligned} \end{array}$

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Problem 1962

$\int_{0}^{4} \frac{p}{\sqrt{9+p^{2}}} d p$

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Problem 1963

Question 1 (Multiple Choice Worth 5 points) (02.07 MC)
Given a function $f(x)=2 x^{2}+3$ , what is the average rate of change of $f$ on the interval $[2,2+h] ?$ 11 $2 h+8$ $2 h^{2}+8 h$ $2 h^{2}+8 h+11$

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Problem 1964

Question 12
You deposit $\$ 5000$ in an account earning $7 \%$ interest compounded continuously. How much will you have in the account in 10 years? \\square$ Next Question

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Problem 1965

Relatel Ratas
17. The volume of a cube decreases at a rate of $10 \mathrm{~m}^{3} / \mathrm{s}$ . Find the rate at which the side of the cube changes when the side of the cube is 2 m .

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Problem 1966

How long will it take for an invertment to triple, if intcrest is compounded continously at 4 t - ?
It will take? years before the invectment trives
Round to the reares tentn of a year

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Problem 1967

16. Determine the value of $f(x)$ when $x \rightarrow-\infty$ if $f(x)=\frac{3 x+2}{2 x+3}$ a) $-\frac{2}{3}$ b) $\frac{2}{3}$ c) $1.5^{+}$ d) $1.5^{-}$

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Problem 1968

Evaluate the definite integral $\int_{2}^{4} x(x-2)^{3} d x$ (Enter a numerical value. Round your answer to 2 decimal places as needed.)

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Problem 1969

oblem \#15: Consider the following integrals. (i) $\int_{0}^{6} \frac{2}{x^{1 / 4}} d x$ (ii) $\int_{0}^{\infty} \frac{2}{x^{4}} d x$ (iii) $\int_{0}^{\infty} 4^{x} d x$ (iv) $\int_{-\infty}^{6} 4^{x} d x$
Determine if the above integrals Converge (1) or Diverge (2). So, for example, if you think that the answers, in the above order, are Converge,Diverge,Diverge,Converge then you would enter ' $1,2,2,1$ ' into the answer box below (without the quotes).

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Problem 1970

A conical tank (vertex down) is 7 meters across the top and 9 meters deep. If the depth of the water (the height) is decreasing at 6.6 meters per minute, what is the change in the volume of the water in the tank when the height of the water in the tank is 4 meters?
Include units on your final answer, and your answer must be entered as number (not 5*7+3).

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Problem 1971

A conical tank (vertex down) is 7 meters across the top and 9 meters deep. If the depth of the water (the height) is decreasing at 6.6 meters per minute, what is the change in the volume of the water in the tank when the height of the water in the tank is 4 meters?
Include units on your final answer, and your answer must be entered as number (not 5*7+3).

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Problem 1972

Find $d y / d x$ by implicit differentiation. $d y / d x=\square \quad \sin x+\cos y=\sin x \cos y$

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Problem 1973

Problem 5. (1 point)
Find the equation of the line tangent to the graph of $y=2 \ln (x)$ at $x=1$ . Tangent Line: $y=$ $\square$

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Problem 1974

For the region formed by the functions $y=x^{3}$ and $y=x$ , use definite integrals to find the area of the region.
Answer: The area is $\square$ - (Use a fraction.)

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Problem 1975

Determine the limit shown below in simplest form. $\lim _{x \rightarrow-6} \frac{5 x+30}{x^{2}+8 x+12}$

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Problem 1976

This is Section 4.4 Problem 54: PREVIOUS ANSWER
Two cars enter a freeway at the same time. The velocities, measured by miles per minute, of Car A and $\mathrm{Car} \mathrm{B} t$ minutes after entering the freeway are given by $\begin{array}{l} v_{A}(t)=0.4+0.2 t-0.02 t^{2}, 0 \leq t \leq 5 \\ v_{B}(t)=0.1+0.3 t-0.02 t^{2}, 0 \leq t \leq 5 \end{array}$
At the end of 5 minutes of driving, Car A has traveled $\square$ miles, and Car B has traveled $\square$ miles. Hence CarA $\checkmark \checkmark$ is ahead of Car B $\checkmark \checkmark$ by $\square$ Hint: Follow Example 5. Additional Materials ebook

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Problem 1977

8. Evaluate $\iiint_{G}\left(x^{2}+y^{2}\right)^{3 / 2} \mathrm{dV}$ , where G is the solid enclosed between the paraboloids $z=2-x^{2}-y^{2}$ and $z=x^{2}+y^{2}$ .

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Problem 1978

$f$ and $g$ are twice differentiable functions such that $g(x)=e^{f(x)}$ and $g^{\prime \prime}(x)=h(x) e^{f(x)}$ , then $h(x)=$ (A) $f^{\prime}(x)+f^{\prime \prime}(x)$ (B) $f^{\prime}(x)+\left(f^{\prime \prime}(x)\right)^{2}$
C $\left(f^{\prime}(x)+f^{\prime \prime}(x)\right)^{2}$ (D) $\left(f^{\prime}(x)\right)^{2}+f^{\prime \prime}(x)$ (E) $2 f^{\prime}(x)+f^{\prime \prime}(x)$

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Problem 1979

Consider the indefinite integral $\int x^{5} \cdot \sqrt[5]{x^{6}-6} d x$ : This can be transformed into a basic integral by letting $u=$ $\square$ and $d u=\square d x$
Performing the substitution yields the integral $\square$ $d u$

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Problem 1980

Consider the function $f(x)$ whose second derivative is $f^{\prime \prime}(x)=6 x+3 \sin (x)$ . If $f(0)=4$ and $f^{\prime}(0)=2$ , what is $f(4)$ ?

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Problem 1981

Calculate the derivative of $y$ with respect to $x$ . Express derivative in terms of $x$ and $y$ . $e^{2 x y}=\sin \left(y^{7}\right)$ (Express numbers in exact form. Use symbolic notation and fractions where needed.) $\frac{d y}{d x}=$ $\square$

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Problem 1982

4. Compute the indefinite integral $\int \sec ^{3} x d x$ .

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Problem 1983

Exercise 10.3
1. Use the integration by parts to evaluate the following integrals. (a) $\int s e^{-2 s} d s$ (b) $\int \ln (x+1) d x$ (c) $\int t \sin 2 t d t$ (d) $\int x 2^{x} d x$ (e) $\int x \cos 5 x d x$ (f) $\int e^{x} \cos x d x$

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Problem 1984

2. The mathematical model: $\frac{d^{2} y}{d t^{2}}+7 \frac{d y}{d t}+y=\sin 2 t$ is a. linear, time-invariant model b. linear, time-varying model c. Nonlinear, time-varying model d. Nonlinear, time-invariant model

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Problem 1985

$\lim _{x \rightarrow-4} \frac{x^{2}+|x|-20}{x+4}$

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Problem 1986

$\lim _{n \rightarrow \infty} \frac{1}{13 \cdot n^{2}+75}+\frac{1}{13 \cdot n^{2}+150}+\ldots+\frac{1}{13 \cdot n^{2}+75 \cdot n}=$

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Problem 1987

$\lim _{n \rightarrow \infty} a_{n}$

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Problem 1988

$: \operatorname{lR} \omega_{c}=20$ -II $b \in \mathbb{R}, \alpha \in \mathbb{R}\left(g(x)=-x^{3}+\alpha x+b\right.$ $A(0 ; 2)$ - (II $D_{f}$ (6) (

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Problem 1989

Trova i punti di massimo e di minimo della funzione $y = \sin 3x + \sqrt{3} \cos 3x$ e traccia il grafico.

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Problem 1990

6. The value of $a$ that makes $\lim _{x \rightarrow \infty} \sqrt{2 x^{2}+x+1}-a x$ equals $\frac{1}{2 \sqrt{2}}$ is A. -2 B. $\sqrt{2}$ C. $-\sqrt{2}$ D. 2

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Problem 1991

Consider the series $\sum_{n=1}^{\infty} \frac{1}{n(n+4)}$
Determine whether the series converges, and if it converges, determine its value. Converges $(\mathrm{y} / \mathrm{n}$ ): $\square$ Value if convergent (blank otherwise): $\square$
Note: You can earn partial credit on this problem.

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Problem 1992

Find all horizontal asymptotes of the function $f(x) = \frac{1-e^{x}}{1+2 e^{x}}$ .

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Problem 1993

```latex \text{The maximum error obtained by doing four bisection steps of a continuous function over } I=[2,4] \text{ is less than} ```

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Problem 1994

The maximum error obtained by doing four bisection steps of a continuous function over $I=[2,4]$ is less than
Answer: $\square$

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Problem 1995

find $y^{\prime}$ given: $y^{3}-x \sin y+\frac{y^{2}}{x}=8$ $y^{\prime}=\frac{x^{2} \cdot \sin y+y^{2}}{3 x^{2} y^{2}+x^{3} \cos y+2 x y}$ $y^{\prime}=\frac{x^{2} \cdot \sin y+y^{2}}{3 x^{2} y^{2}-x^{3} \cos y+2 x y}$ $y^{\prime}=\frac{3 x^{2} y^{2}+x^{3} \cos y+2 x y}{x^{2} \cdot \sin y+y^{2}}$ $y^{\prime}=\frac{3 x^{2} y^{2}-x^{3} \cos y+2 x y}{x^{2} \cdot \sin y+y^{2}}$

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Problem 1996

Find $y^{\prime \prime}$ if $x^{3} y^{3}-4=0$ $\frac{y}{x^{2}}$ $\frac{2 x}{y^{2}}$ $\frac{2 y}{x^{2}}$

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Problem 1997

Find the values of $(x)$ which the function $y=\frac{1}{1+e^{\frac{1}{x}}}$ dicontinuous?

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Problem 1998

Set up the definite integral that gives the area of the shaded region. Do not evaluate the integral
The definite integral is $\int_{\square}^{6}(\square \mathrm{dx}$ .

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Problem 1999

Use a Riemann sum to approximate the area under the graph of $f(x)=4 x^{2}$ on the interval $-2 \leq x \leq 2$ using $n=4$ subintervals with the selected points as the midpoints.
The area is approximately $\square$

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Problem 2000

Use the flux form of Green's Theorem to find the outward flux of $\mathbf{F}=\langle 10 x, 19 y\rangle$ across the boundary of the circle $x^{2}+y^{2}=1$ .
Flux = $\square$

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Calculus

Problem 1901

Problem 1902

Problem 1903

Problem 1904

Problem 1905

Problem 1906

Problem 1907

Problem 1908

Problem 1909

Problem 1910

Problem 1911

Calculate the integral: ∫t2(t3−3)3dt\int t^{2}(t^{3}-3)^{3} dt∫t2(t3−3)3dt

Problem 1912

Find the integral of the inverse cosine function: ∫cos⁡−1x dx\int \cos^{-1} x \, dx∫cos−1xdx.

Problem 1913

Evaluate the integral ∫e2θsin⁡3θdθ\int e^{2 \theta} \sin 3 \theta d \theta∫e2θsin3θdθ.

Problem 1914

Find the tangent line equation for f(x)=3−xf(x)=\sqrt{3-x}f(x)=3−x​ at a given point.

Problem 1915

Find the derivative of the function y=x4+12xy=x^{4}+12xy=x4+12x. What is dydx\frac{dy}{dx}dxdy​?

Problem 1916

Find lim⁡x→∞cos⁡(1+πxx)\lim _{x \rightarrow \infty} \cos \left(\frac{1+\pi x}{x}\right)limx→∞​cos(x1+πx​).

Problem 1917

Find the limit as xxx approaches 0 for cos⁡(1+πxx)\cos \left(\frac{1+\pi x}{x}\right)cos(x1+πx​).

Problem 1918

Calculate the integral ∫02(x2+1)e−xdx\int_{0}^{2}\left(x^{2}+1\right) e^{-x} dx∫02​(x2+1)e−xdx.

Problem 1919

Calculate the integral from 0 to 7 of (x2+1)e−x(x^{2}+1) e^{-x}(x2+1)e−x.

Problem 1920

Find the limit as xxx approaches 4 for the piecewise function f(x)f(x)f(x) defined as: f(x)=−7x+46f(x) = -7x + 46f(x)=−7x+46 if x<4x < 4x<4, 121212 if x=4x = 4x=4, 2x2−142x^2 - 142x2−14 if x>4x > 4x>4. What is lim⁡x→4f(x)\lim_{x \rightarrow 4} f(x)limx→4​f(x)?

Problem 1921

A ball is thrown with a velocity of 41ft/s41 \mathrm{ft} / \mathrm{s}41ft/s. Its height is y=41t−22t2y=41 t-22 t^{2}y=41t−22t2. Find average velocity from t=2t=2t=2 and instantaneous velocity at t=2t=2t=2.

Problem 1922

Find the rate of change of the city's population modeled by P(t)=22e0.08tP(t)=22 e^{0.08 t}P(t)=22e0.08t from 2025 to 2034.

Problem 1923

Calculate the integral from 1 to 2 of w2ln⁡w dww^{2} \ln w \, dww2lnwdw.

Problem 1924

Calculate the integral ∫02πt2sin⁡2t dt\int_{0}^{2 \pi} t^{2} \sin 2 t \, dt∫02π​t2sin2tdt.

Problem 1925

Find the average rate of change of COVID-19 infections from 0 to 8 days, assuming 0 at day 0. Round to two decimals.

Problem 1926

Find the integral of the function exe^{\sqrt{x}}ex​ with respect to xxx: ∫exdx\int e^{\sqrt{x}} d x∫ex​dx.

Problem 1927

A ball is thrown with a velocity of 41ft/s41 \mathrm{ft/s}41ft/s. Height after ttt seconds is y=41t−22t2y=41t-22t^2y=41t−22t2. Find average velocity from t=2t=2t=2 for 0.01, 0.005, and 0.002 seconds. What is the instantaneous velocity at t=2t=2t=2?

Problem 1928

Encuentra la derivada de f(x)=3x3−xf(x) = 3x^3 - xf(x)=3x3−x.

Problem 1929

Approximate the average rate of change of COVID-19 infections in Minnesota from 0 to 8 days after April 1, 2020.

Problem 1930

Find the integral: ∫x2ln⁡(x) dx\int x^{2} \ln (x) \, dx∫x2ln(x)dx.

Problem 1931

Find the rate of change of the city's population, modeled by P(t)=10e0.04tP(t)=10 e^{0.04 t}P(t)=10e0.04t, from 2021 to 2030 in thousand persons/year.

Problem 1932

Calculate the average rates of change for COVID-19 infections in Minnesota over these intervals using ΔyΔx\frac{\Delta y}{\Delta x}ΔxΔy​:1. 0 to 8 days: (calculate)2. 8 to 16 days: 0.59 thousand/day3. 0 to 16 days: 0.46 thousand/day

Problem 1933

Find the average velocity of a ball thrown with y=41t−22t2y=41t-22t^{2}y=41t−22t2 at t=2t=2t=2 for intervals of 0.01, 0.005, 0.002, and 0.001 seconds.

Problem 1934

Calculate the average rate of change of the function k(x)=−16xk(x)=-16 \sqrt{x}k(x)=−16x​ from x=12x=12x=12 to x=15x=15x=15.

Problem 1935

Problem 1936

A particle PPP has a velocity of v=24−4t2v=24-4 t^{2}v=24−4t2 m/s. Find: (a) distance in the first second, (b) when it changes direction, (c) when it returns to start.

Problem 1937

A particle PPP moves along a line with displacement x=13t3−4t2+15tx = \frac{1}{3} t^{3} - 4 t^{2} + 15 tx=31​t3−4t2+15t. Find: (a) when PPP is at rest, (b) distance in 5 seconds.

Problem 1938

Find the distance from the origin when the particle stops, given v=9t−3t2v=9t-3t^{2}v=9t−3t2 m/s and starts at t=0t=0t=0.

Problem 1939

Does the function f(x)=xf(x)=xf(x)=x have a limit as xxx approaches 3 from all real numbers except 3?

Problem 1940

A ball is thrown with a velocity of 41ft/s41 \mathrm{ft/s}41ft/s. Its height after ttt seconds is y=41t−22t2y=41t-22t^2y=41t−22t2. Find average velocity for t=2t=2t=2 over 0.01, 0.005, 0.002, and 0.001 seconds, then determine instantaneous velocity at t=2t=2t=2.

Problem 1941

A ball is thrown with a velocity of 41ft/s41 \mathrm{ft/s}41ft/s. Its height is y=41t−22t2y=41t-22t^2y=41t−22t2. Find average velocity from t=2t=2t=2 for given intervals and the instantaneous velocity at t=2t=2t=2.

Problem 1942

Calculate the integral ∫054x28x dx\int_{0}^{5} 4 x^{2} 8 x \, dx∫05​4x28xdx.

Problem 1943

Find lim⁡x→2f(x)\lim _{x \rightarrow 2} f(x)limx→2​f(x) given that lim⁡x→2[f(x)]2−8x+3x+1=9\lim _{x \rightarrow 2} \sqrt{\frac{[f(x)]^{2}-8 x+3}{x+1}}=9limx→2​x+1[f(x)]2−8x+3​​=9 and f(x)≥0f(x) \geq 0f(x)≥0.

Problem 1944

Find the limit as xxx approaches 3 for the expression x−1/2(5x−7)1/3x^{-1/2}(5x-7)^{1/3}x−1/2(5x−7)1/3.

Problem 1945

Find the limit: lim⁡x→3x2−x−6x−3\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}limx→3​x−3x2−x−6​.

Calculate the integral: $\int t^{2}(t^{3}-3)^{3} dt$

Find the integral of the inverse cosine function: $\int \cos^{-1} x \, dx$ .

Evaluate the integral $\int e^{2 \theta} \sin 3 \theta d \theta$ .

Find the tangent line equation for $f(x)=\sqrt{3-x}$ at a given point.

Find the derivative of the function $y=x^{4}+12x$ . What is $\frac{dy}{dx}$ ?

Find $\lim _{x \rightarrow \infty} \cos \left(\frac{1+\pi x}{x}\right)$ .

Find the limit as $x$ approaches 0 for $\cos \left(\frac{1+\pi x}{x}\right)$ .

Calculate the integral $\int_{0}^{2}\left(x^{2}+1\right) e^{-x} dx$ .

Calculate the integral from 0 to 7 of $(x^{2}+1) e^{-x}$ .

Find the limit as $x$ approaches 4 for the piecewise function $f(x)$ defined as: $f(x) = -7x + 46$ if $x < 4$ , $12$ if $x = 4$ , $2x^2 - 14$ if $x > 4$ . What is $\lim_{x \rightarrow 4} f(x)$ ?

A ball is thrown with a velocity of $41 \mathrm{ft} / \mathrm{s}$ . Its height is $y=41 t-22 t^{2}$ . Find average velocity from $t=2$ and instantaneous velocity at $t=2$ .

Find the rate of change of the city's population modeled by $P(t)=22 e^{0.08 t}$ from 2025 to 2034.

Calculate the integral from 1 to 2 of $w^{2} \ln w \, dw$ .

Calculate the integral $\int_{0}^{2 \pi} t^{2} \sin 2 t \, dt$ .

Find the integral of the function $e^{\sqrt{x}}$ with respect to $x$ : $\int e^{\sqrt{x}} d x$ .

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Height after $t$ seconds is $y=41t-22t^2$ . Find average velocity from $t=2$ for 0.01, 0.005, and 0.002 seconds. What is the instantaneous velocity at $t=2$ ?

Encuentra la derivada de $f(x) = 3x^3 - x$ .

Find the integral: $\int x^{2} \ln (x) \, dx$ .

Find the rate of change of the city's population, modeled by $P(t)=10 e^{0.04 t}$ , from 2021 to 2030 in thousand persons/year.

Calculate the average rates of change for COVID-19 infections in Minnesota over these intervals using $\frac{\Delta y}{\Delta x}$ :
1. 0 to 8 days: (calculate)
2. 8 to 16 days: 0.59 thousand/day
3. 0 to 16 days: 0.46 thousand/day

Find the average velocity of a ball thrown with $y=41t-22t^{2}$ at $t=2$ for intervals of 0.01, 0.005, 0.002, and 0.001 seconds.

Calculate the average rate of change of the function $k(x)=-16 \sqrt{x}$ from $x=12$ to $x=15$ .

A particle $P$ has a velocity of $v=24-4 t^{2}$ m/s. Find: (a) distance in the first second, (b) when it changes direction, (c) when it returns to start.

A particle $P$ moves along a line with displacement $x = \frac{1}{3} t^{3} - 4 t^{2} + 15 t$ . Find: (a) when $P$ is at rest, (b) distance in 5 seconds.

Find the distance from the origin when the particle stops, given $v=9t-3t^{2}$ m/s and starts at $t=0$ .

Does the function $f(x)=x$ have a limit as $x$ approaches 3 from all real numbers except 3?

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Its height after $t$ seconds is $y=41t-22t^2$ . Find average velocity for $t=2$ over 0.01, 0.005, 0.002, and 0.001 seconds, then determine instantaneous velocity at $t=2$ .

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Its height is $y=41t-22t^2$ . Find average velocity from $t=2$ for given intervals and the instantaneous velocity at $t=2$ .

Calculate the integral $\int_{0}^{5} 4 x^{2} 8 x \, dx$ .

Find $\lim _{x \rightarrow 2} f(x)$ given that $\lim _{x \rightarrow 2} \sqrt{\frac{[f(x)]^{2}-8 x+3}{x+1}}=9$ and $f(x) \geq 0$ .

Find the limit as $x$ approaches 3 for the expression $x^{-1/2}(5x-7)^{1/3}$ .

Find the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$ .

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

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}$ .

Find the limit as $x$ approaches 0 for the expression $\frac{\sin x}{x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{5+x}-\frac{1}{5}}{x}$ .

Find the limit as $x$ approaches -1 for the expression $-2x^3 - x + 5$ .

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

Find the limit: $\lim _{x \rightarrow 4} \sqrt{1+2 x}$ .

Find the limit as $x$ approaches 0 for the expression $e^{-x} \cos x$ .

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

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

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

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3}{x^{2}}$ , where $h \neq 0$ .

The function $f$ is shown below. If $g$ is the function defined by $g(x)=\int_{-2}^{x} f(t) d t$ , what is the value of $g(9)$ ?

Given $\frac{d y}{d t}=7 y$ and $y(0)=450$ , determine $y(t)$ $y(t)=$ $\square$

Problem 1. (1 point) $\int_{0}^{2} \int_{0}^{5} e^{5 x+2 y} d x d y=\square$ (Give the exact value, not a decimal approximation)

$\int_{0}^{4} \frac{p}{\sqrt{9+p^{2}}} d p$

Question 1 (Multiple Choice Worth 5 points) (02.07 MC)
Given a function $f(x)=2 x^{2}+3$ , what is the average rate of change of $f$ on the interval $[2,2+h] ?$ 11 $2 h+8$ $2 h^{2}+8 h$ $2 h^{2}+8 h+11$

Question 12
You deposit $\$ 5000$ in an account earning $7 \%$ interest compounded continuously. How much will you have in the account in 10 years? \\square$ Next Question

Relatel Ratas
17. The volume of a cube decreases at a rate of $10 \mathrm{~m}^{3} / \mathrm{s}$ . Find the rate at which the side of the cube changes when the side of the cube is 2 m .

How long will it take for an invertment to triple, if intcrest is compounded continously at 4 t - ?
It will take? years before the invectment trives
Round to the reares tentn of a year

16. Determine the value of $f(x)$ when $x \rightarrow-\infty$ if $f(x)=\frac{3 x+2}{2 x+3}$ a) $-\frac{2}{3}$ b) $\frac{2}{3}$ c) $1.5^{+}$ d) $1.5^{-}$

Evaluate the definite integral $\int_{2}^{4} x(x-2)^{3} d x$ (Enter a numerical value. Round your answer to 2 decimal places as needed.)

Find $d y / d x$ by implicit differentiation. $d y / d x=\square \quad \sin x+\cos y=\sin x \cos y$

Problem 5. (1 point)
Find the equation of the line tangent to the graph of $y=2 \ln (x)$ at $x=1$ . Tangent Line: $y=$ $\square$

For the region formed by the functions $y=x^{3}$ and $y=x$ , use definite integrals to find the area of the region.
Answer: The area is $\square$ - (Use a fraction.)