Calculus

Problem 1901

Find the average growth rate of molar crown length $L(t)=-0.01 t^{2}+0.788 t-7.042$ from weeks 24 to 29. Also, find the instantaneous growth rate at 24 weeks.

See Solution

Problem 1902

Determine if the limit exists: $\lim _{x \rightarrow-10} \frac{x^{2}-100}{x+10}$ . Choose A or B.

See Solution

Problem 1903

Is the statement true or false? The derivative $f^{\prime}(a)$ is the slope of the tangent to $y=f(x)$ at $x=a$ . Choose A, B, C, or D.

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Problem 1904

What does the limit of $f(x)$ as $x \rightarrow a$ describe? Choose true/false options about this statement.

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Problem 1905

Explain the difference between the average rate of change of $y$ from $a$ to $b$ and the instantaneous rate at $x=a$ .

See Solution

Problem 1906

Choose the correct answer: If degree of $p(x)$ > degree of $q(x)$ , what happens to their limits as they grow? A, B, or C?

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Problem 1907

Find the left and right limits of the piecewise function $h(x)=\begin{cases}3x+5 & \text{if } x \leq 0 \\ x^2-5x+5 & \text{if } x>0\end{cases}$ at points of discontinuity.

See Solution

Problem 1908

Find the derivative of $f(x)=6 x^{3}+8$ using the limit $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 1909

Find the derivative $f^{\prime}(x)$ for $f(x)=6x^{3}+8$ and evaluate $f^{\prime}(-1)$ , $f^{\prime}(0)$ , and $f^{\prime}(4)$ .

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Problem 1910

Determine if the limit exists: $\lim _{x \rightarrow \infty} \frac{2 x^{2}+2 x}{4 x^{2}-8 x+1}$ . Find its value or state it does not exist.

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Problem 1911

Evaluate the integral from 2 to 3 of the function $8 x^{3}+3 x^{2}+6 x$ .

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Problem 1912

Find the mosquito population $N(t)$ as a function of time $t$ using $N(t) = N_0 e^{kt}$ for $t=0$ and $k \neq 0$ .

See Solution

Problem 1913

Trish and Sean each have \$6000. After 20 years, who has more money: Trish at 12\% or Sean at 11.6\% compounded continuously?

See Solution

Problem 1914

Trish and Sean invest \$5000. Trish grows at 5\% and Sean at 4.9\% continuously. Who has more after 20 years?

See Solution

Problem 1915

Find the slopes of secant lines and tangent line for $y=f(x)=x^{2}+x$ at points $(-3, f(-3))$ and $(-1, f(-1))$ .

See Solution

Problem 1916

Find slopes of secant lines and tangent line for $y=f(x)=x^{2}+x$ at points $(-4, f(-4))$ and $(-1, f(-1))$ .

See Solution

Problem 1917

Revenue from selling $x$ car seats is $R(x)=32x-0.010x^{2}$ for $0 \leq x \leq 3200$ .
(A) Find average revenue change from 1,000 to 1,050 seats. (B) Use four-step process to find $R^{\prime}(x)$ . (C) Find revenue and rate of change at 1,000 seats and interpret.

See Solution

Problem 1918

Find slopes of secant lines for $f(x)=x^{2}+x$ at given points and the tangent line at $(-3, f(-3))$ .

See Solution

Problem 1919

Find where the function $f(x)=x+\frac{3 \sin (x-3)}{x-3}$ is discontinuous.

See Solution

Problem 1920

Insect population $P(t)=300 e^{0.05 t}$ : (a) Find $P(0)$ . (b) Growth rate? (c) $P(10)$ ? (d) When $P=420$ ? (e) When doubles?

See Solution

Problem 1921

Find the average change in revenue from 1,000 to 1,050 car seats using $R(x)=56x-0.020x^2$ . Also, find $R'(x)$ and evaluate it at $x=1000$ .

See Solution

Problem 1922

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{(x+2)(1-x)}{(x+1)(x-1)}$

See Solution

Problem 1923

Insect population $P(t)=400 e^{0.04 t}$ : (a) Find $P(0)$ , (b) growth rate, (c) $P(10)$ , (d) when $P=680$ , (e) when doubles.

See Solution

Problem 1924

Find the continuity of the function $f(x)=\frac{1}{1+\sin x}$ on the interval $[0,2 \pi]$ .

See Solution

Problem 1925

Find the average velocity of a bowling ball dropped from 300 m between $t=1$ and $t=3$ using $s(t)=-4.9 t^{2}+300$ .

See Solution

Problem 1926

Show that between year 1 (2 ft) and year 2 (5 ft), trees A and B were the same height using the intermediate value theorem.

See Solution

Problem 1927

Show that between year 1 (tree A: $a$ ft, tree B: 5 ft) and year 2 (tree A: 7 ft, tree B: 5 ft), trees were equal height using IVT.

See Solution

Problem 1928

Find the derivative $f^{\prime}(4)$ for the function $f(x)=3x+4$ using the definition of the derivative.

See Solution

Problem 1929

Insect population $P(t)=500 e^{0.05 t}$ : (a) Find $P(0)$ , (b) growth rate, (c) $P(10)$ , (d) when $P(t)=650$ , (e) when $P(t)$ doubles.

See Solution

Problem 1930

Solve the logistic growth function $f(t)=\frac{103,000}{1+4100 e^{-t}}$ . Find: a. Initial cases (t=0) b. Cases at week 4 (t=4) c. Limiting population size.

See Solution

Problem 1931

Insect population $P(t)=600 e^{0.06 t}$ : (a) Find $P(0)$ ; (b) Growth rate?; (c) $P(10)$ ?; (d) When is $P=960$ ?; (e) When doubles?

See Solution

Problem 1932

Given 500g of strontium-90 with a decay rate of $-244\%$ and a half-life of 28.8 years, answer:
(a) Verify the decay rate.
(b) Find remaining strontium-90 after 10 years; round final answer.
(c) Determine when 200g will remain.
(d) Verify the half-life.

See Solution

Problem 1933

Given 800g of strontium-90 decaying at $-2.44\%$ , find: (a) decay rate, (b) amount left after 40 years, (c) time for 200g left, (d) half-life. Use $N = N_0 e^{-\lambda t}$ .

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Problem 1934

Given 500 grams of strontium-90, find:
(a) decay rate, (b) amount left after 20 years, (c) time for 400 grams left, (d) half-life.
Decay rate: $-2.44\%$ , model: $A(t)=a_0e^{-0.0244t}$ . Round final answers as needed.

See Solution

Problem 1935

Find the decay rate, amount left after 40 years, time to reach 600 grams, and half-life of strontium-90 using $A(t)=800e^{-0.0244t}$ .

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Problem 1936

Strontium-90 sample: 800g, decay rate -2.44%/year.
(a) Confirm decay rate. (b) Find remaining after 40 years. (c) When is 600g left? (d) Calculate half-life.

See Solution

Problem 1937

Given 500g of strontium-90 decaying at $-2.44\%$ yearly with $A(t)=a0e^{-0.0244t}$ , solve:
(a) Verify decay rate. (b) Amount after 40 years. (c) Time for 200g left. (d) Half-life.

See Solution

Problem 1938

Given 800g of strontium-90 decaying at $-2.44\%$ , find:
(a) Decay rate. (b) Amount left after 40 years. (c) Time for 600g left. (d) Half-life.
Use $A(t)=800e^{-0.0244t}$ .

See Solution

Problem 1939

Identify the FALSE statement about the concave up function $f(x)$ on the interval $0<x<2$ .

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Problem 1940

Find the average rate of change of $A=2t^{2}$ from $t=1$ to $t=4$ for farming acres over 10 years.

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Problem 1941

Find the limits:
1. $\lim _{x \rightarrow-3^{+}} \frac{|2 x+6|}{x+3}$
2. $\lim _{x \rightarrow-3^{-}} \frac{|2 x+6|}{x+3}$

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Problem 1942

Find the derivatives: (1) $f(x)=x^{8}$ , (2) $g(x)=-8 x^{5}$ , (3) $h(x)=\frac{1}{x^{4}}$ .

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Problem 1943

Find the derivative $f^{\prime}(-7)$ if $f(x)=10$ .

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Problem 1944

Find the derivative of $f(x)=-4 x^{9}+2 x^{7}$ . What is $f^{\prime}(x)=?$

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Problem 1945

Find all points $(x, y)$ on the graph of $f(x)=x^{3}-3 x^{2}-7 x-9$ where the tangent slope is 2.

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Problem 1946

Find points $(x, y)$ on the graph of $f(x)=x^{3}-3 x^{2}-7 x-9$ where the tangent slope is 2.

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Problem 1947

Find the derivative $f'(x)$ of the function $f(x)=(3x^2-7)(2x+3)$ and calculate $f'(1)$ .

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Problem 1948

Find the derivative $f^{\prime}(4)$ for the function $f(x)=-\frac{3}{x}-3 \sqrt{x}$ and simplify the answer.

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Problem 1949

Find the derivative $f'(t)$ of the function $f(t)=(t^{2}+3t+7)(6t^{2}+4)$ and evaluate $f'(4)$ .

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Problem 1950

Find the derivative of the function $f(x)=\frac{5 x+3}{6 x+5}$ . What is $f^{\prime}(x)$ ?

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Problem 1951

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

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Problem 1952

Find the derivative $f^{\prime}(x)$ of the function $f(x)=6+3x-2x^{2}$ and calculate $f^{\prime}(-1)$ .

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Problem 1953

Find the derivative of the function $f(x)=\frac{2-x^{2}}{8+x^{2}}$ : $f^{\prime}(x)=$

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Problem 1954

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{2}-1}{x^{3}-1}$

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Problem 1955

Air is pumped into a balloon at 10 cm³/s. Find the rate of diameter increase in cm/s when the radius is $5 \mathrm{~cm}$ .

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Problem 1956

Find $f^{\prime}(-1)$ if $f(x)=x^{6} h(x)$ , $h(-1)=3$ , and $h^{\prime}(-1)=6$ . Use product and power rules.

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Problem 1957

Find the derivative of $f(x)=-9 \sqrt{x}-\frac{6}{x^{8}}$ using sqrt(x) and avoid fractional or negative exponents. $f^{\prime}(x)=$

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Problem 1958

A curve $y-f(x)$ goes through $P(2,7)$ with gradient $f^{\prime}(x)-4 x-3$ . Find the gradient, tangent equation, and curve equation.

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Problem 1959

Find the cost function $C(x)=8 x^{3}-24 x^{2}+28 x$ for badminton rackets, then determine $C^{\prime}(x)$ , marginal cost at 100 rackets, profit $P(x)$ , $P^{\prime}(x)$ , and optimal production level.

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Problem 1960

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos ^{2}(2 x)}{x^{2}}$ .

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Problem 1961

Find the limit: $\lim _{h \rightarrow 0} \frac{\csc \left(\frac{\pi}{4}+h\right)-\csc \left(\frac{\pi}{4}\right)}{h}$

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Problem 1962

A mosquito colony grows exponentially. If $\mathrm{N}(t) = \mathrm{N}_0 e^{\mathrm{kt}}$ , find $\mathrm{N}(4)$ given $\mathrm{N}_0 = 1000$ , $\mathrm{N}(1) = 1500$ .

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Problem 1963

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{4 x^{2}-5 x+7}{10 x+x^{3}}$

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Problem 1964

Given 800 grams of strontium-90 with decay formula $A(t)=a_0e^{-0.0244t}$ , find:
(a) decay rate as a percentage, (b) amount left after 20 years, (c) time for 200 grams left, (d) half-life.

See Solution

Problem 1965

Insect population $\mathrm{P}(t)=100 e^{0.01 t}$ : Find $\mathrm{P}(0)$ , growth rate, $\mathrm{P}(10)$ , time to reach 140, and doubling time.

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Problem 1966

Find the limit of the population model $N(t)=50-2 e^{-0.5 t} \cos (t)$ as $t$ approaches infinity.

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Problem 1967

Find the left end-behavior of $g(x)=\frac{-3 x^{4}+2 x^{2}+5 x}{4 x^{4}-x^{2}}$ as $x \rightarrow-\infty$ .

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Problem 1968

Find the left end-behavior as $x \rightarrow -\infty$ of $g(x)=\frac{-3 x^{4}+2 x^{2}+5 x}{4 x^{4}-x^{2}}$ .

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Problem 1969

Find the limit of the population function $N(t)=50-2 e^{-0.5 t} \cos (t)$ as $t$ approaches infinity.

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Problem 1970

Find the limit: $\lim _{x \rightarrow \infty} \frac{-2 e^{3 x}+1}{5 e^{3 x}-e^{x}}$

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Problem 1971

Find the limit: $\lim _{x \rightarrow 0} \frac{\cos ^{2}(4 x)-1}{x^{2}}$ .

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Problem 1972

Find the integral: $\int \frac{x^{2}}{\cos ^{2}(x^{3})} dx$ .

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Problem 1973

Evaluate the integral $\int \frac{4}{\sqrt{5-16 x^{2}}} d x$ .

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Problem 1974

Evaluate the integral: $\int \frac{e^{3 x}}{16+16 e^{6 x}} d x$

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Problem 1975

Evaluate the integral $\int x^{4} \cos \left(x^{5}\right) d x$ .

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Problem 1976

Evaluate the integral $\int_{4}^{7} t^{4} \sqrt{t^{5}-1024} \, dt$ .

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Problem 1977

Calculate the integral $\int_{1}^{5} \ln (3 x) \, dx$ .

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Problem 1978

Find the integral of the inverse sine function: $\int \sin^{-1}(5x) \, dx$ .

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Problem 1979

Evaluate the integral from 1 to 6: $\int_{1}^{6} t^{2} \ln (t) d t$ .

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Problem 1980

Find the integral of $x^{2} \cos (x)$ with respect to $x$ .

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Problem 1981

Calculate the integral: $\int e^{x} \cos (x) \, dx$

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Problem 1982

Find the integral of $x \tan^{-1}(4x)$ with respect to $x$ : $\int x \tan^{-1}(4 x) \, dx$ .

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Problem 1983

Find the integral of $x^{3} \cos \left(x^{2}\right)$ with respect to $x$ : $\int x^{3} \cos \left(x^{2}\right) d x$ .

See Solution

Problem 1984

Insect population $P(t)=900 e^{0.06 t}$ : (a) Find $P(0)$ . (b) Growth rate? (c) $P(10)$ ? (d) Time to reach 1170? (e) Time to double?

See Solution

Problem 1985

Evaluate the integral using substitution and integration by parts: $\int x^{3} \cos \left(x^{2}\right) d x.$

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Problem 1986

Find the integral $\int \cos ^{5}(x) d x=\square+C$ .

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Problem 1987

Find the total cost function given the marginal cost $M C=-10 q^{4}+3 q^{2}+80 q$ and fixed cost \R 1000.

See Solution

Problem 1988

Find the total revenue function given $M R=300-13 q^{12}+16 q^{3}$ and $R(1)=3316$ .

See Solution

Problem 1989

Calculate the area under the curve $F(x)=\frac{x}{x^{2}-1}$ from $x=2$ to $x=4$ .

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Problem 1990

Integrate the function $g(t)=12 e^{12 t}+\sqrt[3]{27 t}+\frac{5}{t}+15$ .

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Problem 1991

Analyze the long-term behavior of these functions: $\frac{x^{3}+1}{x^{2}+2}$ , $\frac{x^{2}+1}{x^{2}+2}$ , $\frac{x^{2}+1}{x^{3}+2}$ .

See Solution

Problem 1992

Find $f^{\prime}(x)$ using the difference quotient $\frac{f(x+h)-f(x)}{h}$ and limit as $h \rightarrow 0$ . Given $f(x)=3 \sqrt{x+5}$ .

See Solution

Problem 1993

Find the average rate of change of $f(x)=x^{2}+2x+9$ over the interval $3 \leq x \leq 6$ .

See Solution

Problem 1994

Find the acceleration curves for a hockey puck hit in the positive $x$ -direction, stopped by a net at time $t \approx t_{1}$ . Consider quick stops and constant or changing forces.

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Problem 1995

Express the integral $\int_{6}^{9}\left(x^{2}+\frac{1}{x}\right) d x$ as a limit of Riemann sums using right endpoints.

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Problem 1996

Find the area represented by the integral $\int_{-9}^{4}\left|\frac{1}{2} x\right| d x$ .

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Problem 1997

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{7 x^{4}+3}{x^{4}-x^{3}+x+2}\right)$ .

See Solution

Problem 1998

Find the limit as $x$ approaches $-\infty$ for $\frac{7x^{4}+3}{x^{4}-x^{3}+x+2}$ .

See Solution

Problem 1999

Find the limit as $x$ approaches infinity for the expression $\frac{x+3}{x^{2}+16}$ .

See Solution

Problem 2000

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{2 x^{6}+9 x^{4}-1}{8 x^{3}-3 x+7}\right)$ .

See Solution

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Calculus

Problem 1901

Find the average growth rate of molar crown length L(t)=−0.01t2+0.788t−7.042L(t)=-0.01 t^{2}+0.788 t-7.042L(t)=−0.01t2+0.788t−7.042 from weeks 24 to 29. Also, find the instantaneous growth rate at 24 weeks.

Problem 1902

Determine if the limit exists: lim⁡x→−10x2−100x+10\lim _{x \rightarrow-10} \frac{x^{2}-100}{x+10}x→−10lim​x+10x2−100​. Choose A or B.

Problem 1903

Is the statement true or false? The derivative f′(a)f^{\prime}(a)f′(a) is the slope of the tangent to y=f(x)y=f(x)y=f(x) at x=ax=ax=a. Choose A, B, C, or D.

Problem 1904

What does the limit of f(x)f(x)f(x) as x→ax \rightarrow ax→a describe? Choose true/false options about this statement.

Problem 1905

Explain the difference between the average rate of change of yyy from aaa to bbb and the instantaneous rate at x=ax=ax=a.

Problem 1906

Choose the correct answer: If degree of p(x)p(x)p(x) > degree of q(x)q(x)q(x), what happens to their limits as they grow? A, B, or C?

Problem 1907

Find the left and right limits of the piecewise function h(x)={3x+5if x≤0x2−5x+5if x>0h(x)=\begin{cases}3x+5 & \text{if } x \leq 0 \\ x^2-5x+5 & \text{if } x>0\end{cases}h(x)={3x+5x2−5x+5​if x≤0if x>0​ at points of discontinuity.

Problem 1908

Find the derivative of f(x)=6x3+8f(x)=6 x^{3}+8f(x)=6x3+8 using the limit lim⁡h→0f(x+h)−f(x)h\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}limh→0​hf(x+h)−f(x)​.

Problem 1909

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=6x3+8f(x)=6x^{3}+8f(x)=6x3+8 and evaluate f′(−1)f^{\prime}(-1)f′(−1), f′(0)f^{\prime}(0)f′(0), and f′(4)f^{\prime}(4)f′(4).

Problem 1910

Determine if the limit exists: lim⁡x→∞2x2+2x4x2−8x+1\lim _{x \rightarrow \infty} \frac{2 x^{2}+2 x}{4 x^{2}-8 x+1}x→∞lim​4x2−8x+12x2+2x​. Find its value or state it does not exist.

Problem 1911

Evaluate the integral from 2 to 3 of the function 8x3+3x2+6x8 x^{3}+3 x^{2}+6 x8x3+3x2+6x.

Problem 1912

Find the mosquito population N(t)N(t)N(t) as a function of time ttt using N(t)=N0ektN(t) = N_0 e^{kt}N(t)=N0​ekt for t=0t=0t=0 and k≠0k \neq 0k=0.

Problem 1913

Trish and Sean each have \$6000. After 20 years, who has more money: Trish at 12\% or Sean at 11.6\% compounded continuously?

Problem 1914

Trish and Sean invest \$5000. Trish grows at 5\% and Sean at 4.9\% continuously. Who has more after 20 years?

Problem 1915

Find the slopes of secant lines and tangent line for y=f(x)=x2+xy=f(x)=x^{2}+xy=f(x)=x2+x at points (−3,f(−3))(-3, f(-3))(−3,f(−3)) and (−1,f(−1))(-1, f(-1))(−1,f(−1)).

Problem 1916

Find slopes of secant lines and tangent line for y=f(x)=x2+xy=f(x)=x^{2}+xy=f(x)=x2+x at points (−4,f(−4))(-4, f(-4))(−4,f(−4)) and (−1,f(−1))(-1, f(-1))(−1,f(−1)).

Problem 1917

Problem 1918

Find slopes of secant lines for f(x)=x2+xf(x)=x^{2}+xf(x)=x2+x at given points and the tangent line at (−3,f(−3))(-3, f(-3))(−3,f(−3)).

Problem 1919

Find where the function f(x)=x+3sin⁡(x−3)x−3f(x)=x+\frac{3 \sin (x-3)}{x-3}f(x)=x+x−33sin(x−3)​ is discontinuous.

Problem 1920

Insect population P(t)=300e0.05tP(t)=300 e^{0.05 t}P(t)=300e0.05t: (a) Find P(0)P(0)P(0). (b) Growth rate? (c) P(10)P(10)P(10)? (d) When P=420P=420P=420? (e) When doubles?

Problem 1921

Find the average change in revenue from 1,000 to 1,050 car seats using R(x)=56x−0.020x2R(x)=56x-0.020x^2R(x)=56x−0.020x2. Also, find R′(x)R'(x)R′(x) and evaluate it at x=1000x=1000x=1000.

Problem 1922

Find the limit as xxx approaches infinity: lim⁡x→∞(x+2)(1−x)(x+1)(x−1)\lim _{x \rightarrow \infty} \frac{(x+2)(1-x)}{(x+1)(x-1)}limx→∞​(x+1)(x−1)(x+2)(1−x)​

Problem 1923

Insect population P(t)=400e0.04tP(t)=400 e^{0.04 t}P(t)=400e0.04t: (a) Find P(0)P(0)P(0), (b) growth rate, (c) P(10)P(10)P(10), (d) when P=680P=680P=680, (e) when doubles.

Problem 1924

Find the continuity of the function f(x)=11+sin⁡xf(x)=\frac{1}{1+\sin x}f(x)=1+sinx1​ on the interval [0,2π][0,2 \pi][0,2π].

Problem 1925

Find the average velocity of a bowling ball dropped from 300 m between t=1t=1t=1 and t=3t=3t=3 using s(t)=−4.9t2+300s(t)=-4.9 t^{2}+300s(t)=−4.9t2+300.

Problem 1926

Show that between year 1 (2 ft) and year 2 (5 ft), trees A and B were the same height using the intermediate value theorem.

Problem 1927

Show that between year 1 (tree A: aaa ft, tree B: 5 ft) and year 2 (tree A: 7 ft, tree B: 5 ft), trees were equal height using IVT.

Problem 1928

Find the derivative f′(4)f^{\prime}(4)f′(4) for the function f(x)=3x+4f(x)=3x+4f(x)=3x+4 using the definition of the derivative.

Problem 1929

Insect population P(t)=500e0.05tP(t)=500 e^{0.05 t}P(t)=500e0.05t: (a) Find P(0)P(0)P(0), (b) growth rate, (c) P(10)P(10)P(10), (d) when P(t)=650P(t)=650P(t)=650, (e) when P(t)P(t)P(t) doubles.

Problem 1930

Solve the logistic growth function f(t)=103,0001+4100e−tf(t)=\frac{103,000}{1+4100 e^{-t}}f(t)=1+4100e−t103,000​. Find: a. Initial cases (t=0) b. Cases at week 4 (t=4) c. Limiting population size.

Problem 1931

Insect population P(t)=600e0.06tP(t)=600 e^{0.06 t}P(t)=600e0.06t: (a) Find P(0)P(0)P(0); (b) Growth rate?; (c) P(10)P(10)P(10)?; (d) When is P=960P=960P=960?; (e) When doubles?

Problem 1932

Given 500g of strontium-90 with a decay rate of −244%-244\%−244% and a half-life of 28.8 years, answer:(a) Verify the decay rate.(b) Find remaining strontium-90 after 10 years; round final answer.(c) Determine when 200g will remain.(d) Verify the half-life.

Problem 1933

Given 800g of strontium-90 decaying at −2.44%-2.44\%−2.44%, find: (a) decay rate, (b) amount left after 40 years, (c) time for 200g left, (d) half-life. Use N=N0e−λtN = N_0 e^{-\lambda t}N=N0​e−λt.

Problem 1934

Given 500 grams of strontium-90, find: (a) decay rate, (b) amount left after 20 years, (c) time for 400 grams left, (d) half-life. Decay rate: −2.44%-2.44\%−2.44%, model: A(t)=a0e−0.0244tA(t)=a_0e^{-0.0244t}A(t)=a0​e−0.0244t. Round final answers as needed.

Problem 1935

Find the decay rate, amount left after 40 years, time to reach 600 grams, and half-life of strontium-90 using A(t)=800e−0.0244tA(t)=800e^{-0.0244t}A(t)=800e−0.0244t.

Problem 1936

Strontium-90 sample: 800g, decay rate -2.44%/year. (a) Confirm decay rate. (b) Find remaining after 40 years. (c) When is 600g left? (d) Calculate half-life.

Problem 1937

Given 500g of strontium-90 decaying at −2.44%-2.44\%−2.44% yearly with A(t)=a0e−0.0244tA(t)=a0e^{-0.0244t}A(t)=a0e−0.0244t, solve: (a) Verify decay rate. (b) Amount after 40 years. (c) Time for 200g left. (d) Half-life.

Problem 1938

Given 800g of strontium-90 decaying at −2.44%-2.44\%−2.44%, find:(a) Decay rate. (b) Amount left after 40 years. (c) Time for 600g left. (d) Half-life. Use A(t)=800e−0.0244tA(t)=800e^{-0.0244t}A(t)=800e−0.0244t.

Problem 1939

Identify the FALSE statement about the concave up function f(x)f(x)f(x) on the interval 0<x<20<x<20<x<2.

Problem 1940

Find the average rate of change of A=2t2A=2t^{2}A=2t2 from t=1t=1t=1 to t=4t=4t=4 for farming acres over 10 years.

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Given 500g of strontium-90 with a decay rate of $-244\%$ and a half-life of 28.8 years, answer:
(a) Verify the decay rate.
(b) Find remaining strontium-90 after 10 years; round final answer.
(c) Determine when 200g will remain.
(d) Verify the half-life.

Given 800g of strontium-90 decaying at $-2.44\%$ , find: (a) decay rate, (b) amount left after 40 years, (c) time for 200g left, (d) half-life. Use $N = N_0 e^{-\lambda t}$ .

Given 500 grams of strontium-90, find:
(a) decay rate, (b) amount left after 20 years, (c) time for 400 grams left, (d) half-life.
Decay rate: $-2.44\%$ , model: $A(t)=a_0e^{-0.0244t}$ . Round final answers as needed.

Find the decay rate, amount left after 40 years, time to reach 600 grams, and half-life of strontium-90 using $A(t)=800e^{-0.0244t}$ .

Strontium-90 sample: 800g, decay rate -2.44%/year.
(a) Confirm decay rate. (b) Find remaining after 40 years. (c) When is 600g left? (d) Calculate half-life.

Given 500g of strontium-90 decaying at $-2.44\%$ yearly with $A(t)=a0e^{-0.0244t}$ , solve:
(a) Verify decay rate. (b) Amount after 40 years. (c) Time for 200g left. (d) Half-life.

Given 800g of strontium-90 decaying at $-2.44\%$ , find:
(a) Decay rate. (b) Amount left after 40 years. (c) Time for 600g left. (d) Half-life.
Use $A(t)=800e^{-0.0244t}$ .

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1. $\lim _{x \rightarrow-3^{+}} \frac{|2 x+6|}{x+3}$
2. $\lim _{x \rightarrow-3^{-}} \frac{|2 x+6|}{x+3}$

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Given 800 grams of strontium-90 with decay formula $A(t)=a_0e^{-0.0244t}$ , find:
(a) decay rate as a percentage, (b) amount left after 20 years, (c) time for 200 grams left, (d) half-life.

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