Calculus

Problem 22901

Find the rate of change of the triangle's base when the altitude is $10 \mathrm{~cm}$ and area is $150 \mathrm{~cm}^{2}$ .

See Solution

Problem 22902

Find the tangent line equation for $f(x)=\tan^{2} x$ at the point $\left(-\frac{\pi}{4}, 1\right)$ .

See Solution

Problem 22903

Find the max and min values of $f(x)=x+\frac{9}{x}$ on $[0.2,12]$ . Min: $6$ , Max: $12.75$ .

See Solution

Problem 22904

Evaluate the integral: $\frac{d}{d t} \int_{0}^{t^{10}} \sqrt{u^{7}} d u = \frac{d}{d t}\left(\frac{2 t^{45}}{9}\right)$ .

See Solution

Problem 22905

Find the height increase rate (in $\mathrm{m} / \mathrm{min}$ ) for a tank with radius $3 \mathrm{~m}$ filled at $3 \mathrm{~m}^{3}/\mathrm{min}$ .

See Solution

Problem 22906

Find the derivatives of $f(x)=x^{n} \sin x$ for $n=1,2,3,4$ and derive a general formula for $f^{\prime}(x)$ .

See Solution

Problem 22907

Find the volume increase rate (in $\mathrm{mm}^{3} / \mathrm{s}$ ) when the diameter is $100 \mathrm{~mm}$ and radius grows at $4 \mathrm{~mm/s}$ .

See Solution

Problem 22908

Find the 4th derivative of the function $f^{(3)}(x)=\sqrt[3]{x^{2}}$ .

See Solution

Problem 22909

Find $\frac{d A}{d t}$ for a circle with radius $r$ expanding at $\frac{d r}{d t}$ . Given $\frac{d r}{d t}=2 \mathrm{~m/s}$ and $r=28 \mathrm{~m}$ , calculate the area increase rate in $\mathrm{m}^{2}/\mathrm{s}$ .

See Solution

Problem 22910

At 4 PM, how fast is the distance between ship A (sailing south at 25 km/h) and ship B (sailing north at 35 km/h) changing?

See Solution

Problem 22911

A rectangle's length increases by $5 \mathrm{~cm/s}$ and width by $4 \mathrm{~cm/s}$ . Find area increase rate when length is $13 \mathrm{~cm}$ and width is $5 \mathrm{~cm}$ .

See Solution

Problem 22912

Find $d y / d t$ for $y=4 x^{2}+3$ when $\frac{d x}{d t}=3$ cm/s at $x=-1, 0, 1$ .

See Solution

Problem 22913

Find $d y / d t$ and $d x / d t$ for $y=\sqrt{x}$ given $x=1$ , $d x / d t=5$ and $x=81$ , $d y / d t=6$ .

See Solution

Problem 22914

At 4:00 p.m., what is the distance rate change (in km/h) between ships A and B, given their speeds? Round to one decimal place.

See Solution

Problem 22915

Water leaks from a conical tank at $11,000 \mathrm{~cm}^{3} / \mathrm{min}$ . Find the inflow rate when water rises at $20 \mathrm{~cm} / \mathrm{min}$ at $2 \mathrm{~m}$ .

See Solution

Problem 22916

A sphere's radius $r$ increases at 3 in/min. Find volume change rate when $r=8$ in and $r=36$ in.

See Solution

Problem 22917

A worker pulls a 5m plank up a wall at 0.19 m/s. How fast is the plank's end sliding when 2m from the wall? Round to two decimals. $\mathrm{m} / \mathrm{sec}$

See Solution

Problem 22918

Sand falls onto a conical pile at 20 ft³/min. If the base diameter is 3 times the height, find $h^{\prime}$ when $h=2$ ft. $h^{\prime}=\square \mathrm{ft} / \mathrm{min}$

See Solution

Problem 22919

Two planes are converging at a point. One is 75 miles away at 300 mph, the other 100 miles away at 400 mph. Find: (a) Rate of distance change in $\mathrm{mph}$ (b) Time before a flight path change in h

See Solution

Problem 22920

Evaluate the integral $F(x)=\int_{x^{4}}^{x}(2 t-1)^{3} d t$ .

See Solution

Problem 22921

Find the volume of a cube with edge $15 \mathrm{~cm}$ and error $0.1 \mathrm{~cm}$ . Estimate max error, relative error, and percentage error.

See Solution

Problem 22922

A 12 ft long trough has a 3 ft top and isosceles triangle ends with 3 ft altitudes.
(a) Water is pumped in at 2 ft³/min. Find how fast the water level rises when $h = 1.1$ ft. $\mathrm{ft} / \mathrm{min}$
(b) If water rises at $3/8$ inch/min when $h=2.1$ , find the pumping rate. $\mathrm{ft}^{3} / \mathrm{min}$

See Solution

Problem 22923

Find the derivative at the extremum $\left(-\frac{2}{3}, \frac{10 \sqrt{3}}{9}\right)$ for $f(x)=-5 x \sqrt{x+1}$ .

See Solution

Problem 22924

Estimate the distance traveled from $t = 0$ to $t = 8$ using a left endpoint Riemann sum with given velocities.

See Solution

Problem 22925

Find the Taylor series for $f(x)=\ln x$ around the point $x=3$ .

See Solution

Problem 22926

A substance decays at $13.4\%$ daily. After 5 days, how much remains from an initial $90 \mathrm{mg}$ ? Round to two decimals.

See Solution

Problem 22927

A bear population grew by $50\%$ in 4 years. What formula finds the continuous annual growth rate using $n$ ?

See Solution

Problem 22928

Find the derivative of $g(x)=\int_{5 x}^{4 x} \frac{u+1}{u-5} du$ . What is $g'(x)$ ?

See Solution

Problem 22929

Find the critical numbers of the function $f(x)=6 x^{2}-9 x$ . Enter answers as a comma-separated list.

See Solution

Problem 22930

How many years will it take for a lion pride to grow to $275\%$ of its size at a continuous growth rate of $0.5\%$ ? Round up.

See Solution

Problem 22931

Find the absolute extrema of $y=3 \cos x$ on the interval $[0, 2\pi]$ . Minimum $(x,y)=(\square)$ , maximum $(x,y)=(\square)$ (smaller $x$ ), $(x,y)=(\square)$ (larger $x$ ).

See Solution

Problem 22932

Find the limit as $x$ approaches 2 for the expression $\frac{(x-2)(x-5)}{x^{2}+2x-8}$ .

See Solution

Problem 22933

Find the light intensity $I$ that maximizes photosynthesis given by $P(I)=\frac{100 I}{I^{2}+I+4}$ .

See Solution

Problem 22934

Find the absolute extrema of $f(x)=x^{2}-4 x$ on the intervals: (a) $[-1,4]$ , (b) $(2,5]$ , (c) $(0,4)$ , (d) $[2,6)$ .

See Solution

Problem 22935

Show that the definite integral $\int_{a}^{a} f(x) \, dx = 0$ using its definition.

See Solution

Problem 22936

Check if Rolle's Theorem applies to $f(x)=\cos x$ on $[\pi, 3\pi]$ . If yes, find $c$ where $f'(c)=0$ .

See Solution

Problem 22937

A ball's height after $t$ seconds is $f(t)=-16t^2+64t+5$ .
(a) Show $f(1)=f(3)$ . (b) Find the velocity at some time in $(1,3)$ using Rolle's Theorem. What is that time?

See Solution

Problem 22938

Can Rolle's Theorem apply to $f(x)=x^{2/3}-2$ on $[-27,27]$ ? If yes, find $c$ where $f'(c)=0$ . Enter NA if not applicable.

See Solution

Problem 22939

Find the limit as $u$ approaches $a$ for $\frac{2 \sqrt{u}+\sqrt{a}}{4 u-a}$ .

See Solution

Problem 22940

Find the limit as $x$ approaches infinity for the expression $\frac{2 \ln(x)^{\infty}}{x^{2}}$ .

See Solution

Problem 22941

Find the work to pump oil from a 15 ft parabolic tank ( $y=x^{2}$ ) half-full of oil ( $35 \mathrm{lbs/ft}^3$ ).

See Solution

Problem 22942

Can the Mean Value Theorem apply to $f(x)=\sqrt{9-x}$ on $[-7,9]$ ? If yes, find $c$ where $f'(c)=\frac{f(9)-f(-7)}{16}$ .

See Solution

Problem 22943

Evaluate the integral: $\int_{4}^{8} \frac{1}{8} x^{3} d x$ using given values.

See Solution

Problem 22944

Can the Mean Value Theorem be applied to $f(x)=2x^{3}$ on $[1,2]$ ? If yes, find $c$ where $f'(c)=\frac{f(2)-f(1)}{2-1}$ .

See Solution

Problem 22945

Find the average value of $f(x)=\frac{9}{x}$ on $[7, 7e]$ and graph it with the average value indicated.

See Solution

Problem 22946

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{-1}^{1}(2 x^{3}+4) dx=\square$

See Solution

Problem 22947

Find the area under the curve of $f(x)=8e^{-x}$ from 0 to an unknown upper limit using $\int_{0}^{\square} dx$ .

See Solution

Problem 22948

Evaluate the integral using the limit definition: $\int_{2}^{5} 7 \, dx$ .

See Solution

Problem 22949

Calculate the integral of $x$ from 0 to 8: $\int_{0}^{8} x \, dx$ .

See Solution

Problem 22950

Evaluate the definite integral using limits: $\int_{-7}^{7} x^{3} \, dx$ .

See Solution

Problem 22951

Sketch the area represented by the integral $\int_{0}^{2}(3 x+3) d x$ .

See Solution

Problem 22952

Determine if the function $f(x)=\frac{1}{(x+1)^{2}}$ is increasing or decreasing.

See Solution

Problem 22953

Evaluate the integral using symmetry: $\int_{-2}^{2}(3+x+x^{2}+x^{3}) dx=\square$ (Type an integer or a simplified fraction.)

See Solution

Problem 22954

Calculate the integral from -4 to 4 of the function $\sqrt{16 - x^{2}}$ .

See Solution

Problem 22955

Maximize the function $f(x)=x^{a}(1-x)^{b}$ for $0 \leq x \leq 1$ .

See Solution

Problem 22956

Evaluate the integral $\int_{4}^{8}(9 x^{3}-12 x+2) d x$ using given values.

See Solution

Problem 22957

Find $\frac{d y}{d x}$ using implicit differentiation for $\cos(2x + y) = 5y$ . Result: $\frac{d y}{d x} = \square$ .

See Solution

Problem 22958

Find the intervals where the function $y=x \sqrt{64-x^{2}}$ is increasing or decreasing. Use interval notation.

See Solution

Problem 22959

Evaluate the integral from 1/2 to 1 of $(x^{-3} - 13)$ . Find $\int_{1/2}^{1}(x^{-3}-13) dx=\square$ .

See Solution

Problem 22960

Calculate the integral $\int_{0}^{2}(3 x+3) d x$ .

See Solution

Problem 22961

E. coli grows exponentially. Start with 7000 cells. After 6 hours, predict cells using $N(t)=N_0 e^{(D-m)t}$ .
(a) Find $N(6)$ . (b) With 733,000 cells after 6 hours, can the difference be due to wrong $t_b$ or $t_m$ ? (c) If $t_m$ is correct, estimate $t_b$ to fit the data: $t_b=\square$ hours.

See Solution

Problem 22962

Evaluate the integral using symmetry: $\int_{-2}^{2} \frac{x^{5}+4 x^{7}}{x^{6}+5} d x=\square$ (Simplify your answer.)

See Solution

Problem 22963

Evaluate the integral $\int_{0}^{1}(x^{2}-2x+7)dx$ and check if it matches the graph's behavior. Choose A, B, C, or D.

See Solution

Problem 22964

Set up the integral for net area: $\int_{-3}^{\square}(9 - x^2) dx$ . What is the upper limit?

See Solution

Problem 22965

Find the value of $x$ that minimizes the average cost given $C(x)=10,000+250x+10x^2$ . Round to two decimals.

See Solution

Problem 22966

Calculate the integral from 0 to 8 of the function $\frac{x}{4}$ .

See Solution

Problem 22967

Evaluate the integral $\int_{2}^{2} x d x$ using given values: $\int_{2}^{6} x^{3} d x=320$ , $\int_{2}^{6} x d x=16$ , $\int_{2}^{6} d x=4$ .

See Solution

Problem 22968

Given the function $f(x)=6-|x-2|$ , find critical numbers, intervals of increase/decrease, and use the First Derivative Test for extrema.

See Solution

Problem 22969

Given the function $f(x)=\left\{\begin{array}{ll} 4 x+1, & x \leq-1 \\ x^{2}-4, & x>-1 \end{array}\right.$ , find critical numbers, intervals of increase/decrease, and relative extrema.

See Solution

Problem 22970

Find critical numbers of $f(x)=-2 x^{2}+12 x+2$ , intervals of increase/decrease, and relative extrema.

See Solution

Problem 22971

Find the net area and area above the $x$ -axis for $y=81-x^{2}$ . Set up the integral: $\int_{-9}^{9}(81-x^{2}) dx$ . Graph the function.

See Solution

Problem 22972

Find critical numbers of $f(x)=x^{1/3}+1$ . Determine intervals of increase/decrease and apply the First Derivative Test.

See Solution

Problem 22973

Analyze the function $f(x)=\frac{x}{2}+\cos x$ on $(0,2 \pi)$ .
(a) Determine where it is increasing or decreasing. (b) Use the First Derivative Test to find relative extrema: maximum $(x, y)=(\square)$ , minimum $(x, y)=(\square)$ .

See Solution

Problem 22974

Find the intervals where the graph of $f(x)=\frac{11}{x^{2}+3}$ is concave up or down. Use interval notation.

See Solution

Problem 22975

Find the inflection point of $f(x)=6-5x^{4}$ and describe concavity. $(x, y)=(\square)$ Use interval notation.

See Solution

Problem 22976

Find $\frac{d y}{d x}$ at the point $(1,1)$ for the equation $5 x^{2}+2 x^{2} y+y^{2}=8$ .

See Solution

Problem 22977

Estimate the area under $f(x)=3x+2$ from $x=0$ to $x=3$ using 6 rectangles and right endpoints. Round to 2 decimal places.

See Solution

Problem 22978

Find the intervals where the graph of $y=2 x-2 \tan x$ is concave up or down in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ .

See Solution

Problem 22979

Find the intervals where the graph of $g(x)=3 x^{2}-x^{3}$ is concave up or down. Use interval notation.

See Solution

Problem 22980

Find the area function $A(x)=\int_{-7}^{x} (t+7) dt$ and verify $A'(x)=f(x)$ . Graph $A(x)$ .

See Solution

Problem 22981

Find the inflection point of $f(x)=\sin \frac{x}{2}$ on $[0,4\pi]$ and describe its concavity using interval notation.

See Solution

Problem 22982

Find the inflection points of $f(x)=x+7 \cos x$ on $[0,2 \pi]$ . Also describe concavity in interval notation.

See Solution

Problem 22983

Find the inflection point of $f(x)=x \sqrt{x+9}$ and describe concavity in interval notation. $(x, y)=(\square)$

See Solution

Problem 22984

Estimate the area under $f(x) = 2x + 1$ on $[0, 2]$ using 4 rectangles.

See Solution

Problem 22985

Find relative extrema of $f(x)=x^{2/3}-2$ using the Second Derivative Test. Provide $(x, y)$ for max and min.

See Solution

Problem 22986

Find the area under $g(x) = 5x^2 + 5$ on $[1,3]$ using 8 rectangles with left and right endpoint methods.

See Solution

Problem 22987

How long (in seconds) does it take for Sally to hit the ground if launched from 50m high and lands 128m away?

See Solution

Problem 22988

Calculate $\Delta \mathrm{G}^{\circ}$ at 25°C for $\mathrm{PbCl}_{2}$ with Ksp = $1.7 \times 10^{-5}$ . Given $[\mathrm{Pb}^{2+}] = 0.020 \mathrm{M}$ and $[\mathrm{Cl}^{-}] = 0.65 \mathrm{M}$ , find $\Delta \mathrm{G}$ .

See Solution

Problem 22989

Find the area under $f(x)=2x+6$ on $[0,2]$ using 4 rectangles.

See Solution

Problem 22990

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty}\left(5+\frac{6}{x}\right)$ .

See Solution

Problem 22991

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sin (6 x)}{x}$ .

See Solution

Problem 22992

Find the limits as $x$ approaches infinity: (a) $\lim _{x \rightarrow \infty} \frac{x^{3}+9}{x^{4}-3}$ , (b) $\lim _{x \rightarrow \infty} \frac{x^{3}+9}{x^{3}-3}$ , (c) $\lim _{x \rightarrow \infty} \frac{x^{3}+9}{x^{2}-3}$ .

See Solution

Problem 22993

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x}{\sqrt{x^{2}-x}}$

See Solution

Problem 22994

Find the limit as $x$ approaches infinity: $\lim_{x \rightarrow \infty} \frac{4x^{2}+x}{6x^{3}+7x^{2}+x}$ .

See Solution

Problem 22995

Find $\lim _{x \rightarrow \infty} h(x)$ for: (a) $h(x)=\frac{4x^{2}+2x+4}{x}$ , (b) $h(x)=\frac{4x^{2}+2x+4}{x^{2}}$ , (c) $h(x)=\frac{4x^{2}+2x+4}{x^{3}}$ .

See Solution

Problem 22996

Find the growth rate $\frac{\mathrm{dH}}{\mathrm{dt}}$ for the formula $\mathrm{H}=-30.12+1.874 \mathrm{t}^{2}-0.8461 \mathrm{t}^{2} \log \mathrm{t}$ . Compare rates at $\mathrm{t}=8$ and $\mathrm{t}=36$ weeks. Also, evaluate the fractional growth rate $\frac{1}{\mathrm{H}} \frac{\mathrm{dH}}{\mathrm{dt}}$ at the same times.

See Solution

Problem 22997

Find the limit: $\lim _{x \rightarrow-\infty}\left(4 x+\sqrt{16 x^{2}-x}\right)$ . Use a graphing tool to check.

See Solution

Problem 22998

Analyze the function $y=\frac{x^{2}}{x^{2}+192}$ . Find intercepts, extrema, inflection points, and asymptotes.

See Solution

Problem 22999

Find the growth rate of head circumference $H$ for a fetus at age $t$ using $H=-30.12+1.874 t^2-0.8461 t^2 \log t$ .
(a) Calculate $\frac{dH}{dt}$ . (b) Compare $\frac{dH}{dt}$ at $t=8$ and $t=36$ weeks. (c) Compare fractional growth rate $\frac{1}{H} \frac{dH}{dt}$ at those ages.

See Solution

Problem 23000

Analyze the function $f(x)=\frac{x^{2}-11 x+67}{x-9}$ . Find intercepts, extrema, inflection points, and asymptotes.

See Solution

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Calculus

Problem 22901

Find the rate of change of the triangle's base when the altitude is 10 cm10 \mathrm{~cm}10 cm and area is 150 cm2150 \mathrm{~cm}^{2}150 cm2.

Problem 22902

Find the tangent line equation for f(x)=tan⁡2xf(x)=\tan^{2} xf(x)=tan2x at the point (−π4,1)\left(-\frac{\pi}{4}, 1\right)(−4π​,1).

Problem 22903

Find the max and min values of f(x)=x+9xf(x)=x+\frac{9}{x}f(x)=x+x9​ on [0.2,12][0.2,12][0.2,12]. Min: 666, Max: 12.7512.7512.75.

Problem 22904

Evaluate the integral: ddt∫0t10u7du=ddt(2t459)\frac{d}{d t} \int_{0}^{t^{10}} \sqrt{u^{7}} d u = \frac{d}{d t}\left(\frac{2 t^{45}}{9}\right)dtd​∫0t10​u7​du=dtd​(92t45​).

Problem 22905

Find the height increase rate (in m/min\mathrm{m} / \mathrm{min}m/min) for a tank with radius 3 m3 \mathrm{~m}3 m filled at 3 m3/min3 \mathrm{~m}^{3}/\mathrm{min}3 m3/min.

Problem 22906

Find the derivatives of f(x)=xnsin⁡xf(x)=x^{n} \sin xf(x)=xnsinx for n=1,2,3,4n=1,2,3,4n=1,2,3,4 and derive a general formula for f′(x)f^{\prime}(x)f′(x).

Problem 22907

Find the volume increase rate (in mm3/s\mathrm{mm}^{3} / \mathrm{s}mm3/s) when the diameter is 100 mm100 \mathrm{~mm}100 mm and radius grows at 4 mm/s4 \mathrm{~mm/s}4 mm/s.

Problem 22908

Find the 4th derivative of the function f(3)(x)=x23f^{(3)}(x)=\sqrt[3]{x^{2}}f(3)(x)=3x2​.

Problem 22909

Find dAdt\frac{d A}{d t}dtdA​ for a circle with radius rrr expanding at drdt\frac{d r}{d t}dtdr​. Given drdt=2 m/s\frac{d r}{d t}=2 \mathrm{~m/s}dtdr​=2 m/s and r=28 mr=28 \mathrm{~m}r=28 m, calculate the area increase rate in m2/s\mathrm{m}^{2}/\mathrm{s}m2/s.

Problem 22910

At 4 PM, how fast is the distance between ship A (sailing south at 25 km/h) and ship B (sailing north at 35 km/h) changing?

Problem 22911

A rectangle's length increases by 5 cm/s5 \mathrm{~cm/s}5 cm/s and width by 4 cm/s4 \mathrm{~cm/s}4 cm/s. Find area increase rate when length is 13 cm13 \mathrm{~cm}13 cm and width is 5 cm5 \mathrm{~cm}5 cm.

Problem 22912

Find dy/dtd y / d tdy/dt for y=4x2+3y=4 x^{2}+3y=4x2+3 when dxdt=3\frac{d x}{d t}=3dtdx​=3 cm/s at x=−1,0,1x=-1, 0, 1x=−1,0,1.

Problem 22913

Find dy/dtd y / d tdy/dt and dx/dtd x / d tdx/dt for y=xy=\sqrt{x}y=x​ given x=1x=1x=1, dx/dt=5d x / d t=5dx/dt=5 and x=81x=81x=81, dy/dt=6d y / d t=6dy/dt=6.

Problem 22914

At 4:00 p.m., what is the distance rate change (in km/h) between ships A and B, given their speeds? Round to one decimal place.

Problem 22915

Water leaks from a conical tank at 11,000 cm3/min11,000 \mathrm{~cm}^{3} / \mathrm{min}11,000 cm3/min. Find the inflow rate when water rises at 20 cm/min20 \mathrm{~cm} / \mathrm{min}20 cm/min at 2 m2 \mathrm{~m}2 m.

Problem 22916

A sphere's radius rrr increases at 3 in/min. Find volume change rate when r=8r=8r=8 in and r=36r=36r=36 in.

Problem 22917

A worker pulls a 5m plank up a wall at 0.19 m/s. How fast is the plank's end sliding when 2m from the wall? Round to two decimals. m/sec\mathrm{m} / \mathrm{sec}m/sec

Problem 22918

Sand falls onto a conical pile at 20 ft³/min. If the base diameter is 3 times the height, find h′h^{\prime}h′ when h=2h=2h=2 ft. h′=□ft/min h^{\prime}=\square \mathrm{ft} / \mathrm{min} h′=□ft/min

Problem 22919

Two planes are converging at a point. One is 75 miles away at 300 mph, the other 100 miles away at 400 mph. Find: (a) Rate of distance change in mph\mathrm{mph}mph (b) Time before a flight path change in h

Problem 22920

Evaluate the integral F(x)=∫x4x(2t−1)3dtF(x)=\int_{x^{4}}^{x}(2 t-1)^{3} d tF(x)=∫x4x​(2t−1)3dt.

Problem 22921

Find the volume of a cube with edge 15 cm15 \mathrm{~cm}15 cm and error 0.1 cm0.1 \mathrm{~cm}0.1 cm. Estimate max error, relative error, and percentage error.

Problem 22922

Problem 22923

Find the derivative at the extremum (−23,1039)\left(-\frac{2}{3}, \frac{10 \sqrt{3}}{9}\right)(−32​,9103​​) for f(x)=−5xx+1f(x)=-5 x \sqrt{x+1}f(x)=−5xx+1​.

Problem 22924

Estimate the distance traveled from t=0 t = 0 t=0 to t=8 t = 8 t=8 using a left endpoint Riemann sum with given velocities.

Problem 22925

Find the Taylor series for f(x)=ln⁡xf(x)=\ln xf(x)=lnx around the point x=3x=3x=3.

Problem 22926

A substance decays at 13.4%13.4\%13.4% daily. After 5 days, how much remains from an initial 90mg90 \mathrm{mg}90mg? Round to two decimals.

Problem 22927

A bear population grew by 50%50\%50% in 4 years. What formula finds the continuous annual growth rate using nnn?

Problem 22928

Find the derivative of g(x)=∫5x4xu+1u−5dug(x)=\int_{5 x}^{4 x} \frac{u+1}{u-5} dug(x)=∫5x4x​u−5u+1​du. What is g′(x)g'(x)g′(x)?

Problem 22929

Find the critical numbers of the function f(x)=6x2−9xf(x)=6 x^{2}-9 xf(x)=6x2−9x. Enter answers as a comma-separated list.

Problem 22930

How many years will it take for a lion pride to grow to 275%275\%275% of its size at a continuous growth rate of 0.5%0.5\%0.5%? Round up.

Problem 22931

Find the absolute extrema of y=3cos⁡xy=3 \cos xy=3cosx on the interval [0,2π][0, 2\pi][0,2π]. Minimum (x,y)=(□)(x,y)=(\square)(x,y)=(□), maximum (x,y)=(□)(x,y)=(\square)(x,y)=(□) (smaller xxx), (x,y)=(□)(x,y)=(\square)(x,y)=(□) (larger xxx).

Problem 22932

Find the limit as xxx approaches 2 for the expression (x−2)(x−5)x2+2x−8\frac{(x-2)(x-5)}{x^{2}+2x-8}x2+2x−8(x−2)(x−5)​.

Problem 22933

Find the light intensity III that maximizes photosynthesis given by P(I)=100II2+I+4P(I)=\frac{100 I}{I^{2}+I+4}P(I)=I2+I+4100I​.

Problem 22934

Find the absolute extrema of f(x)=x2−4xf(x)=x^{2}-4 xf(x)=x2−4x on the intervals: (a) [−1,4][-1,4][−1,4], (b) (2,5](2,5](2,5], (c) (0,4)(0,4)(0,4), (d) [2,6)[2,6)[2,6).

Problem 22935

Show that the definite integral ∫aaf(x) dx=0\int_{a}^{a} f(x) \, dx = 0∫aa​f(x)dx=0 using its definition.

Problem 22936

Check if Rolle's Theorem applies to f(x)=cos⁡xf(x)=\cos xf(x)=cosx on [π,3π][\pi, 3\pi][π,3π]. If yes, find ccc where f′(c)=0f'(c)=0f′(c)=0.

Problem 22937

A ball's height after ttt seconds is f(t)=−16t2+64t+5f(t)=-16t^2+64t+5f(t)=−16t2+64t+5. (a) Show f(1)=f(3)f(1)=f(3)f(1)=f(3). (b) Find the velocity at some time in (1,3)(1,3)(1,3) using Rolle's Theorem. What is that time?

Problem 22938

Can Rolle's Theorem apply to f(x)=x2/3−2f(x)=x^{2/3}-2f(x)=x2/3−2 on [−27,27][-27,27][−27,27]? If yes, find ccc where f′(c)=0f'(c)=0f′(c)=0. Enter NA if not applicable.

Problem 22939

Find the limit as u u u approaches a a a for 2u+a4u−a \frac{2 \sqrt{u}+\sqrt{a}}{4 u-a} 4u−a2u​+a​​.

Problem 22940

Find the limit as xxx approaches infinity for the expression 2ln⁡(x)∞x2\frac{2 \ln(x)^{\infty}}{x^{2}}x22ln(x)∞​.

Find the rate of change of the triangle's base when the altitude is $10 \mathrm{~cm}$ and area is $150 \mathrm{~cm}^{2}$ .

Find the tangent line equation for $f(x)=\tan^{2} x$ at the point $\left(-\frac{\pi}{4}, 1\right)$ .

Find the max and min values of $f(x)=x+\frac{9}{x}$ on $[0.2,12]$ . Min: $6$ , Max: $12.75$ .

Evaluate the integral: $\frac{d}{d t} \int_{0}^{t^{10}} \sqrt{u^{7}} d u = \frac{d}{d t}\left(\frac{2 t^{45}}{9}\right)$ .

Find the height increase rate (in $\mathrm{m} / \mathrm{min}$ ) for a tank with radius $3 \mathrm{~m}$ filled at $3 \mathrm{~m}^{3}/\mathrm{min}$ .

Find the derivatives of $f(x)=x^{n} \sin x$ for $n=1,2,3,4$ and derive a general formula for $f^{\prime}(x)$ .

Find the volume increase rate (in $\mathrm{mm}^{3} / \mathrm{s}$ ) when the diameter is $100 \mathrm{~mm}$ and radius grows at $4 \mathrm{~mm/s}$ .

Find the 4th derivative of the function $f^{(3)}(x)=\sqrt[3]{x^{2}}$ .

Find $\frac{d A}{d t}$ for a circle with radius $r$ expanding at $\frac{d r}{d t}$ . Given $\frac{d r}{d t}=2 \mathrm{~m/s}$ and $r=28 \mathrm{~m}$ , calculate the area increase rate in $\mathrm{m}^{2}/\mathrm{s}$ .

A rectangle's length increases by $5 \mathrm{~cm/s}$ and width by $4 \mathrm{~cm/s}$ . Find area increase rate when length is $13 \mathrm{~cm}$ and width is $5 \mathrm{~cm}$ .

Find $d y / d t$ for $y=4 x^{2}+3$ when $\frac{d x}{d t}=3$ cm/s at $x=-1, 0, 1$ .

Find $d y / d t$ and $d x / d t$ for $y=\sqrt{x}$ given $x=1$ , $d x / d t=5$ and $x=81$ , $d y / d t=6$ .

Water leaks from a conical tank at $11,000 \mathrm{~cm}^{3} / \mathrm{min}$ . Find the inflow rate when water rises at $20 \mathrm{~cm} / \mathrm{min}$ at $2 \mathrm{~m}$ .

A sphere's radius $r$ increases at 3 in/min. Find volume change rate when $r=8$ in and $r=36$ in.

A worker pulls a 5m plank up a wall at 0.19 m/s. How fast is the plank's end sliding when 2m from the wall? Round to two decimals. $\mathrm{m} / \mathrm{sec}$

Sand falls onto a conical pile at 20 ft³/min. If the base diameter is 3 times the height, find $h^{\prime}$ when $h=2$ ft. $h^{\prime}=\square \mathrm{ft} / \mathrm{min}$

Two planes are converging at a point. One is 75 miles away at 300 mph, the other 100 miles away at 400 mph. Find: (a) Rate of distance change in $\mathrm{mph}$ (b) Time before a flight path change in h

Evaluate the integral $F(x)=\int_{x^{4}}^{x}(2 t-1)^{3} d t$ .

Find the volume of a cube with edge $15 \mathrm{~cm}$ and error $0.1 \mathrm{~cm}$ . Estimate max error, relative error, and percentage error.

Find the derivative at the extremum $\left(-\frac{2}{3}, \frac{10 \sqrt{3}}{9}\right)$ for $f(x)=-5 x \sqrt{x+1}$ .

Estimate the distance traveled from $t = 0$ to $t = 8$ using a left endpoint Riemann sum with given velocities.

Find the Taylor series for $f(x)=\ln x$ around the point $x=3$ .

A substance decays at $13.4\%$ daily. After 5 days, how much remains from an initial $90 \mathrm{mg}$ ? Round to two decimals.

A bear population grew by $50\%$ in 4 years. What formula finds the continuous annual growth rate using $n$ ?

Find the derivative of $g(x)=\int_{5 x}^{4 x} \frac{u+1}{u-5} du$ . What is $g'(x)$ ?

Find the critical numbers of the function $f(x)=6 x^{2}-9 x$ . Enter answers as a comma-separated list.

How many years will it take for a lion pride to grow to $275\%$ of its size at a continuous growth rate of $0.5\%$ ? Round up.

Find the absolute extrema of $y=3 \cos x$ on the interval $[0, 2\pi]$ . Minimum $(x,y)=(\square)$ , maximum $(x,y)=(\square)$ (smaller $x$ ), $(x,y)=(\square)$ (larger $x$ ).

Find the limit as $x$ approaches 2 for the expression $\frac{(x-2)(x-5)}{x^{2}+2x-8}$ .

Find the light intensity $I$ that maximizes photosynthesis given by $P(I)=\frac{100 I}{I^{2}+I+4}$ .

Find the absolute extrema of $f(x)=x^{2}-4 x$ on the intervals: (a) $[-1,4]$ , (b) $(2,5]$ , (c) $(0,4)$ , (d) $[2,6)$ .

Show that the definite integral $\int_{a}^{a} f(x) \, dx = 0$ using its definition.

Check if Rolle's Theorem applies to $f(x)=\cos x$ on $[\pi, 3\pi]$ . If yes, find $c$ where $f'(c)=0$ .

A ball's height after $t$ seconds is $f(t)=-16t^2+64t+5$ .
(a) Show $f(1)=f(3)$ . (b) Find the velocity at some time in $(1,3)$ using Rolle's Theorem. What is that time?

Can Rolle's Theorem apply to $f(x)=x^{2/3}-2$ on $[-27,27]$ ? If yes, find $c$ where $f'(c)=0$ . Enter NA if not applicable.

Find the limit as $u$ approaches $a$ for $\frac{2 \sqrt{u}+\sqrt{a}}{4 u-a}$ .

Find the limit as $x$ approaches infinity for the expression $\frac{2 \ln(x)^{\infty}}{x^{2}}$ .

Find the work to pump oil from a 15 ft parabolic tank ( $y=x^{2}$ ) half-full of oil ( $35 \mathrm{lbs/ft}^3$ ).

Can the Mean Value Theorem apply to $f(x)=\sqrt{9-x}$ on $[-7,9]$ ? If yes, find $c$ where $f'(c)=\frac{f(9)-f(-7)}{16}$ .

Evaluate the integral: $\int_{4}^{8} \frac{1}{8} x^{3} d x$ using given values.

Can the Mean Value Theorem be applied to $f(x)=2x^{3}$ on $[1,2]$ ? If yes, find $c$ where $f'(c)=\frac{f(2)-f(1)}{2-1}$ .

Find the average value of $f(x)=\frac{9}{x}$ on $[7, 7e]$ and graph it with the average value indicated.

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{-1}^{1}(2 x^{3}+4) dx=\square$

Find the area under the curve of $f(x)=8e^{-x}$ from 0 to an unknown upper limit using $\int_{0}^{\square} dx$ .

Evaluate the integral using the limit definition: $\int_{2}^{5} 7 \, dx$ .

Calculate the integral of $x$ from 0 to 8: $\int_{0}^{8} x \, dx$ .

Evaluate the definite integral using limits: $\int_{-7}^{7} x^{3} \, dx$ .

Sketch the area represented by the integral $\int_{0}^{2}(3 x+3) d x$ .

Determine if the function $f(x)=\frac{1}{(x+1)^{2}}$ is increasing or decreasing.

Evaluate the integral using symmetry: $\int_{-2}^{2}(3+x+x^{2}+x^{3}) dx=\square$ (Type an integer or a simplified fraction.)

Calculate the integral from -4 to 4 of the function $\sqrt{16 - x^{2}}$ .

Maximize the function $f(x)=x^{a}(1-x)^{b}$ for $0 \leq x \leq 1$ .

Evaluate the integral $\int_{4}^{8}(9 x^{3}-12 x+2) d x$ using given values.

Find $\frac{d y}{d x}$ using implicit differentiation for $\cos(2x + y) = 5y$ . Result: $\frac{d y}{d x} = \square$ .

Find the intervals where the function $y=x \sqrt{64-x^{2}}$ is increasing or decreasing. Use interval notation.

Evaluate the integral from 1/2 to 1 of $(x^{-3} - 13)$ . Find $\int_{1/2}^{1}(x^{-3}-13) dx=\square$ .

Calculate the integral $\int_{0}^{2}(3 x+3) d x$ .

Evaluate the integral using symmetry: $\int_{-2}^{2} \frac{x^{5}+4 x^{7}}{x^{6}+5} d x=\square$ (Simplify your answer.)

Evaluate the integral $\int_{0}^{1}(x^{2}-2x+7)dx$ and check if it matches the graph's behavior. Choose A, B, C, or D.

Set up the integral for net area: $\int_{-3}^{\square}(9 - x^2) dx$ . What is the upper limit?

Find the value of $x$ that minimizes the average cost given $C(x)=10,000+250x+10x^2$ . Round to two decimals.

Calculate the integral from 0 to 8 of the function $\frac{x}{4}$ .

Evaluate the integral $\int_{2}^{2} x d x$ using given values: $\int_{2}^{6} x^{3} d x=320$ , $\int_{2}^{6} x d x=16$ , $\int_{2}^{6} d x=4$ .

Given the function $f(x)=6-|x-2|$ , find critical numbers, intervals of increase/decrease, and use the First Derivative Test for extrema.

Given the function $f(x)=\left\{\begin{array}{ll} 4 x+1, & x \leq-1 \\ x^{2}-4, & x>-1 \end{array}\right.$ , find critical numbers, intervals of increase/decrease, and relative extrema.

Find critical numbers of $f(x)=-2 x^{2}+12 x+2$ , intervals of increase/decrease, and relative extrema.

Find the net area and area above the $x$ -axis for $y=81-x^{2}$ . Set up the integral: $\int_{-9}^{9}(81-x^{2}) dx$ . Graph the function.

Find critical numbers of $f(x)=x^{1/3}+1$ . Determine intervals of increase/decrease and apply the First Derivative Test.

Find the intervals where the graph of $f(x)=\frac{11}{x^{2}+3}$ is concave up or down. Use interval notation.