Calculus

Problem 30901

Find the limit: $\lim_{{x \to \frac{\pi}{4}^+}} \frac{\tan(2x)}{1+\sec(2x)}$ .

See Solution

Problem 30902

Find the general antiderivative of $g(\theta)=\cos \theta-3 \sin \theta$ and check by differentiation. Use $C$ for the constant.

See Solution

Problem 30903

Find the electric field $E$ inside and outside a solid sphere of radius $R$ with volume charge density $\rho(s)=k s$ .

See Solution

Problem 30904

Find the age of an artifact with $3.125\%$ C-14, given its half-life is 5730 years.

See Solution

Problem 30905

Find $\frac{d^{2} y}{d x^{2}}$ at the point $(1,3)$ for the equation $x^{2} y + y x^{2} = 6$ . Options: -6, 12, -18, 18, 6.

See Solution

Problem 30906

Find the general antiderivative of $f(u)=\frac{u^{4}+4 \sqrt{u}}{u^{2}}$ and verify by differentiation. Use $C$ .

See Solution

Problem 30907

Find $\lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4}$ using a table of values.

See Solution

Problem 30908

Find the general antiderivative of $f(x)=7 e^{x}+6 \sec ^{2} x$ . Use $C$ for the constant. Verify by differentiation.

See Solution

Problem 30909

Find the general antiderivative of $f(x)=7 \sqrt{x}+5 \cos x$ and verify by differentiation using constant $C$ .

See Solution

Problem 30910

Solve the initial value problem: $\frac{d y}{d t}=t^{2}+1$ , $y(0)=3$ . Find $y(t)$ for $t \geq 0$ .

See Solution

Problem 30911

Find the limit as $x$ approaches -1 for the expression $-\sqrt[3]{x+3}$ .

See Solution

Problem 30912

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{\sqrt[3]{x}-1}$ .

See Solution

Problem 30913

Find $f^{\prime}(1)$ if $f(x)=\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}$ .

See Solution

Problem 30914

Solve the initial value problem: $\frac{dy}{dt} = t^2 + 1$ , $y(0) = 8$ . Find $y(t)$ for $t \geq 0$ .

See Solution

Problem 30915

Identify critical numbers of $g(x)$ from $g^{\prime}(x)$ and $g^{\prime \prime}(x)$ ; classify each as min, max, or neither.

See Solution

Problem 30916

Find the derivative of $y=4 x^{3}-4 x-6$ at $x=1$ . What is $\left.\frac{d y}{d x}\right|_{x=1}=?$

See Solution

Problem 30917

Evaluate the integral $\int x \cos 2 x \, dx$ .

See Solution

Problem 30918

Solve the initial value problem: $\frac{d P}{d t}=2 e^{3 t}$ , $P(0)=4$ . Find $P(t)$ for $t \geq 0$ .

See Solution

Problem 30919

Find all inflection points of $g(x)$ given the behavior of $g'(x)$ and $g''(x)$ across intervals.

See Solution

Problem 30920

Explain how the instantaneous rate of change of $y=\operatorname{alog}(k(x-d))+c$ varies with parameters $(a, k, d, c)$ .

See Solution

Problem 30921

Find $f$ given that $f^{\prime \prime}(x)=12 x+24 x^{2}$ . Use constants $C$ and $D$ .

See Solution

Problem 30922

Find the derivative of the function $f(x)=2 x^{5/2}$ and evaluate it at $x=9$ : $f^{\prime}(9)=$

See Solution

Problem 30923

Use a graph to find the approximate $x$ -coordinates where $y=4 x \sin(x^{2})$ and $y=4 x^{4}$ intersect, then estimate the area between them. Round to two decimal places.

See Solution

Problem 30924

Find all $x$ where the slope of the tangent line to $f(x)=3x^{2}-x+4$ equals 0. What is $x=$ ?

See Solution

Problem 30925

Solve the initial value problem: $\frac{d P}{d t}=2 e^{3 t}$ with $P(0)=4$ . Find $P(t)$ for $t \geq 0$ .

See Solution

Problem 30926

Find the limits:
1. $\lim_{x \rightarrow -2} \frac{x^{5}-3x^{3}-4x}{x^{3}+x^{2}-4x-4}$
2. $\lim_{x \rightarrow +\infty} (x e^{-x})$
3. $\lim_{x \rightarrow 0} \frac{-4x^{3}}{x+\sin 2x}$
4. $\lim_{x \rightarrow 0^{+}} (x \ln x)$

See Solution

Problem 30927

Evaluate the integral $\int_{1}^{2} x \ln x \, dx$ .

See Solution

Problem 30928

Find the distance in 30 seconds for an object with velocity $v(t)=20+8 \cos (t)$ ft/s. Use precalculus or calculus?

See Solution

Problem 30929

Find the function $f$ given that $f''(x) = 4x + \sin x$ . Use constants $C$ and $D$ for the antiderivatives.

See Solution

Problem 30930

Differentiate the function $f(x) = 3x^{2}(2x^{2}+1)^{2}$ .

See Solution

Problem 30931

Find the derivative of $g(t)=3 t^{3}+3 t^{2}$ and evaluate it at $t=-3$ : $g^{\prime}(-3)=$

See Solution

Problem 30932

Find the rate of change of elevation for the function $f(x)=0.09 x$ at $x=4$ miles. Does it need calculus?

See Solution

Problem 30933

Solve the initial value problem: $\frac{d P}{d t}=2 e^{3 t}$ with $P(0)=9$ . Find $P(t)$ for $t \geq 0$ .

See Solution

Problem 30934

A bicyclist's path is given by $f(x)=0.04(10x-x^{2})$ . Find the elevation change rate at $x=1$ . Calculus needed.

See Solution

Problem 30935

Find the limit as $x$ approaches 0 from the left: $\lim _{x \rightarrow 0^{-}}\left(e^{3 x}-1\right) \csc 2 x$ .

See Solution

Problem 30936

Find $f$ given that $f^{\prime \prime}(x)=4 x+\sin x$ . Use $C$ and $D$ for constants of the antiderivatives.

See Solution

Problem 30937

Find the function $f$ given that $f''(\theta) = \sin \theta + \cos \theta$ , $f(0) = 2$ , and $f'(0) = 2$ .

See Solution

Problem 30938

A culture of Rhodobacter sphaeroides starts with 20 bacteria and grows at a rate of $3.4657 e^{0.1386 t}$ . Find the population after 4 hours.

See Solution

Problem 30939

Find the function $f$ given that $f^{\prime \prime}(x)=6-18 x$ , $f(0)=5$ , and $f(2)=5$ .

See Solution

Problem 30940

A stone is dropped from a tower $470 \mathrm{~m}$ high.
(a) Find $s(t)$ , the height at time $t$ , with $g \approx 9.8 \mathrm{~m/s}^2$ . (b) How long to reach the ground? (c) What is the impact velocity?

See Solution

Problem 30941

Determine the intervals where $g(x)$ is concave up and down using the information from $g^{\prime}(x)$ and $g^{\prime \prime}(x)$ .

See Solution

Problem 30942

Evaluate the integral: $\int_{1 / 2}^{e / 2} \frac{1}{x[\ln (2 x)+3]^{2}} d x$

See Solution

Problem 30943

Find the limit: $\lim _{x \rightarrow+\infty} e^{3 x} \ln \left(e^{-2 x}+1\right)$ .

See Solution

Problem 30944

Integrate: $\int x^{8} \sqrt[3]{2 x^{3}+3} d x$

See Solution

Problem 30945

Find the limit: $\lim _{x \rightarrow \infty} \operatorname{sech}(x)$ .

See Solution

Problem 30946

Find the limit as $x$ approaches 0 from the right for $\operatorname{coth}(x)$ .

See Solution

Problem 30947

Evaluate the integral $\int x^{8} \sqrt[3]{2 x^{3}+3} d x$ .

See Solution

Problem 30948

Find the nonzero value of $x$ where the second derivative of $f(x)=4x^{6}-9x^{5}$ equals zero. Provide a decimal answer with three decimal places.

See Solution

Problem 30949

Find the limits: (d) $\lim _{x \rightarrow-\infty} \sinh (x)$ , (e) $\lim _{x \rightarrow \infty} \operatorname{sech}(x)$ , (f) $\lim _{x \rightarrow \infty} \operatorname{coth}(x)$ .

See Solution

Problem 30950

Find the limit as $x$ approaches 0 from the left: $\lim _{x \rightarrow 0^{-}} \operatorname{coth}(x)$ .

See Solution

Problem 30951

Find the function $f$ given that $f''(x)=8-18x$ , $f(0)=5$ , and $f(2)=7$ .

See Solution

Problem 30952

Estimate the area under $f(x)=4 \cos (x)$ from $x=0$ to $x=\pi / 2$ using 4 rectangles and right endpoints. Round to 4 decimal places. Sketch the graph and rectangles.

See Solution

Problem 30953

Find the limits: (a) $\lim _{x \rightarrow \infty} \tanh (x)$ , (b) $\lim _{x \rightarrow-\infty} \tanh (x)$ , (c) $\lim _{x \rightarrow \infty} \sinh (x)$ .

See Solution

Problem 30954

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sinh (x)}{e^{x}}$ .

See Solution

Problem 30955

Find the limit: $\lim _{x \rightarrow-\infty} \operatorname{csch}(x)$ .

See Solution

Problem 30956

Estimate the area under $f(x)=4 \cos (x)$ from $x=0$ to $x=\pi / 2$ using 4 rectangles and right endpoints. Area: $R_{4}=3.1641$ . Sketch the graph and rectangles.

See Solution

Problem 30957

Find the general antiderivative of $f(x)=x^{3}-9$ and the specific one with $F(3)=5$ . What is $F(0)$ ?

See Solution

Problem 30958

Calculate the integral $\int_{1}^{5}(2 \arctan (0.2 x))^{2} d x$ .

See Solution

Problem 30959

Estimate the area under $f(x)=7+4 x^{2}$ from $x=-1$ to $x=2$ using 3 rectangles ( $R_{3}$ ) and then 6 rectangles ( $R_{6}$ ).

See Solution

Problem 30960

Calculate the integral: $\int \sqrt[3]{\cos ^{5}(2 x)} \sin ^{3}(2 x) \, dx$

See Solution

Problem 30961

Find the tangent line equation for $f(x)=x^{4}+2 x^{2}$ at the point where $f^{\prime}(x)=1$ .

See Solution

Problem 30962

Find the area represented by the integral $\int_{0}^{10}|x-5| d x$ . What is its value?

See Solution

Problem 30963

A bacterium culture starts with 25 bacteria. After 9 hours, the growth rate is $3.4657e^{0.1386t}$ . Find the total population.

See Solution

Problem 30964

Calculate the left endpoint Riemann sum for $f(x)=\frac{x^{2}}{9}$ on $[3,7]$ using 8 subintervals. What is the value?

See Solution

Problem 30965

Determine if these functions are valid in quantum mechanics: $x^{2}$ and $e^{-x^{2}}$ . Normalize if valid.

See Solution

Problem 30966

Find right endpoints $x_{1}, x_{2}, x_{3}$ in terms of $h$ , general expression for $x_{k}$ , $f(x_{k})$ , and Riemann sums.

See Solution

Problem 30967

Find the limit $\lim _{x \rightarrow \infty} G(x)$ for the antiderivative $G(x)$ of $g(x)=e^{-7x}$ with $G(0)=6$ . Round to three decimal places.

See Solution

Problem 30968

A stone is dropped from a 560 m tower. Find $s(t)$ , time to hit the ground, and impact velocity. Use $g \approx 9.8 \mathrm{~m/s}^2$ .

See Solution

Problem 30969

Solve the equation $s^{\prime \prime}(t)=\cos (t)-\sin (t)$ with $s(0)=6$ , $s^{\prime}(0)=10$ , then find $s(\pi)$ .

See Solution

Problem 30970

Estimate $\int_{0}^{6} f(x) dx$ using 3 equal intervals: (a) right endpoints $R_{3}$ , (b) left endpoints $L_{3}$ , (c) midpoints $M_{3}$ . If $f(x)$ is decreasing, are your estimates less or greater than the exact value?

See Solution

Problem 30971

Differentiate $y=\sinh^{-1}(x)$ to find $\frac{dy}{dx}$ using $\cosh^2(y)-\sinh^2(y)=1$ . What is $\frac{dy}{dx}$ ?

See Solution

Problem 30972

Find the right endpoints $x_{1}, x_{2}, x_{3}$ for intervals and express in terms of $n$ . Then find $x_{k}$ , $f(x_{k})$ , $f(x_{k}) \Delta x$ , Riemann sum, and its limit.

See Solution

Problem 30973

Guess the instantaneous velocity of a ball at $t=5$ sec given its height function $h(t)=15t-4.9t^{2}$ .

See Solution

Problem 30974

Find the average velocity of the ball for the intervals: $[4.99,5]$ , $[5,5.01]$ , $[4.999,5]$ , $[5,5.001]$ using $S(t) = 200 - 4.9t^2$ and $v_{avg} = \frac{S(t2) - S(t1)}{t2 - t1}$ .

See Solution

Problem 30975

Find the right-endpoint Riemann sum value in terms of $n$ :
$\sum_{k=1}^{n} f\left(x_{k}\right) \Delta x=9+\left(\frac{81 n(n+1)}{n^{2}}\right)+\left(\frac{729 n(n+1)(2 n+1)}{6 n^{3}}\right)$
Then, compute the limit as $n$ approaches infinity:
$\lim _{n \rightarrow \infty}\left(\sum_{k=1}^{n} f\left(x_{k}\right) \Delta x\right)=\square$

See Solution

Problem 30976

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\sqrt{13x}$ , with $h \neq 0$ . Simplify fully.

See Solution

Problem 30977

Evaluate the integral $\int_{0}^{6} f(x) \, dx$ where $f(x)=9-3x$ for $0 \leq x<3$ and $f(x)=3x-9$ for $3 \leq x \leq 6$ .

See Solution

Problem 30978

Evaluate the integral $\int_{0}^{6} f(x) \, dx$ where $f(x)=9-3x$ for $0 \leq x<3$ and $f(x)=3x-9$ for $3 \leq x \leq 6$ . Find the signed area.

See Solution

Problem 30979

Find an integral for $\boldsymbol{k}$ to split area, length of $y=1+6 x^{3/2}$ from $[0,1]$ , and volume of solid with rectangle cross sections.

See Solution

Problem 30980

Estimate the total infection from $f(t) = -i(t - 21)(t + 1)$ using 7 midpoints in $[0, 21]$ ; peak time is $t_p = 14$ .

See Solution

Problem 30981

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}-9x+3$ , where $h \neq 0$ . Simplify your answer.

See Solution

Problem 30982

Calculate the integral of $I=\int x e^{5 x} dx$ .

See Solution

Problem 30983

Evaluate the integral I = ∫₀² t e^{-t} dt.

See Solution

Problem 30984

Find the indefinite integral of $I=\int e^{-x} \sin 3 x \, dx$ .

See Solution

Problem 30985

Evaluate the integral: $\int \sqrt[5]{\cos ^{3}(2 x)} \sin ^{3}(2 x) d x$

See Solution

Problem 30986

Bestimme den Flächeninhalt zwischen dem Graphen von $f(x) = -x^2 + 6x - 8$ und der $x$ -Achse.

See Solution

Problem 30987

Find the value of $\int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} d x$ . Choose from: (A) $\frac{22}{7}-\pi$ , (B) $\frac{2}{105}$ , (C) 0, (D) $\frac{71}{15}-\frac{3 \pi}{2}$ .

See Solution

Problem 30988

Estimate the total infection using the Midpoint Riemann Sum for $f(t) = -t(t - 21)(t + 1)$ over intervals: $[0, 11], [11, 14], [14, 18], [18, 21]$ .

See Solution

Problem 30989

5 a Explain why $8-4+2-1+\frac{1}{2}-\frac{1}{4}+\ldots = 4+1+\frac{1}{4}+\ldots$ without evaluating. b Evaluate both sums to verify. 6 Find the sum of $5+\frac{5}{\sqrt{2}}+\frac{5}{2}+\frac{5}{2\sqrt{2}}+\ldots$ in the form $a+b\sqrt{2}$ . 7 Prove that $\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+\frac{5}{64}+\ldots = 1$ .

See Solution

Problem 30990

Find the general antiderivative of $f(x)=x^{3}-6$ and the specific $F(x)$ with $F(3)=5$ . What is $F(0)$ ?

See Solution

Problem 30991

Evaluate the integral: $\int 2 x \cdot \cos \left(x^{2}-2\right) d x$

See Solution

Problem 30992

Solve the equation $s^{\prime \prime}(t)=\cos (t)-\sin (t)$ with $s(0)=7$ , $s^{\prime}(0)=7$ . Find $s(\pi)$ .

See Solution

Problem 30993

Zeichne die Graphen von $f(x)=x^{3}-3 x^{2}$ und $f(x)=0,25 x^{4}-1,505 x^{2}+1,98 x-3$ sowie deren Ableitungen.

See Solution

Problem 30994

Bestimme die Stammfunktion von $2 x \cdot \cos \left(x^{2}-2\right)$ .

See Solution

Problem 30995

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

See Solution

Problem 30996

Bestimme das Integral von $2 x \cdot \cos(x^{2}-2) \, dx$ .

See Solution

Problem 30997

Calculate the integral: $\int \frac{2 x-1}{x^{2}-x} d x$

See Solution

Problem 30998

Determine if these statements about the function $f(x)=\frac{1}{\sqrt{1-x^{2}}}$ are True (T) or False (F): 1. $f(x)$ is never positive. 2. $f(x)$ is never zero. 3. 0 is in the domain of $f$ . 4. 1 is in the domain of $f$ . 5. All positive real numbers are in the domain of $f$ . 6. $f(x)$ is never negative. 7. All negative real numbers are in the domain of $f$ .

See Solution

Problem 30999

Berechnen Sie die Ableitungen für: a) $f(x)=x^{4}+x^{2}$ , b) $f(x)=x^{n}+2 x^{3}$ , c) $f(x)=0,25(x^{4}-x)$ , d) $f(x)=\frac{4}{x}$ .

See Solution

Problem 31000

Solve the differential equation $s^{\prime \prime}(t)=\cos (t)-\sin (t)$ with $s(0)=10$ , $s^{\prime}(0)=3$ . Find $s(\pi)$ .

See Solution

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Calculus

Problem 30901

Find the limit: lim⁡x→π4+tan⁡(2x)1+sec⁡(2x)\lim_{{x \to \frac{\pi}{4}^+}} \frac{\tan(2x)}{1+\sec(2x)}limx→4π​+​1+sec(2x)tan(2x)​.

Problem 30902

Find the general antiderivative of g(θ)=cos⁡θ−3sin⁡θg(\theta)=\cos \theta-3 \sin \thetag(θ)=cosθ−3sinθ and check by differentiation. Use CCC for the constant.

Problem 30903

Find the electric field EEE inside and outside a solid sphere of radius RRR with volume charge density ρ(s)=ks\rho(s)=k sρ(s)=ks.

Problem 30904

Find the age of an artifact with 3.125%3.125\%3.125% C-14, given its half-life is 5730 years.

Problem 30905

Find d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ at the point (1,3)(1,3)(1,3) for the equation x2y+yx2=6x^{2} y + y x^{2} = 6x2y+yx2=6. Options: -6, 12, -18, 18, 6.

Problem 30906

Find the general antiderivative of f(u)=u4+4uu2f(u)=\frac{u^{4}+4 \sqrt{u}}{u^{2}}f(u)=u2u4+4u​​ and verify by differentiation. Use CCC.

Problem 30907

Find lim⁡x→4x−2x−4\lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4}limx→4​x−4x​−2​ using a table of values.

Problem 30908

Find the general antiderivative of f(x)=7ex+6sec⁡2xf(x)=7 e^{x}+6 \sec ^{2} xf(x)=7ex+6sec2x. Use CCC for the constant. Verify by differentiation.

Problem 30909

Find the general antiderivative of f(x)=7x+5cos⁡xf(x)=7 \sqrt{x}+5 \cos xf(x)=7x​+5cosx and verify by differentiation using constant CCC.

Problem 30910

Solve the initial value problem: dydt=t2+1\frac{d y}{d t}=t^{2}+1dtdy​=t2+1, y(0)=3y(0)=3y(0)=3. Find y(t)y(t)y(t) for t≥0t \geq 0t≥0.

Problem 30911

Find the limit as xxx approaches -1 for the expression −x+33-\sqrt[3]{x+3}−3x+3​.

Problem 30912

Find the limit: lim⁡x→1x−1x3−1\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{\sqrt[3]{x}-1}limx→1​3x​−1x​−1​.

Problem 30913

Find f′(1)f^{\prime}(1)f′(1) if f(x)=1x+1x2+1x3f(x)=\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}f(x)=x1​+x21​+x31​.

Problem 30914

Solve the initial value problem: dydt=t2+1\frac{dy}{dt} = t^2 + 1dtdy​=t2+1, y(0)=8y(0) = 8y(0)=8. Find y(t)y(t)y(t) for t≥0t \geq 0t≥0.

Problem 30915

Identify critical numbers of g(x)g(x)g(x) from g′(x)g^{\prime}(x)g′(x) and g′′(x)g^{\prime \prime}(x)g′′(x); classify each as min, max, or neither.

Problem 30916

Find the derivative of y=4x3−4x−6y=4 x^{3}-4 x-6y=4x3−4x−6 at x=1x=1x=1. What is dydx∣x=1=?\left.\frac{d y}{d x}\right|_{x=1}=?dxdy​∣∣​x=1​=?

Problem 30917

Evaluate the integral ∫xcos⁡2x dx\int x \cos 2 x \, dx∫xcos2xdx.

Problem 30918

Solve the initial value problem: dPdt=2e3t\frac{d P}{d t}=2 e^{3 t}dtdP​=2e3t, P(0)=4P(0)=4P(0)=4. Find P(t)P(t)P(t) for t≥0t \geq 0t≥0.

Problem 30919

Find all inflection points of g(x) g(x) g(x) given the behavior of g′(x) g'(x) g′(x) and g′′(x) g''(x) g′′(x) across intervals.

Problem 30920

Explain how the instantaneous rate of change of y=alog⁡(k(x−d))+cy=\operatorname{alog}(k(x-d))+cy=alog(k(x−d))+c varies with parameters (a,k,d,c)(a, k, d, c)(a,k,d,c).

Problem 30921

Find fff given that f′′(x)=12x+24x2f^{\prime \prime}(x)=12 x+24 x^{2}f′′(x)=12x+24x2. Use constants CCC and DDD.

Problem 30922

Find the derivative of the function f(x)=2x5/2f(x)=2 x^{5/2}f(x)=2x5/2 and evaluate it at x=9x=9x=9: f′(9)=f^{\prime}(9)=f′(9)=

Problem 30923

Use a graph to find the approximate xxx-coordinates where y=4xsin⁡(x2)y=4 x \sin(x^{2})y=4xsin(x2) and y=4x4y=4 x^{4}y=4x4 intersect, then estimate the area between them. Round to two decimal places.

Problem 30924

Find all xxx where the slope of the tangent line to f(x)=3x2−x+4f(x)=3x^{2}-x+4f(x)=3x2−x+4 equals 0. What is x=x=x=?

Problem 30925

Solve the initial value problem: dPdt=2e3t\frac{d P}{d t}=2 e^{3 t}dtdP​=2e3t with P(0)=4P(0)=4P(0)=4. Find P(t)P(t)P(t) for t≥0t \geq 0t≥0.

Problem 30926

Problem 30927

Evaluate the integral ∫12xln⁡x dx\int_{1}^{2} x \ln x \, dx∫12​xlnxdx.

Problem 30928

Find the distance in 30 seconds for an object with velocity v(t)=20+8cos⁡(t)v(t)=20+8 \cos (t)v(t)=20+8cos(t) ft/s. Use precalculus or calculus?

Problem 30929

Find the function fff given that f′′(x)=4x+sin⁡xf''(x) = 4x + \sin xf′′(x)=4x+sinx. Use constants CCC and DDD for the antiderivatives.

Problem 30930

Differentiate the function f(x)=3x2(2x2+1)2f(x) = 3x^{2}(2x^{2}+1)^{2}f(x)=3x2(2x2+1)2.

Problem 30931

Find the derivative of g(t)=3t3+3t2g(t)=3 t^{3}+3 t^{2}g(t)=3t3+3t2 and evaluate it at t=−3t=-3t=−3: g′(−3)=g^{\prime}(-3)=g′(−3)=

Problem 30932

Find the rate of change of elevation for the function f(x)=0.09xf(x)=0.09 xf(x)=0.09x at x=4x=4x=4 miles. Does it need calculus?

Problem 30933

Solve the initial value problem: dPdt=2e3t\frac{d P}{d t}=2 e^{3 t}dtdP​=2e3t with P(0)=9P(0)=9P(0)=9. Find P(t)P(t)P(t) for t≥0t \geq 0t≥0.

Problem 30934

A bicyclist's path is given by f(x)=0.04(10x−x2)f(x)=0.04(10x-x^{2})f(x)=0.04(10x−x2). Find the elevation change rate at x=1x=1x=1. Calculus needed.

Problem 30935

Find the limit as xxx approaches 0 from the left: lim⁡x→0−(e3x−1)csc⁡2x\lim _{x \rightarrow 0^{-}}\left(e^{3 x}-1\right) \csc 2 xlimx→0−​(e3x−1)csc2x.

Problem 30936

Find fff given that f′′(x)=4x+sin⁡xf^{\prime \prime}(x)=4 x+\sin xf′′(x)=4x+sinx. Use CCC and DDD for constants of the antiderivatives.

Problem 30937

Find the function fff given that f′′(θ)=sin⁡θ+cos⁡θf''(\theta) = \sin \theta + \cos \thetaf′′(θ)=sinθ+cosθ, f(0)=2f(0) = 2f(0)=2, and f′(0)=2f'(0) = 2f′(0)=2.

Problem 30938

A culture of Rhodobacter sphaeroides starts with 20 bacteria and grows at a rate of 3.4657e0.1386t3.4657 e^{0.1386 t}3.4657e0.1386t. Find the population after 4 hours.

Problem 30939

Find the function fff given that f′′(x)=6−18xf^{\prime \prime}(x)=6-18 xf′′(x)=6−18x, f(0)=5f(0)=5f(0)=5, and f(2)=5f(2)=5f(2)=5.

Problem 30940

A stone is dropped from a tower 470 m470 \mathrm{~m}470 m high. (a) Find s(t)s(t)s(t), the height at time ttt, with g≈9.8 m/s2g \approx 9.8 \mathrm{~m/s}^2g≈9.8 m/s2. (b) How long to reach the ground? (c) What is the impact velocity?

Find the limit: $\lim_{{x \to \frac{\pi}{4}^+}} \frac{\tan(2x)}{1+\sec(2x)}$ .

Find the general antiderivative of $g(\theta)=\cos \theta-3 \sin \theta$ and check by differentiation. Use $C$ for the constant.

Find the electric field $E$ inside and outside a solid sphere of radius $R$ with volume charge density $\rho(s)=k s$ .

Find the age of an artifact with $3.125\%$ C-14, given its half-life is 5730 years.

Find $\frac{d^{2} y}{d x^{2}}$ at the point $(1,3)$ for the equation $x^{2} y + y x^{2} = 6$ . Options: -6, 12, -18, 18, 6.

Find the general antiderivative of $f(u)=\frac{u^{4}+4 \sqrt{u}}{u^{2}}$ and verify by differentiation. Use $C$ .

Find $\lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4}$ using a table of values.

Find the general antiderivative of $f(x)=7 e^{x}+6 \sec ^{2} x$ . Use $C$ for the constant. Verify by differentiation.

Find the general antiderivative of $f(x)=7 \sqrt{x}+5 \cos x$ and verify by differentiation using constant $C$ .

Solve the initial value problem: $\frac{d y}{d t}=t^{2}+1$ , $y(0)=3$ . Find $y(t)$ for $t \geq 0$ .

Find the limit as $x$ approaches -1 for the expression $-\sqrt[3]{x+3}$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{\sqrt[3]{x}-1}$ .

Find $f^{\prime}(1)$ if $f(x)=\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}$ .

Solve the initial value problem: $\frac{dy}{dt} = t^2 + 1$ , $y(0) = 8$ . Find $y(t)$ for $t \geq 0$ .

Identify critical numbers of $g(x)$ from $g^{\prime}(x)$ and $g^{\prime \prime}(x)$ ; classify each as min, max, or neither.

Find the derivative of $y=4 x^{3}-4 x-6$ at $x=1$ . What is $\left.\frac{d y}{d x}\right|_{x=1}=?$

Evaluate the integral $\int x \cos 2 x \, dx$ .

Solve the initial value problem: $\frac{d P}{d t}=2 e^{3 t}$ , $P(0)=4$ . Find $P(t)$ for $t \geq 0$ .

Find all inflection points of $g(x)$ given the behavior of $g'(x)$ and $g''(x)$ across intervals.

Explain how the instantaneous rate of change of $y=\operatorname{alog}(k(x-d))+c$ varies with parameters $(a, k, d, c)$ .

Find $f$ given that $f^{\prime \prime}(x)=12 x+24 x^{2}$ . Use constants $C$ and $D$ .

Find the derivative of the function $f(x)=2 x^{5/2}$ and evaluate it at $x=9$ : $f^{\prime}(9)=$

Use a graph to find the approximate $x$ -coordinates where $y=4 x \sin(x^{2})$ and $y=4 x^{4}$ intersect, then estimate the area between them. Round to two decimal places.

Find all $x$ where the slope of the tangent line to $f(x)=3x^{2}-x+4$ equals 0. What is $x=$ ?

Solve the initial value problem: $\frac{d P}{d t}=2 e^{3 t}$ with $P(0)=4$ . Find $P(t)$ for $t \geq 0$ .

Evaluate the integral $\int_{1}^{2} x \ln x \, dx$ .

Find the distance in 30 seconds for an object with velocity $v(t)=20+8 \cos (t)$ ft/s. Use precalculus or calculus?

Find the function $f$ given that $f''(x) = 4x + \sin x$ . Use constants $C$ and $D$ for the antiderivatives.

Differentiate the function $f(x) = 3x^{2}(2x^{2}+1)^{2}$ .

Find the derivative of $g(t)=3 t^{3}+3 t^{2}$ and evaluate it at $t=-3$ : $g^{\prime}(-3)=$

Find the rate of change of elevation for the function $f(x)=0.09 x$ at $x=4$ miles. Does it need calculus?

Solve the initial value problem: $\frac{d P}{d t}=2 e^{3 t}$ with $P(0)=9$ . Find $P(t)$ for $t \geq 0$ .

A bicyclist's path is given by $f(x)=0.04(10x-x^{2})$ . Find the elevation change rate at $x=1$ . Calculus needed.

Find the limit as $x$ approaches 0 from the left: $\lim _{x \rightarrow 0^{-}}\left(e^{3 x}-1\right) \csc 2 x$ .

Find $f$ given that $f^{\prime \prime}(x)=4 x+\sin x$ . Use $C$ and $D$ for constants of the antiderivatives.

Find the function $f$ given that $f''(\theta) = \sin \theta + \cos \theta$ , $f(0) = 2$ , and $f'(0) = 2$ .

A culture of Rhodobacter sphaeroides starts with 20 bacteria and grows at a rate of $3.4657 e^{0.1386 t}$ . Find the population after 4 hours.

Find the function $f$ given that $f^{\prime \prime}(x)=6-18 x$ , $f(0)=5$ , and $f(2)=5$ .

A stone is dropped from a tower $470 \mathrm{~m}$ high.
(a) Find $s(t)$ , the height at time $t$ , with $g \approx 9.8 \mathrm{~m/s}^2$ . (b) How long to reach the ground? (c) What is the impact velocity?

Determine the intervals where $g(x)$ is concave up and down using the information from $g^{\prime}(x)$ and $g^{\prime \prime}(x)$ .

Evaluate the integral: $\int_{1 / 2}^{e / 2} \frac{1}{x[\ln (2 x)+3]^{2}} d x$

Find the limit: $\lim _{x \rightarrow+\infty} e^{3 x} \ln \left(e^{-2 x}+1\right)$ .

Integrate: $\int x^{8} \sqrt[3]{2 x^{3}+3} d x$

Find the limit: $\lim _{x \rightarrow \infty} \operatorname{sech}(x)$ .

Find the limit as $x$ approaches 0 from the right for $\operatorname{coth}(x)$ .

Evaluate the integral $\int x^{8} \sqrt[3]{2 x^{3}+3} d x$ .

Find the nonzero value of $x$ where the second derivative of $f(x)=4x^{6}-9x^{5}$ equals zero. Provide a decimal answer with three decimal places.

Find the limit as $x$ approaches 0 from the left: $\lim _{x \rightarrow 0^{-}} \operatorname{coth}(x)$ .

Find the function $f$ given that $f''(x)=8-18x$ , $f(0)=5$ , and $f(2)=7$ .

Estimate the area under $f(x)=4 \cos (x)$ from $x=0$ to $x=\pi / 2$ using 4 rectangles and right endpoints. Round to 4 decimal places. Sketch the graph and rectangles.

Find the limits: (a) $\lim _{x \rightarrow \infty} \tanh (x)$ , (b) $\lim _{x \rightarrow-\infty} \tanh (x)$ , (c) $\lim _{x \rightarrow \infty} \sinh (x)$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sinh (x)}{e^{x}}$ .

Find the limit: $\lim _{x \rightarrow-\infty} \operatorname{csch}(x)$ .

Estimate the area under $f(x)=4 \cos (x)$ from $x=0$ to $x=\pi / 2$ using 4 rectangles and right endpoints. Area: $R_{4}=3.1641$ . Sketch the graph and rectangles.

Find the general antiderivative of $f(x)=x^{3}-9$ and the specific one with $F(3)=5$ . What is $F(0)$ ?

Calculate the integral $\int_{1}^{5}(2 \arctan (0.2 x))^{2} d x$ .

Estimate the area under $f(x)=7+4 x^{2}$ from $x=-1$ to $x=2$ using 3 rectangles ( $R_{3}$ ) and then 6 rectangles ( $R_{6}$ ).

Calculate the integral: $\int \sqrt[3]{\cos ^{5}(2 x)} \sin ^{3}(2 x) \, dx$

Find the tangent line equation for $f(x)=x^{4}+2 x^{2}$ at the point where $f^{\prime}(x)=1$ .

Find the area represented by the integral $\int_{0}^{10}|x-5| d x$ . What is its value?

A bacterium culture starts with 25 bacteria. After 9 hours, the growth rate is $3.4657e^{0.1386t}$ . Find the total population.

Calculate the left endpoint Riemann sum for $f(x)=\frac{x^{2}}{9}$ on $[3,7]$ using 8 subintervals. What is the value?

Determine if these functions are valid in quantum mechanics: $x^{2}$ and $e^{-x^{2}}$ . Normalize if valid.

Find right endpoints $x_{1}, x_{2}, x_{3}$ in terms of $h$ , general expression for $x_{k}$ , $f(x_{k})$ , and Riemann sums.

Find the limit $\lim _{x \rightarrow \infty} G(x)$ for the antiderivative $G(x)$ of $g(x)=e^{-7x}$ with $G(0)=6$ . Round to three decimal places.

A stone is dropped from a 560 m tower. Find $s(t)$ , time to hit the ground, and impact velocity. Use $g \approx 9.8 \mathrm{~m/s}^2$ .

Solve the equation $s^{\prime \prime}(t)=\cos (t)-\sin (t)$ with $s(0)=6$ , $s^{\prime}(0)=10$ , then find $s(\pi)$ .

Estimate $\int_{0}^{6} f(x) dx$ using 3 equal intervals: (a) right endpoints $R_{3}$ , (b) left endpoints $L_{3}$ , (c) midpoints $M_{3}$ . If $f(x)$ is decreasing, are your estimates less or greater than the exact value?

Differentiate $y=\sinh^{-1}(x)$ to find $\frac{dy}{dx}$ using $\cosh^2(y)-\sinh^2(y)=1$ . What is $\frac{dy}{dx}$ ?

Find the right endpoints $x_{1}, x_{2}, x_{3}$ for intervals and express in terms of $n$ . Then find $x_{k}$ , $f(x_{k})$ , $f(x_{k}) \Delta x$ , Riemann sum, and its limit.

Guess the instantaneous velocity of a ball at $t=5$ sec given its height function $h(t)=15t-4.9t^{2}$ .