Calculus

Problem 8801

Find the rate of change of angle $A$ given $3 \cos(A) = 1 - 4 \cos\left(\left(\frac{P+Q}{2}\right)^{2}\right)$ with rates $\frac{\pi}{12}$ for $P$ and $-\frac{\pi}{16}$ for $Q$ .

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Problem 8802

Trouver l'équation de la tangente à $f(x)=9 x+\frac{9}{x^{2}}$ au point $(-3,-26)$ avec $y=m x+b$ , où $m=$ et $b=$ .

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Problem 8803

Which function has the tangent plane $z=0$ at the point $(0,0)$ ? A. $z=\cos (x) \cos (y)$ B. $z=x+y$ C. $z=e^{x y}$ D. $z=x^{2}+y^{2}$ E. None.

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Problem 8804

Find the derivative $d y / d x$ for $y=\ln \left(\frac{2 x^{3}}{3 x^{2}+5}\right)^{4}$ .

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Problem 8805

Show that the polynomial $f(x)=4 x^{3}+2 x^{2}-3 x+2$ has a zero in the interval [-4,-1] using the intermediate value theorem.

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Problem 8806

Find the first and second derivatives of $f(x)=9 \cos^{2}(x)-18 \sin(x)$ .

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Problem 8807

Find the derivative of the function $y=\frac{\sqrt{3x-2}}{x}$ . What is $y'$ ?

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Problem 8808

Calculate the limit: $\lim _{x \rightarrow \infty}\left(e^{x}\right)^{\frac{1}{x^{2}}}$ .

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Problem 8809

Calculez $f^{\prime}(t)$ et $f^{\prime \prime}(t)$ pour la fonction $f(t)=t^{5} e^{2 t}$ .

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Problem 8810

Strontium-90 decays as $A(t)=800 e^{-0.0244 t}$ . Find decay rate, amount left after 30 years, time for 600g, and half-life.

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Problem 8811

Insect population $\mathrm{P}(t)=100 e^{0.02 t}$ : (a) Find $\mathrm{P}(0)$ . (b) What is the growth rate? (c) Find $\mathrm{P}(10)$ . (d) When is $\mathrm{P}=160$ ? (e) When does $\mathrm{P}$ double?

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Problem 8812

Trouvez les valeurs de $x$ où la tangente à la courbe $y=2 x^{3}-9 x^{2}-108 x+16$ est horizontale. $x=$

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Problem 8813

Insect population $\mathrm{P}(t)=600 e^{0.02 t}$ : (a) Find $\mathrm{P}(0)$ . (b) What is the growth rate? (c) Find $\mathrm{P}(10)$ . (d) When is $\mathrm{P}=780$ ? (e) When does $\mathrm{P}$ double?

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Problem 8814

Find the derivative using implicit differentiation for the equation $x^{2}y - y = x$ .

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Problem 8815

A baseball machine throws a ball up at $64 \mathrm{ft/sec}$ from $1.5 \mathrm{ft}$ . Find the height equation and max height/time.

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Problem 8816

Find the tangent line equation for $y=\frac{1}{x^{2}}$ at $x=3$ .

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Problem 8817

Differentiate the function $f(x)=\frac{x-e^{-x}}{1+x e^{-x}}$ .

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Problem 8818

Find the derivative of the function $y=3 x e^{-5 x}+2 e^{x^{2}}$ .

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Problem 8819

Differentiate the function $y=2 x^{4}(2 x^{3}+5)$ with respect to $x$ .

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Problem 8820

Find $g^{\prime}(x)$ using the derivative definition, then calculate $g^{\prime}(-4)$ , $g^{\prime}(0)$ , and $g^{\prime}(2)$ for $g(x)=\sqrt{15 x}$ .

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Problem 8821

Find the derivative of the function $f(x) = \sqrt{x+1}$ .

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Problem 8822

Find the acceleration function for the object with position $f(x)=-2 x^{2}+27 x+10$ .

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Problem 8823

Find the function $f(x)$ used for linear approximation of $\sqrt{50}$ with $L(x)=f(a)+f^{\prime}(a)(x-a)$ .

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Problem 8824

How fast is the volume of a sphere expanding when the radius is 1 cm and increasing at 5 cm/s? Provide the exact answer.

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Problem 8825

Find the derivative $f^{\prime}(x)$ for $f(x)=4 x^{3}+4$ and evaluate $f^{\prime}(-3)$ , $f^{\prime}(0)$ , $f^{\prime}(2)$ .

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Problem 8826

Compound A decomposes in water, starting from 0.30 M to 0.26 M in 30 min. How much remains after 3 hours? When will it reach 0.10 M?

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Problem 8827

Find the tangent line equation for $f(x)=x^{2}$ at $x=1$ where the rate of change is 2.

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Problem 8828

Compound A starts at $0.30 \mathrm{M}$ and goes to $0.26 \mathrm{M}$ in 30 min. Find remaining A after 3 hours and time for $0.10 \mathrm{M}$ .

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Problem 8829

A runner on a 100 m radius track moves at 7 m/s. Find the rate of distance change from a friend 200 m from the center when they are 200 m apart. Round to three decimal places.

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Problem 8830

Find quantities for $f(x)=5x^{2}$ : (A) slope of secant line, (B) slope at $(1, f(1))$ , (C) tangent line equation.

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Problem 8831

Approximate $\sqrt{50}$ using $L(x)=f(a)+f^{\prime}(a)(x-a)$ .
a) What is $f(x)$ ? Options: $2\sqrt{x}$ , $\ln(x)$ , none, $\frac{1}{\sqrt{x}}$ .
b) What is $a$ ? Enter a number.

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Problem 8832

Find the following for $y=f(x)=9-x^{2}$ : (A) $\frac{f(-1)-f(-2)}{(-1)-(-2)}$ , (B) $\frac{f(-2+h)-f(-2)}{h}$ , (C) $\lim _{h \rightarrow 0} \frac{f(-2+h)-f(-2)}{h}$ .

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Problem 8833

Find the number of thousands of sunglasses, $q$ , to maximize profit from $P(q)=-0.03 q^{2}+3 q-25$ . What is the max profit?

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Problem 8834

Find the derivative of $y=xe^{3x}$ .

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Problem 8835

Find the second derivative of $f(x) = x^5 \ln x$ given $f'(x) = 5x^4 \ln x + x^4$ .

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Problem 8836

Find the derivative of the function $s(t)=e^{\sqrt{t}}$ .

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Problem 8837

Find the derivative of the function $s(t)=\frac{e^{2 t}}{1+e^{2 t}}$ .

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Problem 8838

A log falls from the 188 ft Horseshoe Falls. Find the height $h$ after $t$ seconds and time to reach the river. Round to nearest tenth.

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Problem 8839

Find the tangent line equation for $y=x^{2}-1$ with slope -3. Also, find points on $y=3x^{2}+x+1$ where tangent is horizontal.

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Problem 8840

Find points on the curve $y=x^{2} e^{-x}$ where the tangent is horizontal.

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Problem 8841

Find the derivative using the definition of the function $f(x) = x^{2} - 1$ .

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Problem 8842

Find $f^{\prime}(1)$ given $f(x)+x^{2}[f(x)]^{4}=18$ and $f(1)=2$ .

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Problem 8843

Explain how $\frac{1}{x}(3x)$ simplifies to $3$ given $f^{\prime}(x) =\frac{3-3 \ln x}{9 x^{2}}$ .

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Problem 8844

Find local max and min values of $f(x)=x+\sqrt{8-x}$ . If none, enter DNE.

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Problem 8845

Evaluate the limit: $\lim _{x \rightarrow \infty}\left(1+\frac{3}{x}+\frac{5}{x^{2}}\right)^{x}$ .

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Problem 8846

Find the derivative of the function $y=e^{-x} \sin x$ .

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Problem 8847

Find the derivative of $\ln \sqrt{x^{2}+17}$ .

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Problem 8848

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{4-\cos x}{7+\cos x}$ .

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Problem 8849

Find the limit as $x$ approaches 0 for $\frac{\sin 5x}{\sin 8x}$ .

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Problem 8850

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

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Problem 8851

Find the second derivative $y^{\prime \prime}$ for the function $y=e^{x^{5}}+5x$ .

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Problem 8852

Find the derivative of $t=\frac{2 x}{2-x}$ .

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Problem 8853

Determine the interval where the slope of $f(x)=\frac{1}{3 x+6}$ is increasing. Options: $x<-2$ , $x>-2$ , $x>2$ , $x \in \mathbb{R}, x \neq-2$ , $x>-6$ .

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Problem 8854

What happens to $f(x)=-\frac{1}{x^{2}+6 x-7}$ as $x \rightarrow 1^{-}$ ? Choose the correct behavior.

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Problem 8855

What happens to $f(x)=\frac{2 x+5}{x+3}$ as $x \rightarrow -3^{+}$ ? Consider limits from above and below.

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Problem 8856

Differentiate implicitly: $e^{x^{2} y}=8 x+4 y+3$ . Find $\frac{d y}{d x}$ .

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Problem 8857

Find the per capita growth rate for a bacterial colony with population $N(t)=N_{0} 2^{t}$ . Calculate $\frac{1}{N} \frac{dN}{dt}$ .

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Problem 8858

Find the differential equation for radioactive decay $W(t)$ with decay rate $1.7/$ day. Choose A, B, C, or D.

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Problem 8859

Differentiate implicitly to find $\frac{d y}{d x}$ for the equation $y^{2}=5 e^{x^{2}}+2 x$ .

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Problem 8860

Insect population $\mathrm{P}(t)=700 e^{0.03 t}$ : (a) Find $\mathrm{P}(0)$ , (b) growth rate, (c) $\mathrm{P}(10)$ , (d) time to reach 1050, (e) time to double.

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Problem 8861

Find the derivative of $y=\frac{e^{3 x^{3}}\left(5 x^{4}+13\right)^{5}}{\left(24 x^{2}+9 x-23\right)^{3}}$ with respect to $x$ .

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Problem 8862

Invest \$100000 at 5\% continuous compounding. (a) Find value after 9 years. (b) When will it reach \$240000? Round to 3 decimals.

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Problem 8863

Find the intervals of concavity for $f(x)$ given $f^{\prime \prime}(x)=30 x^{6}(x+5)^{3}(x^{2}+81)$ .

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Problem 8864

Find the arc length of $y=\frac{x^{3}}{3}+\frac{1}{4x}$ from $x=1$ to $x=2$ using $s=\int_{1}^{2} \sqrt{1+[f'(x)]^{2}} dx$ .

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Problem 8865

Evaluate the integral: $\int \csc ^{2} 2 x(1+\cos 2 x) d x$

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Problem 8866

Find the range of $f(11) - f(7)$ given that $10 \leq f'(x) \leq 13$ on $(7,11)$ .

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Problem 8867

Find the range of $f(8) - f(1)$ given that $12 \leq f'(x) \leq 13$ for $x \in (1,8)$ .

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Problem 8868

Find all values of $c$ in the interval [-4, 2] where $f'(c) = 0$ for the function $f(x) = 2x^3 + \frac{45x^2}{2} + 21x - 2$ .

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Problem 8869

Determine which statements about Rolle's theorem apply to the function $f(x)=x^{3}-x+3$ on $[-1,0]$ .

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Problem 8870

Find the minimum value of $f(14)$ given $f(7)=-9$ and $f'(x) \geq 9$ for $x \in (7,14)$ .

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Problem 8871

Determine which statements about Rolle's theorem apply to the function $f(x)=\frac{4 x+3}{x^{2}-x-12}$ on $[-9, \frac{6}{11}]$ .

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Problem 8872

Does the extreme value theorem ensure an absolute max and min for $f(x)=\sin^{2}(2x)+3$ on $[-\frac{\pi}{3}, \frac{\pi}{2}]$ ? Yes or No?

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Problem 8873

Find critical points of $f(x)=\frac{x^{2}-9}{x^{2}+7x+12}$ . List them separated by commas or enter $\varnothing$ if none exist.

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Problem 8874

Does the extreme value theorem ensure an absolute max and min for $f(x)=2 \tan (x)-3$ on $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ ? Yes or No?

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Problem 8875

Does the extreme value theorem ensure an absolute max and min for $f(x)=\ln(2-3x)$ on $[3,8]$ ? Yes or No?

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Problem 8876

Does the extreme value theorem ensure an absolute max and min for $f(x)=3 e^{\frac{x}{2}-3}$ on $[1,9]$ ? Yes or No?

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Problem 8877

Does the extreme value theorem ensure an absolute max and min for $f(x)=6 x^{2}+(3 x-3)^{\frac{3}{4}}-3$ on $[3,9]$ ? Yes or No?

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Problem 8878

Does the extreme value theorem ensure an absolute max and min for $f(x)=\frac{3}{\sqrt{x^{2}-25}}$ on $[3,4]$ ? Yes or No?

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Problem 8879

Does the extreme value theorem ensure an absolute max and min for $f(x)=\ln(x+5)$ on [1,9]? Yes or No?

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Problem 8880

Does the extreme value theorem ensure an absolute max and min for $f(x)=\ln(3-4x)$ on [3, 9]? Yes or No?

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Problem 8881

Integrate the function: $\int \frac{x+2}{9\left(x^{2}+1\right)} dx$

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Problem 8882

Insect population $P(t)=200 e^{0.05 t}$ . Find: (a) $P(0)$ , (b) growth rate, (c) $P(10)$ , (d) $P(t)=260$ , (e) when $P$ doubles.

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Problem 8883

How long until the population reaches 900, given $P(t)=\frac{100000}{100+900 e^{-t}}$ with initial population 100?

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Problem 8884

Evaluate the integral from 4 to 9 of $\frac{1}{\sqrt{x}} \, dx$ .

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Problem 8885

Find the electric field $\vec{E}(r)$ and force $\vec{F}(r)$ from $1 \times 10^9$ electrons at $1 \, \mathrm{m}$ .

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Problem 8886

Find the integral of the constant 7: $\int 7 \, dx$ .

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Problem 8887

Gegeben ist $f(x)=\frac{1}{4} x^{4}-\frac{1}{3} x^{3}-x^{2}$ . Analysiere Verhalten, Nullstellen, Extrema, Wendetangente und Dreiecksfläche.

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Problem 8888

Find the volume of a solid with a square cross-section, base enclosed by $y=\sqrt{x(2-x)}$ , $y=\frac{1}{2} \sqrt{x(2-x)}$ .

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Problem 8889

Find the tangential and normal acceleration components for the trajectory $\mathbf{r}(t)=\langle e^{t} \sin t, e^{t} \cos t, 12 e^{t}\rangle$ .

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Problem 8890

Find the tangential and normal components of acceleration for the trajectory $\mathbf{r}(t)=\langle e^{t} \sin t, e^{t} \cos t, 12 e^{t} \rangle$ . Tangential: $a_{T}=\sqrt{146} e^{t}$ . Normal: $a_{N}=$ (exact answer).

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Problem 8891

Investing in an account with continuous compounding at $1.8\%$ : (a) How much from $\$ 3000$ after 10 years? (b) To get $\$ 8000$ in 10 years, how much to invest now?

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Problem 8892

Find the optimal lot size and number of orders per year to minimize costs for selling 100 calculus textbooks.

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Problem 8893

Find the limit as $x$ approaches 4 for the function 5. What is the result?

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Problem 8894

Find the marginal cost if the average cost is given by $A(x)=0.4x+4$ .

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Problem 8895

A bookstore plans to sell 110 calculus textbooks. Storage costs \$. 2.50 per book/year and ordering costs \$. 5.50. Find optimal lot size and orders/year.

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Problem 8896

Find the maximum profit for a product with demand \ $13 and cost function$ C(x)=0.2 x^{2}+4.6 x+7$. Round to the nearest cent.

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Problem 8897

Find $\frac{\partial z}{\partial y}$ for $e^{2 x y z}=8 x^{4} y^{6} z^{7}$ and identify coefficients $a, b, c, d, g, k$ .

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Problem 8898

Find the partial derivative of $f(y, z)=(z-6)^{y}$ with respect to $y$ at the point $(y, z)=(-2, 7)$ .

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Problem 8899

Find the partial derivatives of $t(r, s)$ at $(e, e)$ for the equation $8 r^{3} \ln r + 3 s^{3} \ln s = 11 t^{3} \ln t$ . Calculate: - $t_{1}^{\prime}(e, e)$ - $t_{11}^{\prime \prime}(e, e)$ - $t_{12}^{\prime \prime}(e, e)$

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Problem 8900

Calculate the partial and total derivatives of $R(x, y, t)=x y$ with $x(y)=4 y^{3}$ and $y(t)=3 t^{6}$ . Find $\frac{\partial R}{\partial y}=a+b \cdot x^{c} \cdot y^{d}+h \cdot t^{k}$ and determine coefficients $a, b, c, d, h, k$ .

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Calculus

Problem 8801

Find the rate of change of angle AAA given 3cos⁡(A)=1−4cos⁡((P+Q2)2)3 \cos(A) = 1 - 4 \cos\left(\left(\frac{P+Q}{2}\right)^{2}\right)3cos(A)=1−4cos((2P+Q​)2) with rates π12\frac{\pi}{12}12π​ for PPP and −π16-\frac{\pi}{16}−16π​ for QQQ.

Problem 8802

Trouver l'équation de la tangente à f(x)=9x+9x2f(x)=9 x+\frac{9}{x^{2}}f(x)=9x+x29​ au point (−3,−26)(-3,-26)(−3,−26) avec y=mx+by=m x+by=mx+b, où m=m=m= et b=b=b=.

Problem 8803

Which function has the tangent plane z=0z=0z=0 at the point (0,0)(0,0)(0,0)? A. z=cos⁡(x)cos⁡(y)z=\cos (x) \cos (y)z=cos(x)cos(y) B. z=x+yz=x+yz=x+y C. z=exyz=e^{x y}z=exy D. z=x2+y2z=x^{2}+y^{2}z=x2+y2 E. None.

Problem 8804

Find the derivative dy/dxd y / d xdy/dx for y=ln⁡(2x33x2+5)4y=\ln \left(\frac{2 x^{3}}{3 x^{2}+5}\right)^{4}y=ln(3x2+52x3​)4.

Problem 8805

Show that the polynomial f(x)=4x3+2x2−3x+2f(x)=4 x^{3}+2 x^{2}-3 x+2f(x)=4x3+2x2−3x+2 has a zero in the interval [-4,-1] using the intermediate value theorem.

Problem 8806

Find the first and second derivatives of f(x)=9cos⁡2(x)−18sin⁡(x)f(x)=9 \cos^{2}(x)-18 \sin(x)f(x)=9cos2(x)−18sin(x).

Problem 8807

Find the derivative of the function y=3x−2xy=\frac{\sqrt{3x-2}}{x}y=x3x−2​​. What is y′y'y′?

Problem 8808

Calculate the limit: lim⁡x→∞(ex)1x2\lim _{x \rightarrow \infty}\left(e^{x}\right)^{\frac{1}{x^{2}}}limx→∞​(ex)x21​.

Problem 8809

Calculez f′(t)f^{\prime}(t)f′(t) et f′′(t)f^{\prime \prime}(t)f′′(t) pour la fonction f(t)=t5e2tf(t)=t^{5} e^{2 t}f(t)=t5e2t.

Problem 8810

Strontium-90 decays as A(t)=800e−0.0244tA(t)=800 e^{-0.0244 t}A(t)=800e−0.0244t. Find decay rate, amount left after 30 years, time for 600g, and half-life.

Problem 8811

Insect population P(t)=100e0.02t\mathrm{P}(t)=100 e^{0.02 t}P(t)=100e0.02t: (a) Find P(0)\mathrm{P}(0)P(0). (b) What is the growth rate? (c) Find P(10)\mathrm{P}(10)P(10). (d) When is P=160\mathrm{P}=160P=160? (e) When does P\mathrm{P}P double?

Problem 8812

Trouvez les valeurs de xxx où la tangente à la courbe y=2x3−9x2−108x+16y=2 x^{3}-9 x^{2}-108 x+16y=2x3−9x2−108x+16 est horizontale. x= x= x=

Problem 8813

Insect population P(t)=600e0.02t\mathrm{P}(t)=600 e^{0.02 t}P(t)=600e0.02t: (a) Find P(0)\mathrm{P}(0)P(0). (b) What is the growth rate? (c) Find P(10)\mathrm{P}(10)P(10). (d) When is P=780\mathrm{P}=780P=780? (e) When does P\mathrm{P}P double?

Problem 8814

Find the derivative using implicit differentiation for the equation x2y−y=xx^{2}y - y = xx2y−y=x.

Problem 8815

A baseball machine throws a ball up at 64ft/sec64 \mathrm{ft/sec}64ft/sec from 1.5ft1.5 \mathrm{ft}1.5ft. Find the height equation and max height/time.

Problem 8816

Find the tangent line equation for y=1x2y=\frac{1}{x^{2}}y=x21​ at x=3x=3x=3.

Problem 8817

Differentiate the function f(x)=x−e−x1+xe−xf(x)=\frac{x-e^{-x}}{1+x e^{-x}}f(x)=1+xe−xx−e−x​.

Problem 8818

Find the derivative of the function y=3xe−5x+2ex2y=3 x e^{-5 x}+2 e^{x^{2}}y=3xe−5x+2ex2.

Problem 8819

Differentiate the function y=2x4(2x3+5)y=2 x^{4}(2 x^{3}+5)y=2x4(2x3+5) with respect to xxx.

Problem 8820

Find g′(x)g^{\prime}(x)g′(x) using the derivative definition, then calculate g′(−4)g^{\prime}(-4)g′(−4), g′(0)g^{\prime}(0)g′(0), and g′(2)g^{\prime}(2)g′(2) for g(x)=15xg(x)=\sqrt{15 x}g(x)=15x​.

Problem 8821

Find the derivative of the function f(x)=x+1f(x) = \sqrt{x+1}f(x)=x+1​.

Problem 8822

Find the acceleration function for the object with position f(x)=−2x2+27x+10f(x)=-2 x^{2}+27 x+10f(x)=−2x2+27x+10.

Problem 8823

Find the function f(x)f(x)f(x) used for linear approximation of 50\sqrt{50}50​ with L(x)=f(a)+f′(a)(x−a)L(x)=f(a)+f^{\prime}(a)(x-a)L(x)=f(a)+f′(a)(x−a).

Problem 8824

How fast is the volume of a sphere expanding when the radius is 1 cm and increasing at 5 cm/s? Provide the exact answer.

Problem 8825

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=4x3+4f(x)=4 x^{3}+4f(x)=4x3+4 and evaluate f′(−3)f^{\prime}(-3)f′(−3), f′(0)f^{\prime}(0)f′(0), f′(2)f^{\prime}(2)f′(2).

Problem 8826

Compound A decomposes in water, starting from 0.30 M to 0.26 M in 30 min. How much remains after 3 hours? When will it reach 0.10 M?

Problem 8827

Find the tangent line equation for f(x)=x2f(x)=x^{2}f(x)=x2 at x=1x=1x=1 where the rate of change is 2.

Problem 8828

Compound A starts at 0.30M0.30 \mathrm{M}0.30M and goes to 0.26M0.26 \mathrm{M}0.26M in 30 min. Find remaining A after 3 hours and time for 0.10M0.10 \mathrm{M}0.10M.

Problem 8829

A runner on a 100 m radius track moves at 7 m/s. Find the rate of distance change from a friend 200 m from the center when they are 200 m apart. Round to three decimal places.

Problem 8830

Find quantities for f(x)=5x2f(x)=5x^{2}f(x)=5x2: (A) slope of secant line, (B) slope at (1,f(1))(1, f(1))(1,f(1)), (C) tangent line equation.

Problem 8831

Approximate 50\sqrt{50}50​ using L(x)=f(a)+f′(a)(x−a)L(x)=f(a)+f^{\prime}(a)(x-a)L(x)=f(a)+f′(a)(x−a). a) What is f(x)f(x)f(x)? Options: 2x2\sqrt{x}2x​, ln⁡(x)\ln(x)ln(x), none, 1x\frac{1}{\sqrt{x}}x​1​. b) What is aaa? Enter a number.

Problem 8832

Problem 8833

Find the number of thousands of sunglasses, qqq, to maximize profit from P(q)=−0.03q2+3q−25P(q)=-0.03 q^{2}+3 q-25P(q)=−0.03q2+3q−25. What is the max profit?

Problem 8834

Find the derivative of y=xe3xy=xe^{3x}y=xe3x.

Problem 8835

Find the second derivative of f(x)=x5ln⁡xf(x) = x^5 \ln xf(x)=x5lnx given f′(x)=5x4ln⁡x+x4f'(x) = 5x^4 \ln x + x^4f′(x)=5x4lnx+x4.

Problem 8836

Find the derivative of the function s(t)=ets(t)=e^{\sqrt{t}}s(t)=et​.

Problem 8837

Find the derivative of the function s(t)=e2t1+e2ts(t)=\frac{e^{2 t}}{1+e^{2 t}}s(t)=1+e2te2t​.

Problem 8838

A log falls from the 188 ft Horseshoe Falls. Find the height hhh after ttt seconds and time to reach the river. Round to nearest tenth.

Problem 8839

Find the tangent line equation for y=x2−1y=x^{2}-1y=x2−1 with slope -3. Also, find points on y=3x2+x+1y=3x^{2}+x+1y=3x2+x+1 where tangent is horizontal.

Problem 8840

Find points on the curve y=x2e−xy=x^{2} e^{-x}y=x2e−x where the tangent is horizontal.

Find the rate of change of angle $A$ given $3 \cos(A) = 1 - 4 \cos\left(\left(\frac{P+Q}{2}\right)^{2}\right)$ with rates $\frac{\pi}{12}$ for $P$ and $-\frac{\pi}{16}$ for $Q$ .

Trouver l'équation de la tangente à $f(x)=9 x+\frac{9}{x^{2}}$ au point $(-3,-26)$ avec $y=m x+b$ , où $m=$ et $b=$ .

Which function has the tangent plane $z=0$ at the point $(0,0)$ ? A. $z=\cos (x) \cos (y)$ B. $z=x+y$ C. $z=e^{x y}$ D. $z=x^{2}+y^{2}$ E. None.

Find the derivative $d y / d x$ for $y=\ln \left(\frac{2 x^{3}}{3 x^{2}+5}\right)^{4}$ .

Show that the polynomial $f(x)=4 x^{3}+2 x^{2}-3 x+2$ has a zero in the interval [-4,-1] using the intermediate value theorem.

Find the first and second derivatives of $f(x)=9 \cos^{2}(x)-18 \sin(x)$ .

Find the derivative of the function $y=\frac{\sqrt{3x-2}}{x}$ . What is $y'$ ?

Calculate the limit: $\lim _{x \rightarrow \infty}\left(e^{x}\right)^{\frac{1}{x^{2}}}$ .

Calculez $f^{\prime}(t)$ et $f^{\prime \prime}(t)$ pour la fonction $f(t)=t^{5} e^{2 t}$ .

Strontium-90 decays as $A(t)=800 e^{-0.0244 t}$ . Find decay rate, amount left after 30 years, time for 600g, and half-life.

Insect population $\mathrm{P}(t)=100 e^{0.02 t}$ : (a) Find $\mathrm{P}(0)$ . (b) What is the growth rate? (c) Find $\mathrm{P}(10)$ . (d) When is $\mathrm{P}=160$ ? (e) When does $\mathrm{P}$ double?

Trouvez les valeurs de $x$ où la tangente à la courbe $y=2 x^{3}-9 x^{2}-108 x+16$ est horizontale. $x=$

Insect population $\mathrm{P}(t)=600 e^{0.02 t}$ : (a) Find $\mathrm{P}(0)$ . (b) What is the growth rate? (c) Find $\mathrm{P}(10)$ . (d) When is $\mathrm{P}=780$ ? (e) When does $\mathrm{P}$ double?

Find the derivative using implicit differentiation for the equation $x^{2}y - y = x$ .

A baseball machine throws a ball up at $64 \mathrm{ft/sec}$ from $1.5 \mathrm{ft}$ . Find the height equation and max height/time.

Find the tangent line equation for $y=\frac{1}{x^{2}}$ at $x=3$ .

Differentiate the function $f(x)=\frac{x-e^{-x}}{1+x e^{-x}}$ .

Find the derivative of the function $y=3 x e^{-5 x}+2 e^{x^{2}}$ .

Differentiate the function $y=2 x^{4}(2 x^{3}+5)$ with respect to $x$ .

Find $g^{\prime}(x)$ using the derivative definition, then calculate $g^{\prime}(-4)$ , $g^{\prime}(0)$ , and $g^{\prime}(2)$ for $g(x)=\sqrt{15 x}$ .

Find the derivative of the function $f(x) = \sqrt{x+1}$ .

Find the acceleration function for the object with position $f(x)=-2 x^{2}+27 x+10$ .

Find the function $f(x)$ used for linear approximation of $\sqrt{50}$ with $L(x)=f(a)+f^{\prime}(a)(x-a)$ .

Find the derivative $f^{\prime}(x)$ for $f(x)=4 x^{3}+4$ and evaluate $f^{\prime}(-3)$ , $f^{\prime}(0)$ , $f^{\prime}(2)$ .

Find the tangent line equation for $f(x)=x^{2}$ at $x=1$ where the rate of change is 2.

Compound A starts at $0.30 \mathrm{M}$ and goes to $0.26 \mathrm{M}$ in 30 min. Find remaining A after 3 hours and time for $0.10 \mathrm{M}$ .

Find quantities for $f(x)=5x^{2}$ : (A) slope of secant line, (B) slope at $(1, f(1))$ , (C) tangent line equation.

Approximate $\sqrt{50}$ using $L(x)=f(a)+f^{\prime}(a)(x-a)$ .
a) What is $f(x)$ ? Options: $2\sqrt{x}$ , $\ln(x)$ , none, $\frac{1}{\sqrt{x}}$ .
b) What is $a$ ? Enter a number.

Find the number of thousands of sunglasses, $q$ , to maximize profit from $P(q)=-0.03 q^{2}+3 q-25$ . What is the max profit?

Find the derivative of $y=xe^{3x}$ .

Find the second derivative of $f(x) = x^5 \ln x$ given $f'(x) = 5x^4 \ln x + x^4$ .

Find the derivative of the function $s(t)=e^{\sqrt{t}}$ .

Find the derivative of the function $s(t)=\frac{e^{2 t}}{1+e^{2 t}}$ .

A log falls from the 188 ft Horseshoe Falls. Find the height $h$ after $t$ seconds and time to reach the river. Round to nearest tenth.

Find the tangent line equation for $y=x^{2}-1$ with slope -3. Also, find points on $y=3x^{2}+x+1$ where tangent is horizontal.

Find points on the curve $y=x^{2} e^{-x}$ where the tangent is horizontal.

Find the derivative using the definition of the function $f(x) = x^{2} - 1$ .

Find $f^{\prime}(1)$ given $f(x)+x^{2}[f(x)]^{4}=18$ and $f(1)=2$ .

Explain how $\frac{1}{x}(3x)$ simplifies to $3$ given $f^{\prime}(x) =\frac{3-3 \ln x}{9 x^{2}}$ .

Find local max and min values of $f(x)=x+\sqrt{8-x}$ . If none, enter DNE.

Evaluate the limit: $\lim _{x \rightarrow \infty}\left(1+\frac{3}{x}+\frac{5}{x^{2}}\right)^{x}$ .

Find the derivative of the function $y=e^{-x} \sin x$ .

Find the derivative of $\ln \sqrt{x^{2}+17}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{4-\cos x}{7+\cos x}$ .

Find the limit as $x$ approaches 0 for $\frac{\sin 5x}{\sin 8x}$ .

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

Find the second derivative $y^{\prime \prime}$ for the function $y=e^{x^{5}}+5x$ .

Find the derivative of $t=\frac{2 x}{2-x}$ .

Determine the interval where the slope of $f(x)=\frac{1}{3 x+6}$ is increasing. Options: $x<-2$ , $x>-2$ , $x>2$ , $x \in \mathbb{R}, x \neq-2$ , $x>-6$ .

What happens to $f(x)=-\frac{1}{x^{2}+6 x-7}$ as $x \rightarrow 1^{-}$ ? Choose the correct behavior.

What happens to $f(x)=\frac{2 x+5}{x+3}$ as $x \rightarrow -3^{+}$ ? Consider limits from above and below.

Differentiate implicitly: $e^{x^{2} y}=8 x+4 y+3$ . Find $\frac{d y}{d x}$ .

Find the per capita growth rate for a bacterial colony with population $N(t)=N_{0} 2^{t}$ . Calculate $\frac{1}{N} \frac{dN}{dt}$ .

Find the differential equation for radioactive decay $W(t)$ with decay rate $1.7/$ day. Choose A, B, C, or D.

Differentiate implicitly to find $\frac{d y}{d x}$ for the equation $y^{2}=5 e^{x^{2}}+2 x$ .

Insect population $\mathrm{P}(t)=700 e^{0.03 t}$ : (a) Find $\mathrm{P}(0)$ , (b) growth rate, (c) $\mathrm{P}(10)$ , (d) time to reach 1050, (e) time to double.

Find the derivative of $y=\frac{e^{3 x^{3}}\left(5 x^{4}+13\right)^{5}}{\left(24 x^{2}+9 x-23\right)^{3}}$ with respect to $x$ .

Find the intervals of concavity for $f(x)$ given $f^{\prime \prime}(x)=30 x^{6}(x+5)^{3}(x^{2}+81)$ .

Find the arc length of $y=\frac{x^{3}}{3}+\frac{1}{4x}$ from $x=1$ to $x=2$ using $s=\int_{1}^{2} \sqrt{1+[f'(x)]^{2}} dx$ .

Evaluate the integral: $\int \csc ^{2} 2 x(1+\cos 2 x) d x$

Find the range of $f(11) - f(7)$ given that $10 \leq f'(x) \leq 13$ on $(7,11)$ .

Find the range of $f(8) - f(1)$ given that $12 \leq f'(x) \leq 13$ for $x \in (1,8)$ .

Find all values of $c$ in the interval [-4, 2] where $f'(c) = 0$ for the function $f(x) = 2x^3 + \frac{45x^2}{2} + 21x - 2$ .

Determine which statements about Rolle's theorem apply to the function $f(x)=x^{3}-x+3$ on $[-1,0]$ .

Find the minimum value of $f(14)$ given $f(7)=-9$ and $f'(x) \geq 9$ for $x \in (7,14)$ .

Determine which statements about Rolle's theorem apply to the function $f(x)=\frac{4 x+3}{x^{2}-x-12}$ on $[-9, \frac{6}{11}]$ .

Does the extreme value theorem ensure an absolute max and min for $f(x)=\sin^{2}(2x)+3$ on $[-\frac{\pi}{3}, \frac{\pi}{2}]$ ? Yes or No?

Find critical points of $f(x)=\frac{x^{2}-9}{x^{2}+7x+12}$ . List them separated by commas or enter $\varnothing$ if none exist.

Does the extreme value theorem ensure an absolute max and min for $f(x)=2 \tan (x)-3$ on $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ ? Yes or No?