Calculus

Problem 25701

Find the weight function $W(h)$ given $\frac{d W}{d h}=0.0012 h^{2}$ and $W(80)=224.8$ . What is $W(69)$ ?

See Solution

Problem 25702

Find the derivative of $f(x)=\int_{x}^{-2}(5 t^{2}+\arccos (t)) dt$ .

See Solution

Problem 25703

Apply the Mean Value Theorem to $f(x)=(x-2)^{2}(x-3)$ on $[1,4]$ . Find $c$ such that $f^{\prime}(c)=$ average rate of change.

See Solution

Problem 25704

Find the derivative $f^{\prime}(x)$ and second derivative $f^{\prime \prime}(x)$ for $f(x)=x^{3}-5x$ . Graph them together.

See Solution

Problem 25705

Use implicit differentiation on $y^{3}-y^{2}+y-2=x$ to find $\frac{d y}{d x}$ .

See Solution

Problem 25706

Find the domain of $f(x)=\frac{x+1}{x^{2}}$ , limits at the ends, and intercepts. Domain: (-\infty, 0) U (0, \infty); Limits: ?, Intercepts: (-1, 0).

See Solution

Problem 25707

Find the average slope of $f(x)=\frac{1}{x}$ on $[4,9]$ . Then, find $c$ in $(4,9)$ where $f^{\prime}(c)$ equals this slope.

See Solution

Problem 25708

Graph the function $f(x)=x^{3}+3 x^{2}-9 x-8$ and identify all relative extrema and inflection points.

See Solution

Problem 25709

Approximate $4.6^{5}$ using the tangent line of $f(x)=x^{5}$ at $x=5$ . Find $m$ and $b$ for $y=m x+b$ .

See Solution

Problem 25710

Find the tangent line equation for $y^{5}+2xy^{2}+3x^{2}=4$ at $(1,-1)$ in the form $y=mx+b$ .

See Solution

Problem 25711

Find the average slope of $f(x)=\frac{1}{x}$ on $[5,10]$ and values of $c$ in $(5,10)$ where $f'(c)$ equals this slope.

See Solution

Problem 25712

Find the population $N(9)$ of a city after 9 years if $\frac{d N}{d t}=400+300 \sqrt{t}$ and $N(0)=2000$ .

See Solution

Problem 25713

Find critical numbers, increasing/decreasing regions, and classify extrema for: A. $f(x)=(x+2)x(x-4)$ B. $f(x)=\frac{x^{2}-4x-5}{x-2}$ .

See Solution

Problem 25714

Calculate the integral of $e^{4x} \cdot 4 \, dx$ and verify by differentiation. Result: $\int e^{4x}(4) \, dx = \square$

See Solution

Problem 25715

Gegeben ist die Funktion $f_{a}(x) = 2x^{2} - 4ax + 6a - 4$ . Bestimme Ableitungen, Extrempunkte und zeichne den Graphen für $a=1$ und $a=0,6$ im Intervall $[-3; 3]$ .

See Solution

Problem 25716

Evaluate the integral from 3 to 6 of the function $9x^{2} - 8x + 6$ .

See Solution

Problem 25717

Find the critical numbers of the function $f(z)=\frac{4 z+4}{10 z^{2}+10 z+10}$ in increasing order. Use $\mathrm{N}$ for unused blanks.

See Solution

Problem 25718

Evaluate the integral $\frac{1}{6-x} \int_{x}^{6}(6x) dx$ and solve $\frac{1}{3-0} \int_{0}^{3} 11(t) dt$ .

See Solution

Problem 25719

Find these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{5+5 x^{2}}}{8+6 x}=$ , (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{5+5 x^{2}}}{8+6 x}=$ .

See Solution

Problem 25720

Find the critical numbers of $f(\theta)=12 \cos (\theta)+6 \sin ^{2}(\theta)$ for $-\pi \leq \theta \leq \pi$ . Use $\mathrm{N}$ for blanks.

See Solution

Problem 25721

Evaluate the limits and value of the piecewise function $g(x)$ at specific points: $x=1$ and $x=2$ .

See Solution

Problem 25722

Trouvez la dérivée de la fonction $g(x) = \frac{1}{\sqrt{x}} + \sqrt[4]{x}$ .

See Solution

Problem 25723

Find the derivative of the function $f(x)=5 x^{2}+9 x \ln (x)+1$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 25724

Evaluate the integral or state "IMPOSSIBLE": $\int x^{2} \sqrt[3]{x^{3}-8} d x$ . Use $C$ for the constant.

See Solution

Problem 25725

Find how $dV/dt$ relates to $dr/dt$ (with constant $h$ ), $dh/dt$ (with constant $r$ ), and both $dh/dt$ and $dr/dt$ .

See Solution

Problem 25726

Find the intervals where the function $f(x)=\frac{x+1}{x^{2}}$ is increasing or decreasing based on its derivatives.

See Solution

Problem 25727

Find the limit: $\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+4} d t}{x}$ . What is it equal to?

See Solution

Problem 25728

Find where $f(x)=8 \sqrt{x}-8 x$ is increasing/decreasing and the $x$ -coordinates of relative maxima/minima.

See Solution

Problem 25729

Find the intervals where the function $f(x)=\frac{x+1}{x^{2}}$ is concave up and down using $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ .

See Solution

Problem 25730

Find the absolute min and max of $f(x)=8 x^{3}-24 x$ on the interval $[0,5]$ .

See Solution

Problem 25731

Gegeben ist $f_{a}(x)=2 x^{2}-4 a x+6 a-4,5$ . Bestimmen Sie Ableitungen, Steigung bei $x=0$ , Punkt $S(1,5/0)$ , Extrempunkte und $a$ für Extrempunkt auf $x$ -Achse.

See Solution

Problem 25732

Find the value of the integral $\int_{1}^{\infty} f\left(x^{m}\right) x^{m-1} d x$ given $F(1)=-4$ , $F(0)=1$ , $\lim_{x \rightarrow \infty} F(x)=\infty$ , and $m<0$ . Options: $\frac{5}{m}$ , $\frac{4}{m}$ , 0, diverges, 5.

See Solution

Problem 25733

Calculate the integral $\int_{-1}^{\pi} f(x) dx$ for the piecewise function $f(x)$ defined as $f(x) = -x^2$ for $x \leq 0$ and $f(x) = -\sin x + C$ for $x > 0$ .

See Solution

Problem 25734

Find the concavity and inflection points for $f(x)=2 x^{3}-3 x^{2}-12 x+5$ and $g(x)=-\frac{1}{8}(x+2)^{2}(x-4)^{2}$ .

See Solution

Problem 25735

Find the derivative of $\int_{\sin x}^{\cos x} e^{t} dt$ . Choose from: $e^{\sin (x)}+e^{\cos (x)}$ , $\cos (x) e^{\sin (x)}+\sin (x) e^{\cos (x)}$ , $-\cos (x) e^{\sin (x)}-\sin (x) e^{\cos (x)}$ , or $e^{\cos (x)}-e^{\sin (x)}$ .

See Solution

Problem 25736

Gegeben ist die Funktion $f_{a}(x)=2 x^{2}-4 a x+6 a-4,5$ . Finde die 1. und 2. Ableitung und die Steigung bei $x=0$ .

See Solution

Problem 25737

Given $x y=-3$ and $\frac{d y}{d t}=3$ , find $\frac{d x}{d t}$ when $x=1$ . What is $\frac{d x}{d t}$ ?

See Solution

Problem 25738

Find $\frac{d P}{d t}$ for $b=1.4, P=7 \mathrm{kPa}, V=60 \mathrm{cm}^{2}$ , $\frac{d V}{d t}=30 \mathrm{cm}^{3}/\mathrm{min}$ . $\frac{d P}{d t}=\square \mathrm{kPa}/\mathrm{min}$

See Solution

Problem 25739

Analyze an energy diagram of two molecules.
1. At which separation is the attractive force greatest?
2. Which separation is stable equilibrium?

See Solution

Problem 25740

Let $f$ be a continuous function on $(0, \infty)$ with $\lim _{x \rightarrow 0^{+}} f(x)=\infty$ and $\lim _{x \rightarrow \infty} f(x)=0$ . If $g(x) \geq f(x)$ , which statement is always TRUE?

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Problem 25741

A balloon inflates at $10.5 \mathrm{~cm}^{3}/\mathrm{s}$ . Find the radius growth rate when the radius is $1 \mathrm{~cm}$ , $10 \mathrm{~cm}$ , and $100 \mathrm{~cm}$ .

See Solution

Problem 25742

Gegeben ist die Funktion $f_{a}(x)=2 x^{2}-4 a x+6 a-4,5$ . Bestimme die Extrempunkte in Abhängigkeit von $a$ und den Wert für $a$ , wenn der Extrempunkt auf der $x$ -Achse liegt.

See Solution

Problem 25743

Which statement is TRUE by the Comparison Test? Consider the integrals and their convergence properties.

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Problem 25744

Find the domain of $f(x)=\frac{x+1}{x^{2}}$ , limits at domain ends, and intercepts $(x, y)$ .

See Solution

Problem 25745

Find the volume increase rate in $\mathrm{cm}^{3} / \mathrm{sec}$ for a sphere with radius $3 \mathrm{~cm}$ and rate $3 \mathrm{~cm/sec}$ .

See Solution

Problem 25746

Find the rate of area increase for a rectangle with length $40 \mathrm{~cm}$ , width $20 \mathrm{~cm}$ , length rate $6 \mathrm{~cm/s}$ , and width rate $4 \mathrm{~cm/s}$ . Answer in $\mathrm{cm}^{2} / \mathrm{s}$ .

See Solution

Problem 25747

Find the marginal revenue equations for the revenue function $R(x, y)=70 x+120 y-2 x^{2}-4 y^{2}-x y$ .

See Solution

Problem 25748

Find the marginal revenue equations for $R(x, y)=70x+120y-2x^2-4y^2-xy$ . Set $R_x=0$ and $R_y=0$ to maximize revenue.

See Solution

Problem 25749

Find the partial derivatives of the function $f(x, y)=-6 x^{2}-5 x y^{3}-3 y^{4}$ : $f_{x}(x, y)=$ and $f_{y}(x, y)=$ .

See Solution

Problem 25750

Find the second partial derivatives of the function $f(x, y)=4 x^{3}-2 x y^{2}+5 y^{6}$ : $f_{x x}(x, y)$ and $f_{x y}(x, y)$ .

See Solution

Problem 25751

Find the maximum revenue from the function $R(x)=80 x-0.2 x^{2}$ and the units $x$ needed for it.

See Solution

Problem 25752

Find the time $t$ (in min) when the drug concentration $C(t)=0.06 t-0.0002 t^{2}$ is maximized and its value.

See Solution

Problem 25753

A company's revenue is $R(q)=-q^{3}+320 q^{2}$ and cost is $C(q)=170+14 q$ .
A) Find the marginal profit function $M P(q)$ . B) Determine the quantity (in hundreds) to maximize profits.

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Problem 25754

Evaluate the limits and value of the piecewise function $g(x)$ at specific points. Sketch the graph of $g$ .

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Problem 25755

Find the total cost for producing the first 25 units given the marginal cost function $\frac{9}{\sqrt{x}}$ . Total cost: \$

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Problem 25756

Bestimmen Sie die Extrem- und Wendepunkte der Funktion $f(x)=x \cdot(\ln (x))^{2}$ .

See Solution

Problem 25757

Evaluate the integral: $\int x^{2} \sqrt{10+x^{3}} \, dx$

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Problem 25758

Given $f(x)=3 x^{4}-3 e^{x}$ , find $f'(x)$ and $f'(5)$ , then find $f''(x)$ and $f''(5)$ .

See Solution

Problem 25759

Find the total cost for producing 100 units, given the marginal cost $M C(x)=x^{-1/2}+4$ . Round to the nearest dollar.

See Solution

Problem 25760

Evaluate the integral from 1 to 7 of $\frac{4 x^{2}+9}{\sqrt{x}}$ dx.

See Solution

Problem 25761

Solve the equation $\frac{d x}{d t}=\frac{6}{x}$ with initial condition $x(0)=8$ to find $x(t)=$ .

See Solution

Problem 25762

Evaluate the integral: $\int \frac{2 d x}{x \ln (4 x)}$ .

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Problem 25763

Find the first derivative of $g(x)=4 x^{3}+54 x^{2}+240 x$ , then determine $g^{\prime \prime}(x)$ and $g^{\prime \prime}(-5)$ . Is it concave up or down? Local minimum or maximum at $x=-5$ ?

See Solution

Problem 25764

Find the slope of the tangent line to the curve $3 x^{2}-x y-3 y^{3}=-45$ at the point $(-3,3)$ .

See Solution

Problem 25765

Finde die Punkte, wo die Tangente von $f$ einen Steigungswinkel von 21,8\% hat. Funktionen: a) $f(x)=5 x^{2}$ , b) $f(x)=-\frac{40}{x}$ , c) $f(x)=\frac{5}{6} x^{3}$ , d) $f(x)=0,15 x^{2}-0,2 x$ .

See Solution

Problem 25766

Find the rate of change of the volume $V=\frac{1}{3} \pi (2^{2}) h$ when $h=10$ and $dh/dt=3$ inches/sec.

See Solution

Problem 25767

Compute the integral of the derivative from $x=3$ to $x=5$ : $\int_{3}^{5} f^{\prime}(x) d x$ . Choices: -4, -1, 2, 4.

See Solution

Problem 25768

Evaluate the integral from 1.7 to 5.8 of $(0.2 e^{-0.2 A} + \frac{5}{A}) dA$ . Round to three decimal places.

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Problem 25769

Find the indefinite integral $\int \frac{\ln (x)}{3 x} d x$ using the substitution $u=\ln (x)$ . What does it transform into?

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Problem 25770

Find the limit definition of $f^{\prime}(4)$ for $f(x)=3 x^{2}-\csc x$ . No need to simplify or evaluate.

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Problem 25771

Find the slope of the tangent line to the curve $-x^{2}+2xy+3y^{3}=208$ at the point $(4,4)$ .

See Solution

Problem 25772

Eine Firma modelliert Verkaufszahlen eines Smartphones mit $f_{k}(t)=k \cdot(t-15) \cdot e^{-0,01 t}+15 k$ .
a) Bestimme $k$ für mindestens 750 Verkäufe täglich. b) Zeige, dass der maximale Verkaufszeitpunkt unabhängig von $k$ ist. c) Zeige, dass Verkaufszahlen nach dem Maximum sinken. d) Berechne den Zeitpunkt des stärksten Rückgangs. e) Bestimme die maximale und langfristige Verkaufszahl für $k=100$ und $k=200$ .

See Solution

Problem 25773

Evaluate the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-x-12}{x+3}$ .

See Solution

Problem 25774

Find the first derivative of $g(x)=6 x^{3}+27 x^{2}+36 x$ . Then find the second derivative and evaluate it at $x=-2$ . Is it concave up or down? Local min or max?

See Solution

Problem 25775

Solve the equation $\frac{dx}{dt}=\frac{3}{x}$ with initial condition $x(0)=5$ to find $x(t)$ .

See Solution

Problem 25776

Evaluate the Riemann sum of $f(x)$ on $[-2,4]$ using 2 rectangles with equal width and right endpoints.

See Solution

Problem 25777

Estimate $\lim_{x \rightarrow \infty} f(x)$ using the provided values for $f(x)$ at $x = 10, 100, 1000, 10000, 100000, 1000000$ .

See Solution

Problem 25778

Compute the integral: $\int_{0}^{1}(24 x^{5}+4 x^{3}) e^{4 x^{6}+x^{4}} d x=$

See Solution

Problem 25779

Approximate the root of $f(x)=\sin(x)$ using 5 iterations of Newton's Method starting from $x_0=3.7$ , to 5 decimal places.

See Solution

Problem 25780

Calculate the integral $\int_{0.1}^{3} \frac{3 \ln (5 x)}{2 x} d x$ and round to 4 decimal places.

See Solution

Problem 25781

Evaluate the integral from 2 to 5 of $\frac{10 x^{2}+9}{\sqrt{x}}$ dx.

See Solution

Problem 25782

Evaluate the integral: $\int x^{2} \sqrt{8+x^{3}} \, dx$

See Solution

Problem 25783

Find the derivative function $f^{\prime}$ for each: (a) $f(x)=-3 x^{5}+x^{4}-x+2$ , (b) $f(x)=e^{-5 x}$ , (c) $f(x)=(x \cos x)^{1729}$ .

See Solution

Problem 25784

Given $f(x)=4 x^{4}-2 e^{x}$ , find $f'(x)$ and $f'(5)$ ; also find $f''(x)$ and $f''(5)$ .

See Solution

Problem 25785

A cannon fires a projectile with initial horizontal velocity 866 m/s and vertical velocity 500 m/s. Find the max height: A. $1.54 \times 10^{4} \mathrm{~m}$ B. $4.42 \times 10^{4} \mathrm{~m}$ C. $1.28 \times 10^{4} \mathrm{~m}$ D. $2.50 \times 10^{3} \mathrm{~m}$

See Solution

Problem 25786

Evaluate the integral: $\int_{5}^{10} \frac{6}{x} d x=\square$ (Provide the exact answer.)

See Solution

Problem 25787

Find the total cost for producing the first 64 units given the marginal cost function $\frac{8}{\sqrt{x}}$ . Total cost: \$

See Solution

Problem 25788

Rewrite the integral $\int_{1}^{3}(18 x-1) e^{9 x^{2}-x} d x$ using the substitution $u=9 x^{2}-x$ .

See Solution

Problem 25789

Evaluate the integral: $$\int_{5}^{10} \frac{6}{x} d x = \square($ Type an exact answer.)

See Solution

Problem 25790

A product's monthly sales $q(t)=5000t-170t^{2}$ and price $p(t)=170-t^{2}$ . Find $R^{\prime}(t)$ and its value at $t=6$ .

See Solution

Problem 25791

Find the second partial derivatives of the function $f(x, y)=3 x^{2}+6 x y^{5}+2 y^{6}$ : $f_{x x}(x, y)$ and $f_{x y}(x, y)$ .

See Solution

Problem 25792

Find the first 5 values using Newton's Method for $f(x)=x^{3}-3x^{2}+x+3$ starting with $x_{0}=1$ . Why might it fail?

See Solution

Problem 25793

Determine if the function $h(x)$ is concave up, down, or neither using these values: $(-3, -10)$ , $(0, -3)$ , $(3, 2)$ , $(6, 4)$ , $(9, 5)$ .

See Solution

Problem 25794

Find the critical point of $f(x, y)=-3-8x-3x^{2}+y+6y^{2}$ . What type is it? Select an answer $\vee$

See Solution

Problem 25795

Find the definite integral using the Fundamental Theorem of Calculus: $\int_{-4}^{1}\left(x^{2}-x+1\right) d x$ .

See Solution

Problem 25796

Evaluate the integral: $\int 15 x^{\frac{1}{2}} d x = \square$

See Solution

Problem 25797

Find the derivatives $f_{x}(2,4)$ and $f_{y}(2,4)$ for the function $f(x, y)=6 x^{3}-4 x y^{2}+4 y^{4}$ .

See Solution

Problem 25798

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=3x+1$ , where $h \neq 0$ .

See Solution

Problem 25799

Approximate the solution of $e^{-x}=-3 x+9$ using Newton's method. Start with $x_1=1$ . Find $x_2$ , $x_3$ , and final $x$ .

See Solution

Problem 25800

What are the units of the integral $\int_{0}^{20} f(t) dt$ where $f(x)$ is force in Newtons and $x$ is distance in meters?

See Solution

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Calculus

Problem 25701

Find the weight function W(h)W(h)W(h) given dWdh=0.0012h2\frac{d W}{d h}=0.0012 h^{2}dhdW​=0.0012h2 and W(80)=224.8W(80)=224.8W(80)=224.8. What is W(69)W(69)W(69)?

Problem 25702

Find the derivative of f(x)=∫x−2(5t2+arccos⁡(t))dtf(x)=\int_{x}^{-2}(5 t^{2}+\arccos (t)) dtf(x)=∫x−2​(5t2+arccos(t))dt.

Problem 25703

Apply the Mean Value Theorem to f(x)=(x−2)2(x−3)f(x)=(x-2)^{2}(x-3)f(x)=(x−2)2(x−3) on [1,4][1,4][1,4]. Find ccc such that f′(c)=f^{\prime}(c)=f′(c)= average rate of change.

Problem 25704

Find the derivative f′(x)f^{\prime}(x)f′(x) and second derivative f′′(x)f^{\prime \prime}(x)f′′(x) for f(x)=x3−5xf(x)=x^{3}-5xf(x)=x3−5x. Graph them together.

Problem 25705

Use implicit differentiation on y3−y2+y−2=xy^{3}-y^{2}+y-2=xy3−y2+y−2=x to find dydx\frac{d y}{d x}dxdy​.

Problem 25706

Find the domain of f(x)=x+1x2f(x)=\frac{x+1}{x^{2}}f(x)=x2x+1​, limits at the ends, and intercepts. Domain: (-\infty, 0) U (0, \infty); Limits: ?, Intercepts: (-1, 0).

Problem 25707

Find the average slope of f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ on [4,9][4,9][4,9]. Then, find ccc in (4,9)(4,9)(4,9) where f′(c)f^{\prime}(c)f′(c) equals this slope.

Problem 25708

Graph the function f(x)=x3+3x2−9x−8f(x)=x^{3}+3 x^{2}-9 x-8f(x)=x3+3x2−9x−8 and identify all relative extrema and inflection points.

Problem 25709

Approximate 4.654.6^{5}4.65 using the tangent line of f(x)=x5f(x)=x^{5}f(x)=x5 at x=5x=5x=5. Find mmm and bbb for y=mx+by=m x+by=mx+b.

Problem 25710

Find the tangent line equation for y5+2xy2+3x2=4y^{5}+2xy^{2}+3x^{2}=4y5+2xy2+3x2=4 at (1,−1)(1,-1)(1,−1) in the form y=mx+by=mx+by=mx+b.

Problem 25711

Find the average slope of f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ on [5,10][5,10][5,10] and values of ccc in (5,10)(5,10)(5,10) where f′(c)f'(c)f′(c) equals this slope.

Problem 25712

Find the population N(9)N(9)N(9) of a city after 9 years if dNdt=400+300t\frac{d N}{d t}=400+300 \sqrt{t}dtdN​=400+300t​ and N(0)=2000N(0)=2000N(0)=2000.

Problem 25713

Find critical numbers, increasing/decreasing regions, and classify extrema for: A. f(x)=(x+2)x(x−4)f(x)=(x+2)x(x-4)f(x)=(x+2)x(x−4) B. f(x)=x2−4x−5x−2f(x)=\frac{x^{2}-4x-5}{x-2}f(x)=x−2x2−4x−5​.

Problem 25714

Calculate the integral of e4x⋅4 dxe^{4x} \cdot 4 \, dxe4x⋅4dx and verify by differentiation. Result: ∫e4x(4) dx=□\int e^{4x}(4) \, dx = \square∫e4x(4)dx=□

Problem 25715

Gegeben ist die Funktion fa(x)=2x2−4ax+6a−4f_{a}(x) = 2x^{2} - 4ax + 6a - 4fa​(x)=2x2−4ax+6a−4. Bestimme Ableitungen, Extrempunkte und zeichne den Graphen für a=1a=1a=1 und a=0,6a=0,6a=0,6 im Intervall [−3;3][-3; 3][−3;3].

Problem 25716

Evaluate the integral from 3 to 6 of the function 9x2−8x+69x^{2} - 8x + 69x2−8x+6.

Problem 25717

Find the critical numbers of the function f(z)=4z+410z2+10z+10f(z)=\frac{4 z+4}{10 z^{2}+10 z+10}f(z)=10z2+10z+104z+4​ in increasing order. Use N\mathrm{N}N for unused blanks.

Problem 25718

Evaluate the integral 16−x∫x6(6x)dx\frac{1}{6-x} \int_{x}^{6}(6x) dx6−x1​∫x6​(6x)dx and solve 13−0∫0311(t)dt\frac{1}{3-0} \int_{0}^{3} 11(t) dt3−01​∫03​11(t)dt.

Problem 25719

Find these limits: (a) lim⁡x→∞5+5x28+6x=\lim _{x \rightarrow \infty} \frac{\sqrt{5+5 x^{2}}}{8+6 x}=limx→∞​8+6x5+5x2​​=, (b) lim⁡x→−∞5+5x28+6x=\lim _{x \rightarrow-\infty} \frac{\sqrt{5+5 x^{2}}}{8+6 x}=limx→−∞​8+6x5+5x2​​=.

Problem 25720

Find the critical numbers of f(θ)=12cos⁡(θ)+6sin⁡2(θ)f(\theta)=12 \cos (\theta)+6 \sin ^{2}(\theta)f(θ)=12cos(θ)+6sin2(θ) for −π≤θ≤π-\pi \leq \theta \leq \pi−π≤θ≤π. Use N\mathrm{N}N for blanks.

Problem 25721

Evaluate the limits and value of the piecewise function g(x)g(x)g(x) at specific points: x=1x=1x=1 and x=2x=2x=2.

Problem 25722

Trouvez la dérivée de la fonction g(x)=1x+x4g(x) = \frac{1}{\sqrt{x}} + \sqrt[4]{x}g(x)=x​1​+4x​.

Problem 25723

Find the derivative of the function f(x)=5x2+9xln⁡(x)+1f(x)=5 x^{2}+9 x \ln (x)+1f(x)=5x2+9xln(x)+1. What is f′(x)f^{\prime}(x)f′(x)?

Problem 25724

Evaluate the integral or state "IMPOSSIBLE": ∫x2x3−83dx\int x^{2} \sqrt[3]{x^{3}-8} d x∫x23x3−8​dx. Use CCC for the constant.

Problem 25725

Find how dV/dtdV/dtdV/dt relates to dr/dtdr/dtdr/dt (with constant hhh), dh/dtdh/dtdh/dt (with constant rrr), and both dh/dtdh/dtdh/dt and dr/dtdr/dtdr/dt.

Problem 25726

Find the intervals where the function f(x)=x+1x2f(x)=\frac{x+1}{x^{2}}f(x)=x2x+1​ is increasing or decreasing based on its derivatives.

Problem 25727

Find the limit: lim⁡x→0∫x0t2+4dtx\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+4} d t}{x}x→0lim​x∫x0​t2+4​dt​. What is it equal to?

Problem 25728

Find where f(x)=8x−8xf(x)=8 \sqrt{x}-8 xf(x)=8x​−8x is increasing/decreasing and the xxx-coordinates of relative maxima/minima.

Problem 25729

Find the intervals where the function f(x)=x+1x2f(x)=\frac{x+1}{x^{2}}f(x)=x2x+1​ is concave up and down using f′(x)f^{\prime}(x)f′(x) and f′′(x)f^{\prime \prime}(x)f′′(x).

Problem 25730

Find the absolute min and max of f(x)=8x3−24xf(x)=8 x^{3}-24 xf(x)=8x3−24x on the interval [0,5][0,5][0,5].

Problem 25731

Gegeben ist fa(x)=2x2−4ax+6a−4,5f_{a}(x)=2 x^{2}-4 a x+6 a-4,5fa​(x)=2x2−4ax+6a−4,5. Bestimmen Sie Ableitungen, Steigung bei x=0x=0x=0, Punkt S(1,5/0)S(1,5/0)S(1,5/0), Extrempunkte und aaa für Extrempunkt auf xxx-Achse.

Problem 25732

Problem 25733

Calculate the integral ∫−1πf(x)dx\int_{-1}^{\pi} f(x) dx∫−1π​f(x)dx for the piecewise function f(x)f(x)f(x) defined as f(x)=−x2f(x) = -x^2f(x)=−x2 for x≤0x \leq 0x≤0 and f(x)=−sin⁡x+Cf(x) = -\sin x + Cf(x)=−sinx+C for x>0x > 0x>0.

Problem 25734

Find the concavity and inflection points for f(x)=2x3−3x2−12x+5f(x)=2 x^{3}-3 x^{2}-12 x+5f(x)=2x3−3x2−12x+5 and g(x)=−18(x+2)2(x−4)2g(x)=-\frac{1}{8}(x+2)^{2}(x-4)^{2}g(x)=−81​(x+2)2(x−4)2.

Problem 25735

Problem 25736

Gegeben ist die Funktion fa(x)=2x2−4ax+6a−4,5f_{a}(x)=2 x^{2}-4 a x+6 a-4,5fa​(x)=2x2−4ax+6a−4,5. Finde die 1. und 2. Ableitung und die Steigung bei x=0x=0x=0.

Problem 25737

Given xy=−3x y=-3xy=−3 and dydt=3\frac{d y}{d t}=3dtdy​=3, find dxdt\frac{d x}{d t}dtdx​ when x=1x=1x=1. What is dxdt\frac{d x}{d t}dtdx​?

Problem 25738

Find dPdt\frac{d P}{d t}dtdP​ for b=1.4,P=7kPa,V=60cm2b=1.4, P=7 \mathrm{kPa}, V=60 \mathrm{cm}^{2}b=1.4,P=7kPa,V=60cm2, dVdt=30cm3/min\frac{d V}{d t}=30 \mathrm{cm}^{3}/\mathrm{min}dtdV​=30cm3/min. dPdt=□kPa/min\frac{d P}{d t}=\square \mathrm{kPa}/\mathrm{min}dtdP​=□kPa/min

Problem 25739

Analyze an energy diagram of two molecules. 1. At which separation is the attractive force greatest?2. Which separation is stable equilibrium?

Problem 25740

Let fff be a continuous function on (0,∞)(0, \infty)(0,∞) with lim⁡x→0+f(x)=∞\lim _{x \rightarrow 0^{+}} f(x)=\inftylimx→0+​f(x)=∞ and lim⁡x→∞f(x)=0\lim _{x \rightarrow \infty} f(x)=0limx→∞​f(x)=0. If g(x)≥f(x)g(x) \geq f(x)g(x)≥f(x), which statement is always TRUE?

Problem 25741

Find the weight function $W(h)$ given $\frac{d W}{d h}=0.0012 h^{2}$ and $W(80)=224.8$ . What is $W(69)$ ?

Find the derivative of $f(x)=\int_{x}^{-2}(5 t^{2}+\arccos (t)) dt$ .

Apply the Mean Value Theorem to $f(x)=(x-2)^{2}(x-3)$ on $[1,4]$ . Find $c$ such that $f^{\prime}(c)=$ average rate of change.

Find the derivative $f^{\prime}(x)$ and second derivative $f^{\prime \prime}(x)$ for $f(x)=x^{3}-5x$ . Graph them together.

Use implicit differentiation on $y^{3}-y^{2}+y-2=x$ to find $\frac{d y}{d x}$ .

Find the domain of $f(x)=\frac{x+1}{x^{2}}$ , limits at the ends, and intercepts. Domain: (-\infty, 0) U (0, \infty); Limits: ?, Intercepts: (-1, 0).

Find the average slope of $f(x)=\frac{1}{x}$ on $[4,9]$ . Then, find $c$ in $(4,9)$ where $f^{\prime}(c)$ equals this slope.

Graph the function $f(x)=x^{3}+3 x^{2}-9 x-8$ and identify all relative extrema and inflection points.

Approximate $4.6^{5}$ using the tangent line of $f(x)=x^{5}$ at $x=5$ . Find $m$ and $b$ for $y=m x+b$ .

Find the tangent line equation for $y^{5}+2xy^{2}+3x^{2}=4$ at $(1,-1)$ in the form $y=mx+b$ .

Find the average slope of $f(x)=\frac{1}{x}$ on $[5,10]$ and values of $c$ in $(5,10)$ where $f'(c)$ equals this slope.

Find the population $N(9)$ of a city after 9 years if $\frac{d N}{d t}=400+300 \sqrt{t}$ and $N(0)=2000$ .

Find critical numbers, increasing/decreasing regions, and classify extrema for: A. $f(x)=(x+2)x(x-4)$ B. $f(x)=\frac{x^{2}-4x-5}{x-2}$ .

Calculate the integral of $e^{4x} \cdot 4 \, dx$ and verify by differentiation. Result: $\int e^{4x}(4) \, dx = \square$

Gegeben ist die Funktion $f_{a}(x) = 2x^{2} - 4ax + 6a - 4$ . Bestimme Ableitungen, Extrempunkte und zeichne den Graphen für $a=1$ und $a=0,6$ im Intervall $[-3; 3]$ .

Evaluate the integral from 3 to 6 of the function $9x^{2} - 8x + 6$ .

Find the critical numbers of the function $f(z)=\frac{4 z+4}{10 z^{2}+10 z+10}$ in increasing order. Use $\mathrm{N}$ for unused blanks.

Evaluate the integral $\frac{1}{6-x} \int_{x}^{6}(6x) dx$ and solve $\frac{1}{3-0} \int_{0}^{3} 11(t) dt$ .

Find these limits: (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{5+5 x^{2}}}{8+6 x}=$ , (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{5+5 x^{2}}}{8+6 x}=$ .

Find the critical numbers of $f(\theta)=12 \cos (\theta)+6 \sin ^{2}(\theta)$ for $-\pi \leq \theta \leq \pi$ . Use $\mathrm{N}$ for blanks.

Evaluate the limits and value of the piecewise function $g(x)$ at specific points: $x=1$ and $x=2$ .

Trouvez la dérivée de la fonction $g(x) = \frac{1}{\sqrt{x}} + \sqrt[4]{x}$ .

Find the derivative of the function $f(x)=5 x^{2}+9 x \ln (x)+1$ . What is $f^{\prime}(x)$ ?

Evaluate the integral or state "IMPOSSIBLE": $\int x^{2} \sqrt[3]{x^{3}-8} d x$ . Use $C$ for the constant.

Find how $dV/dt$ relates to $dr/dt$ (with constant $h$ ), $dh/dt$ (with constant $r$ ), and both $dh/dt$ and $dr/dt$ .

Find the intervals where the function $f(x)=\frac{x+1}{x^{2}}$ is increasing or decreasing based on its derivatives.

Find the limit: $\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+4} d t}{x}$ . What is it equal to?

Find where $f(x)=8 \sqrt{x}-8 x$ is increasing/decreasing and the $x$ -coordinates of relative maxima/minima.

Find the intervals where the function $f(x)=\frac{x+1}{x^{2}}$ is concave up and down using $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ .

Find the absolute min and max of $f(x)=8 x^{3}-24 x$ on the interval $[0,5]$ .

Gegeben ist $f_{a}(x)=2 x^{2}-4 a x+6 a-4,5$ . Bestimmen Sie Ableitungen, Steigung bei $x=0$ , Punkt $S(1,5/0)$ , Extrempunkte und $a$ für Extrempunkt auf $x$ -Achse.

Calculate the integral $\int_{-1}^{\pi} f(x) dx$ for the piecewise function $f(x)$ defined as $f(x) = -x^2$ for $x \leq 0$ and $f(x) = -\sin x + C$ for $x > 0$ .

Find the concavity and inflection points for $f(x)=2 x^{3}-3 x^{2}-12 x+5$ and $g(x)=-\frac{1}{8}(x+2)^{2}(x-4)^{2}$ .

Gegeben ist die Funktion $f_{a}(x)=2 x^{2}-4 a x+6 a-4,5$ . Finde die 1. und 2. Ableitung und die Steigung bei $x=0$ .

Given $x y=-3$ and $\frac{d y}{d t}=3$ , find $\frac{d x}{d t}$ when $x=1$ . What is $\frac{d x}{d t}$ ?

Find $\frac{d P}{d t}$ for $b=1.4, P=7 \mathrm{kPa}, V=60 \mathrm{cm}^{2}$ , $\frac{d V}{d t}=30 \mathrm{cm}^{3}/\mathrm{min}$ . $\frac{d P}{d t}=\square \mathrm{kPa}/\mathrm{min}$

Analyze an energy diagram of two molecules.
1. At which separation is the attractive force greatest?
2. Which separation is stable equilibrium?

Let $f$ be a continuous function on $(0, \infty)$ with $\lim _{x \rightarrow 0^{+}} f(x)=\infty$ and $\lim _{x \rightarrow \infty} f(x)=0$ . If $g(x) \geq f(x)$ , which statement is always TRUE?

A balloon inflates at $10.5 \mathrm{~cm}^{3}/\mathrm{s}$ . Find the radius growth rate when the radius is $1 \mathrm{~cm}$ , $10 \mathrm{~cm}$ , and $100 \mathrm{~cm}$ .

Gegeben ist die Funktion $f_{a}(x)=2 x^{2}-4 a x+6 a-4,5$ . Bestimme die Extrempunkte in Abhängigkeit von $a$ und den Wert für $a$ , wenn der Extrempunkt auf der $x$ -Achse liegt.

Find the domain of $f(x)=\frac{x+1}{x^{2}}$ , limits at domain ends, and intercepts $(x, y)$ .

Find the volume increase rate in $\mathrm{cm}^{3} / \mathrm{sec}$ for a sphere with radius $3 \mathrm{~cm}$ and rate $3 \mathrm{~cm/sec}$ .

Find the rate of area increase for a rectangle with length $40 \mathrm{~cm}$ , width $20 \mathrm{~cm}$ , length rate $6 \mathrm{~cm/s}$ , and width rate $4 \mathrm{~cm/s}$ . Answer in $\mathrm{cm}^{2} / \mathrm{s}$ .

Find the marginal revenue equations for the revenue function $R(x, y)=70 x+120 y-2 x^{2}-4 y^{2}-x y$ .

Find the marginal revenue equations for $R(x, y)=70x+120y-2x^2-4y^2-xy$ . Set $R_x=0$ and $R_y=0$ to maximize revenue.

Find the partial derivatives of the function $f(x, y)=-6 x^{2}-5 x y^{3}-3 y^{4}$ : $f_{x}(x, y)=$ and $f_{y}(x, y)=$ .

Find the second partial derivatives of the function $f(x, y)=4 x^{3}-2 x y^{2}+5 y^{6}$ : $f_{x x}(x, y)$ and $f_{x y}(x, y)$ .

Find the maximum revenue from the function $R(x)=80 x-0.2 x^{2}$ and the units $x$ needed for it.

Find the time $t$ (in min) when the drug concentration $C(t)=0.06 t-0.0002 t^{2}$ is maximized and its value.

A company's revenue is $R(q)=-q^{3}+320 q^{2}$ and cost is $C(q)=170+14 q$ .
A) Find the marginal profit function $M P(q)$ . B) Determine the quantity (in hundreds) to maximize profits.

Evaluate the limits and value of the piecewise function $g(x)$ at specific points. Sketch the graph of $g$ .

Find the total cost for producing the first 25 units given the marginal cost function $\frac{9}{\sqrt{x}}$ . Total cost: \$

Bestimmen Sie die Extrem- und Wendepunkte der Funktion $f(x)=x \cdot(\ln (x))^{2}$ .

Evaluate the integral: $\int x^{2} \sqrt{10+x^{3}} \, dx$

Given $f(x)=3 x^{4}-3 e^{x}$ , find $f'(x)$ and $f'(5)$ , then find $f''(x)$ and $f''(5)$ .

Find the total cost for producing 100 units, given the marginal cost $M C(x)=x^{-1/2}+4$ . Round to the nearest dollar.

Evaluate the integral from 1 to 7 of $\frac{4 x^{2}+9}{\sqrt{x}}$ dx.

Solve the equation $\frac{d x}{d t}=\frac{6}{x}$ with initial condition $x(0)=8$ to find $x(t)=$ .

Evaluate the integral: $\int \frac{2 d x}{x \ln (4 x)}$ .

Find the first derivative of $g(x)=4 x^{3}+54 x^{2}+240 x$ , then determine $g^{\prime \prime}(x)$ and $g^{\prime \prime}(-5)$ . Is it concave up or down? Local minimum or maximum at $x=-5$ ?

Find the slope of the tangent line to the curve $3 x^{2}-x y-3 y^{3}=-45$ at the point $(-3,3)$ .

Finde die Punkte, wo die Tangente von $f$ einen Steigungswinkel von 21,8\% hat. Funktionen: a) $f(x)=5 x^{2}$ , b) $f(x)=-\frac{40}{x}$ , c) $f(x)=\frac{5}{6} x^{3}$ , d) $f(x)=0,15 x^{2}-0,2 x$ .

Find the rate of change of the volume $V=\frac{1}{3} \pi (2^{2}) h$ when $h=10$ and $dh/dt=3$ inches/sec.

Compute the integral of the derivative from $x=3$ to $x=5$ : $\int_{3}^{5} f^{\prime}(x) d x$ . Choices: -4, -1, 2, 4.

Evaluate the integral from 1.7 to 5.8 of $(0.2 e^{-0.2 A} + \frac{5}{A}) dA$ . Round to three decimal places.

Find the indefinite integral $\int \frac{\ln (x)}{3 x} d x$ using the substitution $u=\ln (x)$ . What does it transform into?

Find the limit definition of $f^{\prime}(4)$ for $f(x)=3 x^{2}-\csc x$ . No need to simplify or evaluate.

Find the slope of the tangent line to the curve $-x^{2}+2xy+3y^{3}=208$ at the point $(4,4)$ .

Evaluate the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-x-12}{x+3}$ .

Find the first derivative of $g(x)=6 x^{3}+27 x^{2}+36 x$ . Then find the second derivative and evaluate it at $x=-2$ . Is it concave up or down? Local min or max?