Calculus

Problem 901

Find $f(3)$ given that $f^{\prime}(x) = \frac{x-8}{x-4}$ for $x \neq 4$ .

See Solution

Problem 902

Find the derivative of the function $f(x)=2 x^{-10}+7 x^{-2}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 903

Simplify the expression: $\frac{(x+\Delta x)^{2}-9(x+\Delta x)+8-(x^{2}-9 x+8)}{\Delta x}$ .

See Solution

Problem 904

Find the net electric flux from a cube with edge $6.6 \mathrm{~m}$ in the field $\vec{E}=2 y \hat{j}$ N/C. Options: a. 1030, b. 575, c. 862, d. 287, e. 460 N.m²/C.

See Solution

Problem 905

1. Calculate the volume of a solid with a base between $y=x^{2}$ and the $x$ -axis from $x=0$ to $x=2$ , with square cross sections.
2. Calculate the volume of a solid with a base between $y=\sec x$ and the $x$ -axis from $x=\pi / 4$ to $x=\pi / 3$ , with square cross sections.

See Solution

Problem 906

Find the limit as $x$ approaches infinity: $x\left\{\tan ^{-1}\left[\frac{(x+1)}{(x-2)}\right]-\frac{\pi}{4}\right\}$ .

See Solution

Problem 907

Find the volume when the regions are revolved around the $y$ -axis for the curves: (a) $y=\ln x$ , $x=0$ , $y=0$ , $y=1$ . (b) $x=1-y^{2}$ , $x=2+y^{2}$ , $y=-1$ , $y=1$ . (c) $x=y^{2}$ , $x=y+2$ . (d) $x=\csc y$ , $y=\pi / 4$ , $x=0$ , $y=3\pi / 4$ .

See Solution

Problem 908

Find the limit as $n \to \infty$ of $\frac{\lfloor x \rfloor + \lfloor 2^2 x \rfloor + \ldots + \lfloor n^2 x \rfloor}{n^3}$ .

See Solution

Problem 909

Find $b$ (0 < $b$ < 2) such that the volume $V$ of the solid formed by revolving the region bounded by $y=1/x$ , $y=0$ , $x=2$ , and $x=b$ about the $x$ -axis equals 3.

See Solution

Problem 910

Find the limit: $\lim _{n \rightarrow \infty}(1+x)(1+x^{2})(1+x^{4}) \cdots(1+x^{2^{n}})$ for $|x|<1$ .

See Solution

Problem 911

Find the integrals:
1. $\int (x^{4}+3x-9) \, dx$
2. $\int (x^{4}+3x) \, dx - 9$
3. $\int x^{4} \, dx + 3x - 9$

See Solution

Problem 912

Find the partial derivative of $f(x, y)=x^3 y^2+3x e^y$ with respect to $y$ . Options: a, b, c.

See Solution

Problem 913

Find the first order partial derivative of $f(x, y)=x^3 y^2 + 3x e^y$ with respect to $y$ .

See Solution

Problem 914

Calculate the integral: $\int \frac{1+x}{1+x^{2}} \, dx$

See Solution

Problem 915

Evaluate the integral $\int_{0}^{\pi / 6} \frac{\sin t}{\cos ^{2} t} d t$ .

See Solution

Problem 916

곡선 $y=\frac{4}{x}$ 위의 점 A(1,4)와 B(t, $\frac{4}{t}$ )를 지나는 직선의 삼각형 OPB 넓이 $S(t)$ 의 $\lim _{t \rightarrow \infty} S(t)$ 값은?

See Solution

Problem 917

Evaluate the integral $\int_{0}^{a} x \sqrt{a^{2}-x^{2}} \, dx$ .

See Solution

Problem 918

Find the time $t$ for a vehicle to accelerate from $50 \mathrm{~km/h}$ to $140 \mathrm{~km/h}$ using $t=\int_{50}^{140} \frac{2}{700-3t} dt$ .

See Solution

Problem 919

Find the integral $\int \frac{x^{2}}{x^{3}+9} d x$ and provide the result in terms of $c$ .

See Solution

Problem 920

Find the integral of $\sqrt{x} \ln x$ with respect to $x$ : $\int \sqrt{x} \ln x \, dx$ .

See Solution

Problem 921

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}$ . Options: (A) $\frac{1}{2 \pi}$ (B) $\frac{1}{\pi}$ (C) 1 (D) nonexistent.

See Solution

Problem 922

Find the derivative of $y=(2 x+9)(x^{3}-1)$ , i.e., calculate $\frac{d y}{d x}$ .

See Solution

Problem 923

Calculate the derivative $\frac{d y}{d x}$ for the function $y=\frac{x}{1-5 x}$ .

See Solution

Problem 924

Find the derivative $\frac{d y}{d x}$ for $y=(3 x^{2}+5)^{10}$ .

See Solution

Problem 925

Find the derivative of $y=\frac{3}{(2 x-1)^{4}}$ with respect to $x$ : $\frac{d y}{d x}$ .

See Solution

Problem 926

Differentiate the function $y=(3x+1)^{5}$ .

See Solution

Problem 927

Differentiate $y=\left(3 x^{3}-\frac{2}{x}\right)^{2}$ with respect to $x$ .

See Solution

Problem 928

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

See Solution

Problem 929

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

See Solution

Problem 930

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

See Solution

Problem 931

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

See Solution

Problem 932

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

See Solution

Problem 933

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

See Solution

Problem 934

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

See Solution

Problem 935

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

See Solution

Problem 936

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

See Solution

Problem 937

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 $\lim_{x \rightarrow 4} f(x)$ ?

See Solution

Problem 938

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 $t=2$ .

See Solution

Problem 939

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

See Solution

Problem 940

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

See Solution

Problem 941

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

See Solution

Problem 942

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

See Solution

Problem 943

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

See Solution

Problem 944

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 $t=2$ ?

See Solution

Problem 945

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

See Solution

Problem 946

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

See Solution

Problem 947

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

See Solution

Problem 948

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.

See Solution

Problem 949

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

See Solution

Problem 950

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.

See Solution

Problem 951

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

See Solution

Problem 952

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.

See Solution

Problem 953

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.

See Solution

Problem 954

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.

See Solution

Problem 955

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

See Solution

Problem 956

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

See Solution

Problem 957

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 $t=2$ .

See Solution

Problem 958

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 $t=2$ .

See Solution

Problem 959

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

See Solution

Problem 960

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 $f(x) \geq 0$ .

See Solution

Problem 961

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

See Solution

Problem 962

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

See Solution

Problem 963

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

See Solution

Problem 964

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

See Solution

Problem 965

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

See Solution

Problem 966

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

See Solution

Problem 967

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

See Solution

Problem 968

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

See Solution

Problem 969

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

See Solution

Problem 970

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

See Solution

Problem 971

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

See Solution

Problem 972

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

See Solution

Problem 973

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

See Solution

Problem 974

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

See Solution

Problem 975

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

See Solution

Problem 976

Bestimme die erste und zweite Ableitung der Funktionen: a) $f(x)=e^{x}+1$ , b) $f(x)=e^{x}+x$ , c) $f(x)=e^{x}+2 x^{2}$ , d) $f(x)=-e^{x}+1$ , e) $f(x)=2 e^{x}+3 x^{2}$ , f) $f(x)=-5 e^{x}-0,5 x^{3}$ , g) $f(x)=-\frac{1}{2}(e^{x}-x^{3})$ , h) $f(x)=\frac{1}{4}e^{x}+\sin(x)$ .

See Solution

Problem 977

Bestimmen Sie die erste Ableitung für die folgenden Funktionen und überprüfen Sie mit dem GTR: a) $2 x \cdot(4 x-1)$ , b) $(5 x+3) \cdot(x+2)$ , c) $(2-5 x) \cdot(x+2)$ , d) $2 x \cdot e^{x}$ , e) $(4 x+2) \cdot e^{x}$ , f) $(6 x+1) \cdot e^{x}$ .

See Solution

Problem 978

Find the marginal cost $M C$ and average cost $A C$ from the total cost function $T C=300 \ln (q+30)+150$ .

See Solution

Problem 979

Find the tangent line equation for $f(x) = \sqrt{x}$ at the point (1,1).

See Solution

Problem 980

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{[8-9(4+\Delta x)]-(-28)}{\Delta x}$ .

See Solution

Problem 981

Find the function $f(x)$ and the number $c$ given the limit: $\lim _{\Delta x \rightarrow 0} \frac{[8-9(4+\Delta x)]-(-28)}{\Delta x}$

See Solution

Problem 982

An object is launched with an initial velocity of 48 ft/s from 80 ft. Find average velocity from $t=0$ to $t=2$ and $t=2$ to $t=4$ . Use $h(t)=-16 t^{2}+48 t+80$ .

See Solution

Problem 983

연속 함수 $f(x)$ 가 주어지고, $g(x)=\int_{0}^{x} f(t) d t-\int_{x}^{4} f(t) d t$ 가 $x=2$ 에서 0일 때, $\int_{\frac{1}{2}}^{4} f(x) d x$ 의 값은?

See Solution

Problem 984

Find the limit as $x$ approaches $\frac{11}{10}$ from the right: $\lim _{x \rightarrow \frac{11}{10}^{+}}\left(\frac{15 x}{11-10 x}\right)$ .

See Solution

Problem 985

Find $F^{\prime}(x)$ using the first principle if $F(x)=x^{2}-3x$ .

See Solution

Problem 986

Find the average rate of change of $h(t)=\cot t$ over the intervals: a. $\left[\frac{3 \pi}{4}, \frac{5 \pi}{4}\right]$ b. $\left[\frac{5 \pi}{6}, \frac{3 \pi}{2}\right]$

See Solution

Problem 987

Calculate the average rate of change of $h(t)=\cot t$ over the intervals: a. $\left[\frac{3 \pi}{4}, \frac{5 \pi}{4}\right]$ , b. $\left[\frac{5 \pi}{6}, \frac{3 \pi}{2}\right]$ .

See Solution

Problem 988

Find the limits: 1) $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}-3 x}$ 2) $\lim _{x \rightarrow 3} \frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}}$

See Solution

Problem 989

Find the average rate of change of $h(t)=\cot t$ over $\left[\frac{5 \pi}{6}, \frac{3 \pi}{2}\right]$ .

See Solution

Problem 990

Find the limit as $x$ approaches 3 for the expression $\frac{x^{2}-9}{x^{2}-3x}$ .

See Solution

Problem 991

Simplify the expression: $\frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}}$

See Solution

Problem 992

Differentiate $3xy = \sqrt{16x} - \frac{3}{x^3}$ with respect to $x$ .

See Solution

Problem 993

Differentiate $y=\frac{6 x^{4}-x+5}{3 x^{2}}$ and express the answer with positive exponents.

See Solution

Problem 994

Find $\lim _{x \rightarrow 3} f(x)$ for the piecewise function: $f(x) = -9 + 4x$ (for $x<3$ ) and $f(x) = -6 + x^{2}$ (for $x>3$ ).

See Solution

Problem 995

Find $\lim _{x \rightarrow 3} f(x)$ for the piecewise function $f(x)=\begin{cases} x^{2}-7 & x>3 \\ -4+2x & x<3 \end{cases}$ .

See Solution

Problem 996

You have \$400,000 saved at 4\% interest. How much can you withdraw monthly for 20 years?

See Solution

Problem 997

Test if the integral $\int_{1}^{\infty} \frac{2+\cos x}{x} d x$ converges.

See Solution

Problem 998

Find the sum of the series: $\sum_{n=0}^{\infty} \frac{n^{10}}{10^{n}}$ .

See Solution

Problem 999

Find the limit: $\lim _{x \rightarrow 0^{+}} x^{x}$ .

See Solution

Problem 1000

Find the volume of the solid formed by rotating the area between $y=x^{3}+1$ and $y=1$ around the $y$ -axis.

See Solution

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Calculus

Problem 901

Find f(3)f(3)f(3) given that f′(x)=x−8x−4f^{\prime}(x) = \frac{x-8}{x-4}f′(x)=x−4x−8​ for x≠4x \neq 4x=4.

Problem 902

Find the derivative of the function f(x)=2x−10+7x−2f(x)=2 x^{-10}+7 x^{-2}f(x)=2x−10+7x−2. What is f′(x)f^{\prime}(x)f′(x)?

Problem 903

Simplify the expression: (x+Δx)2−9(x+Δx)+8−(x2−9x+8)Δx\frac{(x+\Delta x)^{2}-9(x+\Delta x)+8-(x^{2}-9 x+8)}{\Delta x}Δx(x+Δx)2−9(x+Δx)+8−(x2−9x+8)​.

Problem 904

Find the net electric flux from a cube with edge 6.6 m6.6 \mathrm{~m}6.6 m in the field E⃗=2yj^\vec{E}=2 y \hat{j}E=2yj^​ N/C. Options: a. 1030, b. 575, c. 862, d. 287, e. 460 N.m²/C.

Problem 905

Problem 906

Find the limit as xxx approaches infinity: x{tan⁡−1[(x+1)(x−2)]−π4}x\left\{\tan ^{-1}\left[\frac{(x+1)}{(x-2)}\right]-\frac{\pi}{4}\right\}x{tan−1[(x−2)(x+1)​]−4π​}.

Problem 907

Problem 908

Find the limit as n→∞n \to \inftyn→∞ of ⌊x⌋+⌊22x⌋+…+⌊n2x⌋n3\frac{\lfloor x \rfloor + \lfloor 2^2 x \rfloor + \ldots + \lfloor n^2 x \rfloor}{n^3}n3⌊x⌋+⌊22x⌋+…+⌊n2x⌋​.

Problem 909

Find bbb (0 < bbb < 2) such that the volume VVV of the solid formed by revolving the region bounded by y=1/xy=1/xy=1/x, y=0y=0y=0, x=2x=2x=2, and x=bx=bx=b about the xxx-axis equals 3.

Problem 910

Find the limit: lim⁡n→∞(1+x)(1+x2)(1+x4)⋯(1+x2n)\lim _{n \rightarrow \infty}(1+x)(1+x^{2})(1+x^{4}) \cdots(1+x^{2^{n}})limn→∞​(1+x)(1+x2)(1+x4)⋯(1+x2n) for ∣x∣<1|x|<1∣x∣<1.

Problem 911

Find the integrals: 1. ∫(x4+3x−9) dx\int (x^{4}+3x-9) \, dx∫(x4+3x−9)dx 2. ∫(x4+3x) dx−9\int (x^{4}+3x) \, dx - 9∫(x4+3x)dx−9 3. ∫x4 dx+3x−9\int x^{4} \, dx + 3x - 9∫x4dx+3x−9

Problem 912

Find the partial derivative of f(x,y)=x3y2+3xeyf(x, y)=x^3 y^2+3x e^yf(x,y)=x3y2+3xey with respect to yyy. Options: a, b, c.

Problem 913

Find the first order partial derivative of f(x,y)=x3y2+3xeyf(x, y)=x^3 y^2 + 3x e^yf(x,y)=x3y2+3xey with respect to yyy.

Problem 914

Calculate the integral: ∫1+x1+x2 dx\int \frac{1+x}{1+x^{2}} \, dx∫1+x21+x​dx

Problem 915

Evaluate the integral ∫0π/6sin⁡tcos⁡2tdt\int_{0}^{\pi / 6} \frac{\sin t}{\cos ^{2} t} d t∫0π/6​cos2tsint​dt.

Problem 916

곡선 y=4xy=\frac{4}{x}y=x4​ 위의 점 A(1,4)와 B(t, 4t\frac{4}{t}t4​)를 지나는 직선의 삼각형 OPB 넓이 S(t)S(t)S(t)의 lim⁡t→∞S(t)\lim _{t \rightarrow \infty} S(t)limt→∞​S(t) 값은?

Problem 917

Evaluate the integral ∫0axa2−x2 dx\int_{0}^{a} x \sqrt{a^{2}-x^{2}} \, dx∫0a​xa2−x2​dx.

Problem 918

Find the time ttt for a vehicle to accelerate from 50 km/h50 \mathrm{~km/h}50 km/h to 140 km/h140 \mathrm{~km/h}140 km/h using t=∫501402700−3tdtt=\int_{50}^{140} \frac{2}{700-3t} dtt=∫50140​700−3t2​dt.

Problem 919

Find the integral ∫x2x3+9dx\int \frac{x^{2}}{x^{3}+9} d x∫x3+9x2​dx and provide the result in terms of ccc.

Problem 920

Find the integral of xln⁡x\sqrt{x} \ln xx​lnx with respect to xxx: ∫xln⁡x dx\int \sqrt{x} \ln x \, dx∫x​lnxdx.

Problem 921

Find the limit: lim⁡x→πcos⁡x+sin⁡(2x)+1x2−π2\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}limx→π​x2−π2cosx+sin(2x)+1​. Options: (A) 12π\frac{1}{2 \pi}2π1​ (B) 1π\frac{1}{\pi}π1​ (C) 1 (D) nonexistent.

Problem 922

Find the derivative of y=(2x+9)(x3−1)y=(2 x+9)(x^{3}-1)y=(2x+9)(x3−1), i.e., calculate dydx\frac{d y}{d x}dxdy​.

Problem 923

Calculate the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x1−5xy=\frac{x}{1-5 x}y=1−5xx​.

Problem 924

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=(3x2+5)10y=(3 x^{2}+5)^{10}y=(3x2+5)10.

Problem 925

Find the derivative of y=3(2x−1)4y=\frac{3}{(2 x-1)^{4}}y=(2x−1)43​ with respect to xxx: dydx\frac{d y}{d x}dxdy​.

Problem 926

Differentiate the function y=(3x+1)5y=(3x+1)^{5}y=(3x+1)5.

Problem 927

Differentiate y=(3x3−2x)2y=\left(3 x^{3}-\frac{2}{x}\right)^{2}y=(3x3−x2​)2 with respect to xxx.

Problem 928

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

Problem 929

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

Problem 930

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

Problem 931

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

Problem 932

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

Problem 933

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

Problem 934

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

Problem 935

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 936

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

Problem 937

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 938

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 939

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 940

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

Problem 941

Find $f(3)$ given that $f^{\prime}(x) = \frac{x-8}{x-4}$ for $x \neq 4$ .

Find the derivative of the function $f(x)=2 x^{-10}+7 x^{-2}$ . What is $f^{\prime}(x)$ ?

Simplify the expression: $\frac{(x+\Delta x)^{2}-9(x+\Delta x)+8-(x^{2}-9 x+8)}{\Delta x}$ .

Find the net electric flux from a cube with edge $6.6 \mathrm{~m}$ in the field $\vec{E}=2 y \hat{j}$ N/C. Options: a. 1030, b. 575, c. 862, d. 287, e. 460 N.m²/C.

Find the limit as $x$ approaches infinity: $x\left\{\tan ^{-1}\left[\frac{(x+1)}{(x-2)}\right]-\frac{\pi}{4}\right\}$ .

Find the limit as $n \to \infty$ of $\frac{\lfloor x \rfloor + \lfloor 2^2 x \rfloor + \ldots + \lfloor n^2 x \rfloor}{n^3}$ .

Find $b$ (0 < $b$ < 2) such that the volume $V$ of the solid formed by revolving the region bounded by $y=1/x$ , $y=0$ , $x=2$ , and $x=b$ about the $x$ -axis equals 3.

Find the limit: $\lim _{n \rightarrow \infty}(1+x)(1+x^{2})(1+x^{4}) \cdots(1+x^{2^{n}})$ for $|x|<1$ .

Find the integrals:
1. $\int (x^{4}+3x-9) \, dx$
2. $\int (x^{4}+3x) \, dx - 9$
3. $\int x^{4} \, dx + 3x - 9$

Find the partial derivative of $f(x, y)=x^3 y^2+3x e^y$ with respect to $y$ . Options: a, b, c.

Find the first order partial derivative of $f(x, y)=x^3 y^2 + 3x e^y$ with respect to $y$ .

Calculate the integral: $\int \frac{1+x}{1+x^{2}} \, dx$

Evaluate the integral $\int_{0}^{\pi / 6} \frac{\sin t}{\cos ^{2} t} d t$ .

곡선 $y=\frac{4}{x}$ 위의 점 A(1,4)와 B(t, $\frac{4}{t}$ )를 지나는 직선의 삼각형 OPB 넓이 $S(t)$ 의 $\lim _{t \rightarrow \infty} S(t)$ 값은?

Evaluate the integral $\int_{0}^{a} x \sqrt{a^{2}-x^{2}} \, dx$ .

Find the time $t$ for a vehicle to accelerate from $50 \mathrm{~km/h}$ to $140 \mathrm{~km/h}$ using $t=\int_{50}^{140} \frac{2}{700-3t} dt$ .

Find the integral $\int \frac{x^{2}}{x^{3}+9} d x$ and provide the result in terms of $c$ .

Find the integral of $\sqrt{x} \ln x$ with respect to $x$ : $\int \sqrt{x} \ln x \, dx$ .

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x+\sin (2 x)+1}{x^{2}-\pi^{2}}$ . Options: (A) $\frac{1}{2 \pi}$ (B) $\frac{1}{\pi}$ (C) 1 (D) nonexistent.

Find the derivative of $y=(2 x+9)(x^{3}-1)$ , i.e., calculate $\frac{d y}{d x}$ .

Calculate the derivative $\frac{d y}{d x}$ for the function $y=\frac{x}{1-5 x}$ .

Find the derivative $\frac{d y}{d x}$ for $y=(3 x^{2}+5)^{10}$ .

Find the derivative of $y=\frac{3}{(2 x-1)^{4}}$ with respect to $x$ : $\frac{d y}{d x}$ .

Differentiate the function $y=(3x+1)^{5}$ .

Differentiate $y=\left(3 x^{3}-\frac{2}{x}\right)^{2}$ with respect to $x$ .

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 $h \neq 0$ .