Calculus

Problem 29901

Find the derivative of $\cos^3 x$ with respect to $x$ .

See Solution

Problem 29902

Find the derivative of the function $f(x) = \sqrt{x^{2} - 7x}$ .

See Solution

Problem 29903

Find the tangent line equation to the curve $y=8 e^{x} \cos (x)$ at the point $(0,8)$ .

See Solution

Problem 29904

Find the derivative of the function $y=\sqrt{6+6 e^{7 x}}$ .

See Solution

Problem 29905

Find the derivative of $y=(\sqrt{x})^{3 x}$ using logarithmic differentiation.

See Solution

Problem 29906

Find the limit using the squeeze theorem: $\lim _{x \rightarrow 0} x \sin \left(\frac{9}{x}\right)$ .

See Solution

Problem 29907

Differentiate $y=\ln \left(e^{3 x}+\frac{11}{x}\right)$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 29908

Find the limit: $\lim _{x \rightarrow 0} x^{3} \cot h x$ .

See Solution

Problem 29909

Differentiate $y \sec (x) = 6 x \tan (y)$ implicitly to find $\frac{dx}{dy}$ . What is $x'$ ?

See Solution

Problem 29910

Given the function $B(x)=3 x^{2 / 3}-x$ , find the intervals of increase, decrease, local max/min values, and concavity.

See Solution

Problem 29911

Find $\lim _{x \rightarrow 1} f(x)$ for $f(x)=\frac{x-7}{(x-1)^{2}}$ by computing values in the table.

See Solution

Problem 29912

Differentiate the function $f(x)=x \tan(2x^{4})$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 29913

Find the velocity of $s(t)=5t+2t^{2}$ after 2 seconds and time to reach $30 \mathrm{~m/s}$ .

See Solution

Problem 29914

Differentiate the function $f(x)=\pi^{e x}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 29915

Differentiate the function $f(x)=\frac{6^{x}}{x^{6}}$ . Find $f^{\prime}(x)$ .

See Solution

Problem 29916

Compute $f(x) = \frac{x-7}{(x-1)^2}$ and find $\lim_{x \rightarrow 1} f(x)$ .

See Solution

Problem 29917

A square's side increases at $3 \mathrm{~cm/s}$ . Find the area increase rate when the area is $81 \mathrm{~cm}^{2}$ . $\mathrm{cm}^{2} / \mathrm{s}$

See Solution

Problem 29918

Find the area represented by the integral $\int_{0}^{18}|x-9| d x$ .

See Solution

Problem 29919

Find the derivative of $g(r)=\int_{0}^{r} \sqrt{x^{2}+5} \, dx$ . What is $g^{\prime}(r)$ ?

See Solution

Problem 29920

Calculate the indefinite integral: $\int \frac{5 x^{3}-2 \sqrt{x}}{x} d x$ (use $C$ for the constant).

See Solution

Problem 29921

Evaluate the integral from 1 to 8: $\int_{1}^{8}\left(\frac{x}{8}-\frac{4}{x}\right) d x$ .

See Solution

Problem 29922

Evaluate the integral: $\int e^{\cos (34 t)} \sin (34 t) d t$ (include constant $C$ ).

See Solution

Problem 29923

Estimate $(1.999)^{5}$ using linear approximation or differentials.

See Solution

Problem 29924

Evaluate the integral: $\int_{0}^{14} \frac{d x}{\sqrt[3]{(8+4 x)^{2}}}$

See Solution

Problem 29925

Find the pulse rate change for heights 51 inches ( $y=594 \cdot 51^{-1/2}$ ) and 68 inches ( $y=594 \cdot 68^{-1/2}$ ).

See Solution

Problem 29926

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{2 x-9}{18 x^{5}}$ .

See Solution

Problem 29927

Find the limit: $\lim _{x \rightarrow 0^{+}} \ln (x)=$

See Solution

Problem 29928

Find the derivative $S^{\prime}(t)$ of sales function $S(t)=0.03 t^{3}+0.8 t^{2}+5 t+3$ . Then compute $S(7)$ , $S^{\prime}(7)$ , and interpret $S(13)=269.11$ , $S^{\prime}(13)=41.01$ .

See Solution

Problem 29929

Find the rate of change of pulse rate $y=594 x^{-1/2}$ for heights 51 and 68 inches. Calculate for 51 inches.

See Solution

Problem 29930

Find the instantaneous rate of change of pulse rate $y=590 x^{-1/2}$ for heights 31 and 55 inches.

See Solution

Problem 29931

Find the derivative of $\frac{7 x^{3}}{6}-\frac{8}{5 x^{3}}$ . What is $\frac{d}{d x}\left(\frac{7 x^{3}}{6}-\frac{8}{5 x^{3}}\right)$ ?

See Solution

Problem 29932

Find the tangent line equation for $f(x)=\frac{1}{4+x^{2}}$ at $x=1$ where the slope is $-\frac{2}{25}$ .

See Solution

Problem 29933

Find the marginal cost function for $C(x)=210+6.2 x-0.04 x^{2}$ . What is $C^{\prime}(x)=?$

See Solution

Problem 29934

Find $f^{\prime}(x)$ using four steps, then calculate $f^{\prime}(5)$ , $f^{\prime}(7)$ , and $f^{\prime}(9)$ for $f(x)=12\sqrt{x+7}$ .

See Solution

Problem 29935

Find the marginal cost function for $C(x)=174+1.4 x$ . What is $C^{\prime}(x)=\square$ ?

See Solution

Problem 29936

Find the marginal revenue function for $R(x)=x(13-0.08 x)$ . What is $R^{\prime}(x)$ ?

See Solution

Problem 29937

Find the limit of postage cost $C(x)$ as $x$ approaches 3 oz from the left and right. Options: A. \$0.91, B. Does not exist.

See Solution

Problem 29938

Find the marginal profit function given $C(x)=180+0.6 x$ and $R(x)=3 x-0.02 x^{2}$ . $P^{\prime}(x)=\square$

See Solution

Problem 29939

Find the derivative $h^{\prime}(t)$ for the function $h(t)=\frac{2}{t^{1/2}}-\frac{9}{t^{2/7}}$ .

See Solution

Problem 29940

Find $f^{\prime}(x)$ for $f(x)=\frac{4 x}{5+x}$ , then calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , and $f^{\prime}(3)$ .

See Solution

Problem 29941

Postage costs \ $0.39 for the first ounce and \$0.26 for each additional ounce. Find the limits as$ x$ approaches 3.

See Solution

Problem 29942

Differentiate $y=\frac{3x^{4}-4x^{3}}{x^{2}}$

See Solution

Problem 29943

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{4 x}{5+x}$ using the four-step process.

See Solution

Problem 29944

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{\pi}{\sqrt{x^{3}}}$ .

See Solution

Problem 29945

Find the derivative of the function: $\frac{1}{2x^2} + \sqrt{x}$ .

See Solution

Problem 29946

Find the marginal average cost function given $C(x)=128+2.7 x$ and $R(x)=7 x-0.09 x^{2}$ .

See Solution

Problem 29947

Find the derivative of $f(x)=\left(\frac{2}{\sqrt{x}}\right)^{3}$ .

See Solution

Problem 29948

Find values of $x$ where $r(x)=\ln \left|\frac{3 x}{x-6}\right|$ is discontinuous and determine the limits.

See Solution

Problem 29949

Find the cost of producing the 71st food processor using $C(x)=2200+80x-0.2x^2$ . Exact cost: \$\square.

See Solution

Problem 29950

Find $f^{\prime}(x)$ using the four-step process, then calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , and $f^{\prime}(3)$ for $f(x)=\frac{4 x}{5+x}$ .

See Solution

Problem 29951

Find the derivative of $f(x)=\sqrt{x}(2x+1)$ and select all correct options (MSQ) from the given choices.

See Solution

Problem 29952

Compute the limit as $x$ approaches -1 for the expression $\frac{x^{3}-3 x^{2}-4 x}{x^{2}+6 x+5}$ and explain your steps.

See Solution

Problem 29953

Find the derivative $f'(x)$ of the function $f(x) = 1 + \frac{1}{x^3} + \frac{5}{\sqrt{x^3}} - 2e$ .

See Solution

Problem 29954

Find the cost of producing the 41st food processor using $C(x)=1600+50x-0.6x^2$ . Exact cost: \$\square.

See Solution

Problem 29955

Find the derivative of $x^{\frac{1}{2}}\left(3x + \frac{1}{2}\right)$ .

See Solution

Problem 29956

Find the derivative of $f(x)=\sqrt{x}(2x+1)$ . Select all correct options (MSQ): $x^{-\frac{1}{2}}(3x+\frac{1}{2})$ , $\frac{3}{2}x+x^{\frac{1}{2}}$ .

See Solution

Problem 29957

Estimate the slope of the tangent line for $v(T)=70 \sqrt{T}$ at $T=400$ . $\text { slope }=$

See Solution

Problem 29958

Find the derivative of $f(x)=\sqrt{x}(2x+1)$ . Select all correct options (MSQ):
1. $x^{-\frac{1}{2}}(3x+\frac{1}{2})$
2. $\frac{3}{2}x+x^{\frac{1}{2}}$
3. $3x^{\frac{1}{2}}+\frac{1}{2}x^{-\frac{1}{2}}$
4. $x^{\frac{1}{2}}(3x+\frac{1}{2})$

See Solution

Problem 29959

Find the derivative of $f(x)=\sqrt{x}(2 x+1)$ . Select all correct options (MSQ):
1. $x^{-\frac{1}{2}}\left(3 x+\frac{1}{2}\right)$
2. $\frac{3}{2} x+x^{\frac{1}{2}}$
3. $\frac{3 x^{\frac{1}{2}}+\frac{1}{2} x^{-\frac{1}{2}}}{x^{\frac{1}{2}}\left(3 x+\frac{1}{2}\right)}$

See Solution

Problem 29960

Estimate the slope of the tangent line of $y(x)=\frac{1}{x+1}$ at $x=0$ .

See Solution

Problem 29961

Find where $R(x)$ has a horizontal tangent, profit function $P(x)$ , its tangent, and graph $C(x), R(x), P(x)$ for $0 \leq x \leq 2000$ .

See Solution

Problem 29962

The cost function is $C(x)=2400+80 x-0.3 x^{2}$ . Find the cost of the 61st processor exactly and using marginal cost.

See Solution

Problem 29963

Find the limit: $\lim _{x \rightarrow 1} \frac{x-\frac{1}{x}}{x^{2}-8 x+7}$ .

See Solution

Problem 29964

Find $x$ where $R(x)$ has a horizontal tangent, profit function $P(x)$ , and graph $C(x), R(x), P(x)$ for $0 \leq x \leq 2000$ .

See Solution

Problem 29965

Evaluate the limits and value of $f(x)=x^{2}-1$ at $x=1$ . Find: a. $\lim _{x \rightarrow 1^{-}} f(x)$ , b. $\lim _{x \rightarrow 1} f(x)$ , c. $\lim _{x \rightarrow 1} f(x)$ , d. $f(1)$ .

See Solution

Problem 29966

Evaluate the limits and value for $f(x)=x^{2}-1$ when $x \neq 1$ : a. $\lim _{x \rightarrow 1^{-}} f(x)=$ b. $\lim _{x \rightarrow 1} f(x)=$ c. $\lim _{x \rightarrow 1} f(x)=$ d. $f(1)=$

See Solution

Problem 29967

Find the limits and values for the piecewise function $g(x)$ defined as: $g(x)=\left\{\begin{array}{cc}3 x+1, & x \leq-1 \\ 5, & -1<x<2 \\ x, & x>2\end{array}\right.$ a. $\lim _{x \rightarrow-1} g(x)$ , b. $\lim _{x \rightarrow 2} g(x)$ , c. $g(-1)$ , d. $g(2)$ .

See Solution

Problem 29968

Find the derivative of the function $f(x)=2 x^{2}-7 x+9$ .

See Solution

Problem 29969

Differentiate the function $f(x) = \frac{-3}{(2x^2 + 1)^4}$ .

See Solution

Problem 29970

Find the derivative of f(x) = $\sqrt{x}(2x+1)$ .

See Solution

Problem 29971

Evaluate the limits and value of the function $f(x)=x^{2}-1$ for $x \neq 1$ : a. $\lim _{x \rightarrow 1^{-}} f(x)=$ , b. $\lim _{x \rightarrow 1} f(x)=$ , c. $\lim _{x \rightarrow 1} f(x)=$ , d. $f(1)=$ .

See Solution

Problem 29972

Find the limits and values of the piecewise function:
a. $\lim _{x \rightarrow-2} h(x)$ , b. $\lim _{x \rightarrow 1} h(x)$ , c. $h(-2)$ , d. $h(1)$ .

See Solution

Problem 29973

Find the limits and values for the piecewise function $g(x)$ defined as:
$g(x)=\begin{cases}3 x+1, & x \leq-1 \\ 5, & -1<x<2 \\ x, & x>2\end{cases}$
a. $\lim _{x \rightarrow-1} g(x)$ , b. $\lim _{x \rightarrow 2} g(x)$ , c. $g(-1)$ , d. $g(2)$ .

See Solution

Problem 29974

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

See Solution

Problem 29975

Berechnen Sie die mittlere Änderungsrate für $f$ in den Intervallen $I=[1;3]$ , $J=[-3;-1]$ , $K=[-1;2]$ für a) $f(x)=x^2$ , b) $f(x)=x^3+2x$ , c) $f(x)=2x^2-x$ .

See Solution

Problem 29976

Find the limit: $\lim _{x \rightarrow 0} \frac{\ln (x+5)-\ln 5}{x}$ .

See Solution

Problem 29977

Berechne die Steigung von $f$ an $x_{0}$ mit der $h$ -Methode für: b) $f(x)=2 x^{2}+1, x_{0}=-2$ und c) $f(x)=3 x+2, x_{0}=2$ .

See Solution

Problem 29978

Pour tout $x>0$ , montrer que si $f(x)=\ln x$ , alors $f'(x)=\frac{1}{x}$ .

See Solution

Problem 29979

Die Funktion $f(t) = \frac{-}{3} t^{3} - 15 t^{2} + 200 t$ beschreibt die Änderungsrate der Infektionen.
a. Berechne $f(10)$ und erkläre den Wert.
b. Finde den Zeitpunkt, an dem die meisten Menschen sich anstecken.

See Solution

Problem 29980

In einem Tank sind 20.000 l Benzin. Der Zu/Abfluss wird durch $f(x)=0,1 x^{4}-2 x^{3}+9,6 x^{2}$ (0 $\leq x \leq$ 12) modelliert.
a. Benzinmenge nach 5 Minuten? b. Zeitpunkt maximaler Benzinmenge? c. Wann steigt die Benzinmenge am stärksten und was ist die Zuflussrate?

See Solution

Problem 29981

Find the following limits: (a) $\lim _{x \rightarrow+\infty} \frac{\ln x}{x}$ , (b) $\lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}$ , (c) $\lim _{x \rightarrow 0^{+}} x \cdot \ln x$ .

See Solution

Problem 29982

Si $u$ est dérivable et positif sur I, quel est le signe de la dérivée de $\ln (u(x))$ ?

See Solution

Problem 29983

Gegeben ist die Funktion $f(x)=(1-2 x) \cdot e^{x}$ . Finde die Achsenschnittpunkte, waagrechte Tangenten und die Tangentengleichung bei $P(0 \mid 1)$ .

See Solution

Problem 29984

Soit $f(x)=\ln(1+e^{x})$ , calculez $f^{\prime}(x)$ . Trouvez le signe de $\ln(2x-4)$ sur $[12;+\infty)$ .

See Solution

Problem 29985

The area of a rectangle grows at 300 cm²/hour. What is this rate in m²/min? $(1$ m $=100$ cm)

See Solution

Problem 29986

Find $f^{\prime}(1)$ for $f(x)=2^{\sqrt{x}}+\ln 2^{x}$ . Choose from: (a) $\ln 4$ , (b) $\ln 8$ , (c) $\ln 32$ , (d) $\ln 64$ .

See Solution

Problem 29987

Calculate the integral: $\int \frac{\cos x d x}{1+2 \sin x}$ . Choose the correct answer from the options provided.

See Solution

Problem 29988

Find the limit: $\lim _{n \rightarrow \infty} \frac{(e-1) \sum_{k=0}^{\infty} \frac{1}{e^{k}}}{\left(1+\frac{2}{n}\right)^{n}}$

See Solution

Problem 29989

Evaluate $\int_{1}^{3} 2 e^{x+5} d x$ using $\int_{1}^{3} e^{x} d x=e^{3}-e$ . Rewrite $2 e^{x+5}$ .

See Solution

Problem 29990

Find the derivative of the function $f(x) = x^{2} + 3x + 5$ .

See Solution

Problem 29991

Leite den Ausdruck $\frac{1}{x^{3}}$ ab und erkläre den Prozess in einem kurzen Text.

See Solution

Problem 29992

Find where the tangent line to $f(x)=x^{3} \ln (x)$ is horizontal, and if it intersects at $(1,0)$ for $x \in(1, \infty)$ .

See Solution

Problem 29993

Find the derivative of the function $(4 x^{3}-\frac{2}{3} x^{2}-14 x+5)^{9}$ .

See Solution

Problem 29994

Find the values of $x \in \mathbb{R}$ where $f(x)=\sum_{n=0}^{\infty} e^{-2 n x}$ is defined and compute $f^{\prime}(x)$ .

See Solution

Problem 29995

Gegeben sind die Funktionen $f(x)=3^{x}$ und $g(x)=\left(\frac{1}{3}\right)^{x}$ . Zeichne die Graphen und vergleiche sie. Bestimme $f'(x)$ und die Steigung bei $x_{0}=0$ . Finde $a$ für $h(x)=a^{x}$ mit Steigung bei $x_{0}=0$ .

See Solution

Problem 29996

Bestimme die 1. und 2. Ableitung der Funktion $f$ für: a) $f(x)=e^{2 x}$ , b) $f(x)=2 e^{-4 x+4}$ , c) $f(x)=e^{0,5 x+e \cdot x}$ , d) $f(x)=1,5 e^{-x}+e^{-2 x}$ , e) $f(x)=4 e^{0,25 x}-8 e$ , f) $f(x)=e^{2 x+2 e^{x}+2 x}$ .

See Solution

Problem 29997

Differentiate $\ln(1+y)$ with respect to $x$ and set it equal to $\frac{a}{1+y} \frac{dy}{dx}$ .

See Solution

Problem 29998

Evaluate the integral: $\int \frac{d x}{x \ln x}$ .

See Solution

Problem 29999

Find the derivative of $y=\frac{3x+9}{2-x}$ .

See Solution

Problem 30000

Calculate the integral $\int_{0}^{1} \frac{x^{3}}{\sqrt{x^{4}+9}} d x$ .

See Solution

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Calculus

Problem 29901

Find the derivative of cos⁡3x\cos^3 xcos3x with respect to xxx.

Problem 29902

Find the derivative of the function f(x)=x2−7xf(x) = \sqrt{x^{2} - 7x}f(x)=x2−7x​.

Problem 29903

Find the tangent line equation to the curve y=8excos⁡(x)y=8 e^{x} \cos (x)y=8excos(x) at the point (0,8)(0,8)(0,8).

Problem 29904

Find the derivative of the function y=6+6e7xy=\sqrt{6+6 e^{7 x}}y=6+6e7x​.

Problem 29905

Find the derivative of y=(x)3xy=(\sqrt{x})^{3 x}y=(x​)3x using logarithmic differentiation.

Problem 29906

Find the limit using the squeeze theorem: lim⁡x→0xsin⁡(9x)\lim _{x \rightarrow 0} x \sin \left(\frac{9}{x}\right)limx→0​xsin(x9​).

Problem 29907

Differentiate y=ln⁡(e3x+11x)y=\ln \left(e^{3 x}+\frac{11}{x}\right)y=ln(e3x+x11​). Find dydx\frac{d y}{d x}dxdy​.

Problem 29908

Find the limit: lim⁡x→0x3cot⁡hx\lim _{x \rightarrow 0} x^{3} \cot h xlimx→0​x3cothx.

Problem 29909

Differentiate ysec⁡(x)=6xtan⁡(y)y \sec (x) = 6 x \tan (y)ysec(x)=6xtan(y) implicitly to find dxdy\frac{dx}{dy}dydx​. What is x′x'x′?

Problem 29910

Given the function B(x)=3x2/3−xB(x)=3 x^{2 / 3}-xB(x)=3x2/3−x, find the intervals of increase, decrease, local max/min values, and concavity.

Problem 29911

Find lim⁡x→1f(x)\lim _{x \rightarrow 1} f(x)limx→1​f(x) for f(x)=x−7(x−1)2f(x)=\frac{x-7}{(x-1)^{2}}f(x)=(x−1)2x−7​ by computing values in the table.

Problem 29912

Differentiate the function f(x)=xtan⁡(2x4)f(x)=x \tan(2x^{4})f(x)=xtan(2x4). What is f′(x)f^{\prime}(x)f′(x)?

Problem 29913

Find the velocity of s(t)=5t+2t2s(t)=5t+2t^{2}s(t)=5t+2t2 after 2 seconds and time to reach 30 m/s30 \mathrm{~m/s}30 m/s.

Problem 29914

Differentiate the function f(x)=πexf(x)=\pi^{e x}f(x)=πex. What is f′(x)f^{\prime}(x)f′(x)?

Problem 29915

Differentiate the function f(x)=6xx6f(x)=\frac{6^{x}}{x^{6}}f(x)=x66x​. Find f′(x)f^{\prime}(x)f′(x).

Problem 29916

Compute f(x)=x−7(x−1)2f(x) = \frac{x-7}{(x-1)^2}f(x)=(x−1)2x−7​ and find lim⁡x→1f(x)\lim_{x \rightarrow 1} f(x)limx→1​f(x).

Problem 29917

A square's side increases at 3 cm/s3 \mathrm{~cm/s}3 cm/s. Find the area increase rate when the area is 81 cm281 \mathrm{~cm}^{2}81 cm2. cm2/s\mathrm{cm}^{2} / \mathrm{s}cm2/s

Problem 29918

Find the area represented by the integral ∫018∣x−9∣dx\int_{0}^{18}|x-9| d x∫018​∣x−9∣dx.

Problem 29919

Find the derivative of g(r)=∫0rx2+5 dxg(r)=\int_{0}^{r} \sqrt{x^{2}+5} \, dxg(r)=∫0r​x2+5​dx. What is g′(r)g^{\prime}(r)g′(r)?

Problem 29920

Calculate the indefinite integral: ∫5x3−2xxdx\int \frac{5 x^{3}-2 \sqrt{x}}{x} d x∫x5x3−2x​​dx (use CCC for the constant).

Problem 29921

Evaluate the integral from 1 to 8: ∫18(x8−4x)dx\int_{1}^{8}\left(\frac{x}{8}-\frac{4}{x}\right) d x∫18​(8x​−x4​)dx.

Problem 29922

Evaluate the integral: ∫ecos⁡(34t)sin⁡(34t)dt\int e^{\cos (34 t)} \sin (34 t) d t∫ecos(34t)sin(34t)dt (include constant CCC).

Problem 29923

Estimate (1.999)5(1.999)^{5}(1.999)5 using linear approximation or differentials.

Problem 29924

Evaluate the integral: ∫014dx(8+4x)23\int_{0}^{14} \frac{d x}{\sqrt[3]{(8+4 x)^{2}}}∫014​3(8+4x)2​dx​

Problem 29925

Find the pulse rate change for heights 51 inches (y=594⋅51−1/2y=594 \cdot 51^{-1/2}y=594⋅51−1/2) and 68 inches (y=594⋅68−1/2y=594 \cdot 68^{-1/2}y=594⋅68−1/2).

Problem 29926

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=2x−918x5y=\frac{2 x-9}{18 x^{5}}y=18x52x−9​.

Problem 29927

Find the limit: lim⁡x→0+ln⁡(x)=\lim _{x \rightarrow 0^{+}} \ln (x)=limx→0+​ln(x)=

Problem 29928

Problem 29929

Find the rate of change of pulse rate y=594x−1/2y=594 x^{-1/2}y=594x−1/2 for heights 51 and 68 inches. Calculate for 51 inches.

Problem 29930

Find the instantaneous rate of change of pulse rate y=590x−1/2y=590 x^{-1/2}y=590x−1/2 for heights 31 and 55 inches.

Problem 29931

Find the derivative of 7x36−85x3\frac{7 x^{3}}{6}-\frac{8}{5 x^{3}}67x3​−5x38​. What is ddx(7x36−85x3)\frac{d}{d x}\left(\frac{7 x^{3}}{6}-\frac{8}{5 x^{3}}\right)dxd​(67x3​−5x38​)?

Problem 29932

Find the tangent line equation for f(x)=14+x2f(x)=\frac{1}{4+x^{2}}f(x)=4+x21​ at x=1x=1x=1 where the slope is −225-\frac{2}{25}−252​.

Problem 29933

Find the marginal cost function for C(x)=210+6.2x−0.04x2C(x)=210+6.2 x-0.04 x^{2}C(x)=210+6.2x−0.04x2. What is C′(x)=?C^{\prime}(x)=?C′(x)=?

Problem 29934

Find f′(x)f^{\prime}(x)f′(x) using four steps, then calculate f′(5)f^{\prime}(5)f′(5), f′(7)f^{\prime}(7)f′(7), and f′(9)f^{\prime}(9)f′(9) for f(x)=12x+7f(x)=12\sqrt{x+7}f(x)=12x+7​.

Problem 29935

Find the marginal cost function for C(x)=174+1.4xC(x)=174+1.4 xC(x)=174+1.4x. What is C′(x)=□C^{\prime}(x)=\squareC′(x)=□?

Problem 29936

Find the marginal revenue function for R(x)=x(13−0.08x)R(x)=x(13-0.08 x)R(x)=x(13−0.08x). What is R′(x)R^{\prime}(x)R′(x)?

Problem 29937

Find the limit of postage cost C(x)C(x)C(x) as xxx approaches 3 oz from the left and right. Options: A. \$0.91, B. Does not exist.

Problem 29938

Find the marginal profit function given C(x)=180+0.6xC(x)=180+0.6 xC(x)=180+0.6x and R(x)=3x−0.02x2R(x)=3 x-0.02 x^{2}R(x)=3x−0.02x2. P′(x)=□P^{\prime}(x)=\squareP′(x)=□

Problem 29939

Find the derivative h′(t)h^{\prime}(t)h′(t) for the function h(t)=2t1/2−9t2/7h(t)=\frac{2}{t^{1/2}}-\frac{9}{t^{2/7}}h(t)=t1/22​−t2/79​.

Problem 29940

Find f′(x)f^{\prime}(x)f′(x) for f(x)=4x5+xf(x)=\frac{4 x}{5+x}f(x)=5+x4x​, then calculate f′(1)f^{\prime}(1)f′(1), f′(2)f^{\prime}(2)f′(2), and f′(3)f^{\prime}(3)f′(3).

Find the derivative of $\cos^3 x$ with respect to $x$ .

Find the derivative of the function $f(x) = \sqrt{x^{2} - 7x}$ .

Find the tangent line equation to the curve $y=8 e^{x} \cos (x)$ at the point $(0,8)$ .

Find the derivative of the function $y=\sqrt{6+6 e^{7 x}}$ .

Find the derivative of $y=(\sqrt{x})^{3 x}$ using logarithmic differentiation.

Find the limit using the squeeze theorem: $\lim _{x \rightarrow 0} x \sin \left(\frac{9}{x}\right)$ .

Differentiate $y=\ln \left(e^{3 x}+\frac{11}{x}\right)$ . Find $\frac{d y}{d x}$ .

Find the limit: $\lim _{x \rightarrow 0} x^{3} \cot h x$ .

Differentiate $y \sec (x) = 6 x \tan (y)$ implicitly to find $\frac{dx}{dy}$ . What is $x'$ ?

Given the function $B(x)=3 x^{2 / 3}-x$ , find the intervals of increase, decrease, local max/min values, and concavity.

Find $\lim _{x \rightarrow 1} f(x)$ for $f(x)=\frac{x-7}{(x-1)^{2}}$ by computing values in the table.

Differentiate the function $f(x)=x \tan(2x^{4})$ . What is $f^{\prime}(x)$ ?

Find the velocity of $s(t)=5t+2t^{2}$ after 2 seconds and time to reach $30 \mathrm{~m/s}$ .

Differentiate the function $f(x)=\pi^{e x}$ . What is $f^{\prime}(x)$ ?

Differentiate the function $f(x)=\frac{6^{x}}{x^{6}}$ . Find $f^{\prime}(x)$ .

Compute $f(x) = \frac{x-7}{(x-1)^2}$ and find $\lim_{x \rightarrow 1} f(x)$ .

A square's side increases at $3 \mathrm{~cm/s}$ . Find the area increase rate when the area is $81 \mathrm{~cm}^{2}$ . $\mathrm{cm}^{2} / \mathrm{s}$

Find the area represented by the integral $\int_{0}^{18}|x-9| d x$ .

Find the derivative of $g(r)=\int_{0}^{r} \sqrt{x^{2}+5} \, dx$ . What is $g^{\prime}(r)$ ?

Calculate the indefinite integral: $\int \frac{5 x^{3}-2 \sqrt{x}}{x} d x$ (use $C$ for the constant).

Evaluate the integral from 1 to 8: $\int_{1}^{8}\left(\frac{x}{8}-\frac{4}{x}\right) d x$ .

Evaluate the integral: $\int e^{\cos (34 t)} \sin (34 t) d t$ (include constant $C$ ).

Estimate $(1.999)^{5}$ using linear approximation or differentials.

Evaluate the integral: $\int_{0}^{14} \frac{d x}{\sqrt[3]{(8+4 x)^{2}}}$

Find the pulse rate change for heights 51 inches ( $y=594 \cdot 51^{-1/2}$ ) and 68 inches ( $y=594 \cdot 68^{-1/2}$ ).

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{2 x-9}{18 x^{5}}$ .

Find the limit: $\lim _{x \rightarrow 0^{+}} \ln (x)=$

Find the rate of change of pulse rate $y=594 x^{-1/2}$ for heights 51 and 68 inches. Calculate for 51 inches.

Find the instantaneous rate of change of pulse rate $y=590 x^{-1/2}$ for heights 31 and 55 inches.

Find the derivative of $\frac{7 x^{3}}{6}-\frac{8}{5 x^{3}}$ . What is $\frac{d}{d x}\left(\frac{7 x^{3}}{6}-\frac{8}{5 x^{3}}\right)$ ?

Find the tangent line equation for $f(x)=\frac{1}{4+x^{2}}$ at $x=1$ where the slope is $-\frac{2}{25}$ .

Find the marginal cost function for $C(x)=210+6.2 x-0.04 x^{2}$ . What is $C^{\prime}(x)=?$

Find $f^{\prime}(x)$ using four steps, then calculate $f^{\prime}(5)$ , $f^{\prime}(7)$ , and $f^{\prime}(9)$ for $f(x)=12\sqrt{x+7}$ .

Find the marginal cost function for $C(x)=174+1.4 x$ . What is $C^{\prime}(x)=\square$ ?

Find the marginal revenue function for $R(x)=x(13-0.08 x)$ . What is $R^{\prime}(x)$ ?

Find the limit of postage cost $C(x)$ as $x$ approaches 3 oz from the left and right. Options: A. \$0.91, B. Does not exist.

Find the marginal profit function given $C(x)=180+0.6 x$ and $R(x)=3 x-0.02 x^{2}$ . $P^{\prime}(x)=\square$

Find the derivative $h^{\prime}(t)$ for the function $h(t)=\frac{2}{t^{1/2}}-\frac{9}{t^{2/7}}$ .

Find $f^{\prime}(x)$ for $f(x)=\frac{4 x}{5+x}$ , then calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , and $f^{\prime}(3)$ .

Postage costs \ $0.39 for the first ounce and \$0.26 for each additional ounce. Find the limits as$ x$ approaches 3.

Differentiate $y=\frac{3x^{4}-4x^{3}}{x^{2}}$

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{4 x}{5+x}$ using the four-step process.

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{\pi}{\sqrt{x^{3}}}$ .

Find the derivative of the function: $\frac{1}{2x^2} + \sqrt{x}$ .

Find the marginal average cost function given $C(x)=128+2.7 x$ and $R(x)=7 x-0.09 x^{2}$ .

Find the derivative of $f(x)=\left(\frac{2}{\sqrt{x}}\right)^{3}$ .

Find values of $x$ where $r(x)=\ln \left|\frac{3 x}{x-6}\right|$ is discontinuous and determine the limits.

Find the cost of producing the 71st food processor using $C(x)=2200+80x-0.2x^2$ . Exact cost: \$\square.

Find $f^{\prime}(x)$ using the four-step process, then calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , and $f^{\prime}(3)$ for $f(x)=\frac{4 x}{5+x}$ .

Find the derivative of $f(x)=\sqrt{x}(2x+1)$ and select all correct options (MSQ) from the given choices.

Compute the limit as $x$ approaches -1 for the expression $\frac{x^{3}-3 x^{2}-4 x}{x^{2}+6 x+5}$ and explain your steps.

Find the derivative $f'(x)$ of the function $f(x) = 1 + \frac{1}{x^3} + \frac{5}{\sqrt{x^3}} - 2e$ .

Find the cost of producing the 41st food processor using $C(x)=1600+50x-0.6x^2$ . Exact cost: \$\square.

Find the derivative of $x^{\frac{1}{2}}\left(3x + \frac{1}{2}\right)$ .

Find the derivative of $f(x)=\sqrt{x}(2x+1)$ . Select all correct options (MSQ): $x^{-\frac{1}{2}}(3x+\frac{1}{2})$ , $\frac{3}{2}x+x^{\frac{1}{2}}$ .

Estimate the slope of the tangent line for $v(T)=70 \sqrt{T}$ at $T=400$ . $\text { slope }=$

Estimate the slope of the tangent line of $y(x)=\frac{1}{x+1}$ at $x=0$ .

Find where $R(x)$ has a horizontal tangent, profit function $P(x)$ , its tangent, and graph $C(x), R(x), P(x)$ for $0 \leq x \leq 2000$ .

The cost function is $C(x)=2400+80 x-0.3 x^{2}$ . Find the cost of the 61st processor exactly and using marginal cost.

Find the limit: $\lim _{x \rightarrow 1} \frac{x-\frac{1}{x}}{x^{2}-8 x+7}$ .

Find $x$ where $R(x)$ has a horizontal tangent, profit function $P(x)$ , and graph $C(x), R(x), P(x)$ for $0 \leq x \leq 2000$ .

Find the derivative of the function $f(x)=2 x^{2}-7 x+9$ .

Differentiate the function $f(x) = \frac{-3}{(2x^2 + 1)^4}$ .

Find the derivative of f(x) = $\sqrt{x}(2x+1)$ .

Find the limits and values of the piecewise function:
a. $\lim _{x \rightarrow-2} h(x)$ , b. $\lim _{x \rightarrow 1} h(x)$ , c. $h(-2)$ , d. $h(1)$ .

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