Calculus

Problem 30101

Find the derivative $\frac{d y}{d x}$ of $y=\frac{1}{x^{3}}$ at $x=-2$ . What is the value?

See Solution

Problem 30102

Find $\frac{d y}{d x}$ for $y=9 x+\frac{9}{x^{6}}$ at $x=1$ . What is the simplified value?

See Solution

Problem 30103

Find the derivative $f^{\prime}(2)$ for the function $f(x)=2 x^{2}-6 x+9$ . Simplify your answer.

See Solution

Problem 30104

Find the intervals where the function $f(x)=e^{x} \cos (x)$ is increasing or decreasing.

See Solution

Problem 30105

Find where the tangent line is horizontal for the function $y = x^{2} - 5$ .
A. The point(s) at which the tangent line is horizontal is(are) $\square$ . B. The tangent line is horizontal at all points of the graph. C. There are no points on the graph where the tangent line is horizontal.

See Solution

Problem 30106

Find where the tangent line is horizontal for the function $y=x^{2}-5$ .
A. The point(s) is(are) $\square$ . B. The tangent line is horizontal at all points. C. No points have a horizontal tangent line.

See Solution

Problem 30107

Aufgabe: Analysiere die Stückkosten $k(x)$ und $h(x)$ für große $x$ und finde die Stückzahl, bei der $h(x) < k(300)$ .

See Solution

Problem 30108

Gegeben ist die Funktion $f(x)=-0,5 x^{2}+4 x$ . Zeichne den Graphen für $x \in [0; 5]$ und bestimme die Änderungsrate für $x=3$ .

See Solution

Problem 30109

Find the points of discontinuity for $f(x)=\frac{x^{2}+1}{1-x^{2}}$ and their one-sided limits.

See Solution

Problem 30110

Find where the tangent line to the graph of $y=x^{2}+2$ is horizontal.
A. The point(s) is(are) $\square$ . B. The tangent line is horizontal at all points. C. No points have a horizontal tangent line.

See Solution

Problem 30111

Find the point on the curve $y=3x^{2}-2x$ where the tangent slope is -26. The point is $\square$ .

See Solution

Problem 30112

Plot the line segment on $f(x)=x^{3}+11x^{2}+23x-35$ between $x=-8$ and $x=0$ . Find all $c$ for the Mean Value Theorem on $[-8, 0]$ .

See Solution

Problem 30113

Find the limit: $\lim _{x \rightarrow-3} \frac{2 x^{2}-x-21}{x+3}$ .

See Solution

Problem 30114

Find where the tangent line of $f(x)=5 x^{2}-4 x+4$ is horizontal. State if none exist.

See Solution

Problem 30115

Plot the line segment between $f(-5)$ and $f(0)$ , then find all $c$ satisfying the Mean Value Theorem on $[-5, 0]$ .

See Solution

Problem 30116

Find the weight change rate for a boy aged $t$ months using $w(t)=8.71+1.69t-0.0095t^2+0.000035t^3$ . Calculate $w'(t)$ .

See Solution

Problem 30117

Find the median weight function $w(t)=8.85+1.76 t-0.0018 t^{2}+0.000686 t^{3}$ for a boy aged 0-36 months.
a) Find $w^{\prime}(t)$ .
b) Calculate weight at 8 months: $w(8)=\square$ lbs.
c) Find $w^{\prime}(8)=\square$ lbs/month.

See Solution

Problem 30118

Graph the function $f(x)=x^{3}-13x^{2}+44x-42$ . Find $c$ values for the Mean Value Theorem on $[0, 8]$ .

See Solution

Problem 30119

A city’s population grows from 800,000 to $P(t)=800,000+1000 t^{2}$ in $t$ years. Find: a) $\frac{d P}{d t}$ b) $P(10)$ c) $\frac{d P}{d t}$ at $t=10$ d) Meaning of part (c): A, B, C, or D?

See Solution

Problem 30120

Find the derivative of $f(x)=(x-5)(5x+2)$ using the Product Rule and by expanding the product.

See Solution

Problem 30121

Find all values of $c$ satisfying the Mean Value Theorem for $f(x)=x^{3}-10x^{2}+13x+24$ on $[-1, 7]$ .

See Solution

Problem 30122

Plot the line segment between $f(0)$ and $f(8)$ for $f(x)=x^{3}-13x^{2}+44x-42$ . Find all $c$ satisfying the Mean Value Theorem on $[0, 8]$ .

See Solution

Problem 30123

Find where the function $f(x)=e^{x} \cos (x)$ is increasing or decreasing. Also, find the edge length of a square-based container with volume 0.5 m³ that minimizes material use.

See Solution

Problem 30124

Find the derivative of $y=x^{4} \cdot x^{6}$ using the Product Rule and by multiplying first. Verify both results.

See Solution

Problem 30125

Plot the line segment on $f(x)=x^{3}+2 x^{2}-11 x-22$ between $x=-5$ and $x=3$ . Find all $c$ for the Mean Value Theorem on $[-5, 3]$ .

See Solution

Problem 30126

Gegeben ist die Funktion $f_{a}(x)=e^{2 x}-2 a \cdot e^{x}+a^{2}, a>0$ . a) Bestimme die Schnittpunkte mit den Achsen. b) Finde die Funktion, die durch den Ursprung geht.
Gegeben ist die Funktion $f(x)=\ln(x^{2}+1)$ . a) Zeige, dass es eine waagerechte Tangente gibt. b) Bestimme die Extrempunkte. c) Finde die Wendetangente bei $x_{w}=1$ . d) Nenne die Koordinaten des zweiten Wendepunkts und die Wendetangente ohne Rechnung.

See Solution

Problem 30127

Plot a line segment on $f(x)=x^{3}+8x^{2}+9x-28$ between $x=-7$ and $x=1$ . Find all $c$ for the Mean Value Theorem.

See Solution

Problem 30128

Calculate the integral: $\int 6 x^{2} \sin 3 x \, dx$

See Solution

Problem 30129

Find the second derivative $y^{\prime \prime}$ for the function $y=\frac{3 x-4}{2 x+5}$ .

See Solution

Problem 30130

Find $c$ in the interval $[-6, -2]$ using the Mean Value Theorem for the function $y=(4x+24)^{\frac{1}{2}}$ .

See Solution

Problem 30131

Find where the instantaneous rate of change equals the average rate of change using the Mean Value Theorem for:
1. $y = \cos(x)$ on $\left[\frac{\pi}{4}, \frac{3\pi}{4}\right]$
2. $y = (4x + 24)^{\frac{1}{2}}$ on $[-6, -2]$

See Solution

Problem 30132

Find the derivative of $h(z)=(4-z^{2})(z^{3}-3z+1)$ using the product rule and by expanding the product.

See Solution

Problem 30133

Scrieți diferențialele de ordinul întâi și doi pentru funcțiile date și calculați-le în punctele specificate.

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Problem 30134

Find the second derivative $f^{\prime \prime}(x)$ for $f(x)=\sqrt[4]{(x^{2}+9)^{3}}$ .

See Solution

Problem 30135

Find the derivative of $y=\frac{x^{9}}{x^{5}}$ using the Quotient Rule and by simplifying first.

See Solution

Problem 30136

Find the fourth derivative of the function $y=7 x^{5}$ . What is $\frac{d^{4} y}{d x^{4}}$ ?

See Solution

Problem 30137

Calculaţi derivatele parţiale pentru funcţiile: (a) $f(x, y)=x y^{3} e^{y}$ , (b) $f(p, q)=3 p^{3} q^{2}$ , (c) $f(k, l)=5 \sqrt{k} l^{3}$ .

See Solution

Problem 30138

Find the second derivative of $y=2x+6$ . What is $\frac{d^{2} y}{d x^{2}}$ ?

See Solution

Problem 30139

Find the fourth derivative $t^{(4)}(n)$ of $t(n)=4 n^{-1/4}+5 n^{5/4}$ . Simplify your answer.

See Solution

Problem 30140

Find the derivative of $F(x)=8 x^{4}(x^{2}-5 x)$ using the Product Rule and by multiplying first. Check if both methods give the same result.

See Solution

Problem 30141

Find the slope of the tangent for each curve at the specified point: a. $y=x^{2}-3 x,(2,-2)$ b. $f(x)=\frac{4}{x},(-2,-2)$ c. $y=3 x^{3},(1,3)$ d. $y=\sqrt{x-7},(16,3)$ e. $y=\sqrt{25-x^{2}},(3,4)$ f. $y=\frac{4+x}{x-2},(8,2)$

See Solution

Problem 30142

Find the derivative of $y=\frac{x^{9}}{x^{4}}$ using the Quotient Rule or by simplifying first. Choose the correct option.

See Solution

Problem 30143

Evaluate the integral $\int_{0}^{a} x^{2} \sin ^{2}\left(\frac{n \pi x}{a}\right) d x$ for constants $n$ , $\pi$ , and $a$ .

See Solution

Problem 30144

Find the rate of change of average cost when 174 belts are produced, given $C(x)=770+32x-0.067x^2$ . Calculate $\overline{C}'(174)$ .

See Solution

Problem 30145

Divide $x^{9}$ by $x^{4}$ , simplify to get $x^{5}$ , then find $\frac{d y}{d x}$ and verify both results equal.

See Solution

Problem 30146

Find the rate of change of the particle's position for $s=\sqrt{37+4t}$ at $t=3$ sec. Answer: $\square \mathrm{m} / \mathrm{sec}$ .

See Solution

Problem 30147

Given the temperature function $T(t)=\frac{8 t}{t^{2}+2}+986$ , find:
(a) $T^{\prime}(t)$ , (b) $T(1)$ , (c) $T^{\prime}(1)$ .

See Solution

Problem 30148

Find the derivatives of these functions: 1) $y=2 x^{4}+6 x+\ln(x^{2}+5 x)$ 2) $y=(x^{5}+x) \cos x$ 3) $y=\tan^{2}(x+1)$ 4) $y=\frac{x^{2}-2 x}{x^{2}+1}$ 5) $y=\sqrt{7+\ln x}$ .

See Solution

Problem 30149

Evaluate the integral $f(x)=\int \frac{d x}{(2 x+1) \sqrt{4 x+4 x^{2}}}$ .

See Solution

Problem 30150

Differentiate the function $y=(2x^{2}-6)^{-10}$ . Find $\frac{dy}{dx}=\square$ .

See Solution

Problem 30151

Differentiate the function $y=\frac{1}{(3 x+7)^{2}}$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 30152

Differentiate the function $y=(7 x-2)^{3}$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 30153

Differentiate the function $f(x)=\left(\frac{x+3}{x-6}\right)^{8}$ . Find $f^{\prime}(x)=$ .

See Solution

Problem 30154

Differentiate the function $y=(18-x)^{65}$ . Find $\frac{d y}{d x}=\square$ .

See Solution

Problem 30155

If $f^{\prime \prime}(x)>0$ , what can we conclude about $f(x)$ ? Choose the correct option.

See Solution

Problem 30156

Given $s(t)=4t+8$ , find: a) $v(t)$ , b) $a(t)$ , c) values at $t=2$ hr, d) meaning of uniform motion.

See Solution

Problem 30157

A company's sales function is $S(t)=2t^{3}-45t^{2}+270t+180$ . Find $S'(2)$ , $S'(3)$ , $S'(5)$ and $S''(2)$ , $S''(3)$ , $S''(5)$ .

See Solution

Problem 30158

An object is dropped from 200 m on the moon. After 8 sec, find: a) distance fallen, b) speed, c) acceleration, d) meaning of second derivative.

See Solution

Problem 30159

Find the mean of the density function $p(t)=0.2 e^{-0.2 t}$ for $0 \leq t \leq 30$ , rounded to 2 decimal places.

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Problem 30160

Given $s(t)=t^{2}+t$ , find the velocity $v(t)$ , acceleration $a(t)$ , and their values at $t=4$ sec.

See Solution

Problem 30161

Differentiate $G(x)=(4 x^{2}+5)(2 x+\sqrt{x})$ . Find $G^{\prime}(x)$ .

See Solution

Problem 30162

Find the derivative of $f(s)=\frac{\sqrt{s}-5}{\sqrt{s}+5}$ . Choose the correct answer: A, B, C, or D.

See Solution

Problem 30163

Differentiate the function $g(x)=\frac{9 x-7}{7 x+6}+x^{3}$ . Find $g^{\prime}(x)=$ .

See Solution

Problem 30164

Find the general solution to $(D^{2}-4 D+5)^{2} y=0$ : $y(x)=c_{1} e^{2 x} \cos (x)+c_{2} e^{2 x} \sin (x)+c_{3} x e^{2 x} \cos (x)+c_{4} x e^{2 x} \sin (x)$ .

See Solution

Problem 30165

An object falls on a planet with $s(t)=12 t^{2}$ . Find (a) distance in 3 sec, (b) speed in 3 sec, (c) acceleration after 3 sec.

See Solution

Problem 30166

An object falls a distance of $s(t)=12t^{2}$ feet. Find (a) distance in 3 sec, (b) speed in 3 sec, (c) acceleration after 3 sec.

See Solution

Problem 30167

Find the derivative using the quotient rule for $y=\frac{x^{2}-3 x+7}{x^{2}+9}$ . What is $y^{\prime}=$ ?

See Solution

Problem 30168

Find the derivative using the quotient rule for $y=\frac{x^{2}-3 x+7}{x^{2}+9}$ . What is $y^{\prime}=\square$ ?

See Solution

Problem 30169

Differentiate the function $g(x)=5 x^{-3}(x^{4}-5 x^{3}+12 x-7)$ . Find $g^{\prime}(x)$ .

See Solution

Problem 30170

Find the horizontal asymptote of $f(x)=\frac{2 x^{4}+5 x^{3}+3 x^{2}+2 x-50}{x^{4}+3}$ as $x \to +\infty$ .

See Solution

Problem 30171

What is the behavior of $f'(x)$ if $f''(x) > 0$ ? a) Decreasing b) Increasing c) Stationary d) More info needed.

See Solution

Problem 30172

What are the possible values for $f^{\prime}(x)$ if $f(x_1) > f(x_2)$ for every $x_1 < x_2$ ?

See Solution

Problem 30173

Find the rate of change of average cost when 172 belts are produced, given $C(x)=780+40x-0.064x^2$ . $\overline{C}'(x)=\square$

See Solution

Problem 30174

Evaluate these integrals: 1) $\int\left(x^{5}+4 x^{3}-\frac{x}{3}+2\right) d x$ 2) $\int(\sin (x)-\sec (x) \tan (x)) d x$ 3) $\int(8 x-12)\left(4 x^{2}-12 x\right)^{4} d x$

See Solution

Problem 30175

Find the critical numbers of the function $f(x)=x^{3}+3x+4$ . Options: a) $x=-1, 1$ b) None c) $x=-1$ d) $x=1$

See Solution

Problem 30176

Determine where the function $f(x)=x^{3}+3 x-1$ is increasing. Options: a) $-1<x<1$ , b) $x<-1$ and $x>1$ , c) all real numbers, d) $x>1$ .

See Solution

Problem 30177

1. A projectile's height is given by $h(t)=25t-4.9t^{2}$ . Find $\Delta h$ , $\Delta t$ , and $\frac{\Delta h}{\Delta t}$ for intervals near $t=4$ . Estimate velocity at $t=4$ .
2. Bacteria growth is modeled by $N(t)=75000+64t^{3}$ . Find average growth rate for first 6 hours and instantaneous rate at 6 hours.
3. Particle displacement is $s(t)=4t^{2}-10t+13$ . Calculate average rate from $1s$ to $4s$ and estimate instantaneous rate at $1s$ . Interpret results.
4. Ferris wheel height is $h=18\sin\left(\frac{\pi t}{100}\right)+20$ . Find average rate of change for specified intervals around $t=12$ .

See Solution

Problem 30178

Determine the type of critical point at $(c, f(c))$ if $f'(x)>0$ for $x>c$ and $f'(x)<0$ for $x<c$ .

See Solution

Problem 30179

Evaluate the integral $\int x\left(2 x^{2}+3\right)^{4} dx$ using the substitution $u=2 x^{2}+3$ .

See Solution

Problem 30180

Evaluate the integral $\int \sin^{2} \theta \cos \theta \, d\theta$ using the substitution $u = \sin \theta$ .

See Solution

Problem 30181

Find the asymptotes of $f(x) = \frac{x - 2}{x^2 - 3x - 4}$ and analyze their behavior and the end behavior.

See Solution

Problem 30182

Evaluate the integral $\int_{-2}^{-1} \frac{4}{(2-x)^{2}} d x$ .

See Solution

Problem 30183

Berechnen Sie das Integral $\int_{-2}^{-1} \frac{4}{(2-x)^{2}} d x$ mit einer Substitution.

See Solution

Problem 30184

Find $f^{\prime}(x)$ for $f(x)=\cot(5x)-5\sin(2x)$ .

See Solution

Problem 30185

Calculate the average rate of change of $f(x) = 2x^2 - 1$ on the interval $[-1, 0]$ .

See Solution

Problem 30186

Find the average rate of change of height $h(t)=-5t^{2}+20t+1$ on $0 \leq t \leq 2$ and $2 \leq t \leq 4$ . Then, find when the instantaneous rate of change is 0.

See Solution

Problem 30187

Find $c$ values for the Mean Value Theorem on $R(x)=2x^2+5x-3$ for $1 \leq x \leq 4$ .

See Solution

Problem 30188

Find the rate of change of the function $f(t)=2(2.3)^{2t}$ .

See Solution

Problem 30189

Find the rate of change of the average cost function $C(x)=\frac{0.2 x^{2}-4 x+140}{x}$ at $x=7$ . Options: $-2.59$ , 2.66, $-2.66$ , 2.59, None.

See Solution

Problem 30190

Calculate the integral: $\int_{1}^{4} \cos (x)-\frac{3}{x^{5}} \, dx$ .

See Solution

Problem 30191

Find the instantaneous rate of change of $f(x)=\frac{x}{x+6}$ at $(-3,-1)$ . Options: $1/3$ , $2/3$ , $-2/3$ , None, $-1/3$ .

See Solution

Problem 30192

Find the drug concentration $\mathrm{C}(13) = 0.09(1 - e^{-0.2 \cdot 13})$ . Round to three decimal places.

See Solution

Problem 30193

Determine if the function $f(x)=\left\{\begin{array}{ll}5 x-2, & \text { if } x<2 \\ \frac{4}{x+5}, & \text { if } x \geq 2\end{array}\right.$ has a jump discontinuity at $x=2$ by finding the limits: $\lim _{x \rightarrow 2^{-}} f(x)$ and $\lim _{x \rightarrow 2^{+}} f(x)$ .

See Solution

Problem 30194

Graph $f(x)=0.5 x^{2}+5 x-15$ and estimate the instantaneous rate of change at $x=-5$ , $x=-1$ , $x=0$ , $x=3$ .

See Solution

Problem 30195

Find the one-sided limits of $f(x)$ at $x=4$ , check continuity, and sketch the graph with appropriate circles.
Limits: $\lim _{x \rightarrow 4^{-}} f(x)=$ $\lim _{x \rightarrow 4^{+}} f(x)=$ $\lim _{x \rightarrow 4} f(x)=$ $f(4)=$
Is $f$ continuous at $x=4$ ? (YES/NO)

See Solution

Problem 30196

Find the general antiderivative of $f(x)=\frac{5}{6}+\frac{3}{4} x^{2}-\frac{3}{4} x^{3}$ , $F(x)=\square$ .

See Solution

Problem 30197

Expand and simplify: $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\sqrt{3x-7}$ .

See Solution

Problem 30198

Identify the intervals where the function is increasing, decreasing, constant, and state its domain and range based on points $(-1,-3)$ and $(4,-3)$ .

See Solution

Problem 30199

Determine if the function $f(x)$ has a jump discontinuity at $x=4$ by calculating $\lim _{x \rightarrow 4^{-}} f(x)$ and $\lim _{x \rightarrow 4^{+}} f(x)$ .

See Solution

Problem 30200

Given a continuous function $f$ with $f(-3)=-1$ and $f(3)=1$ , classify the statements:
1. $f(0)=0$ : (A, B, or C)
2. For some $c$ where $-3 \leq c \leq 3, f(c)=0$ : (A, B, or C)

See Solution

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Calculus

Problem 30101

Find the derivative dydx\frac{d y}{d x}dxdy​ of y=1x3y=\frac{1}{x^{3}}y=x31​ at x=−2x=-2x=−2. What is the value?

Problem 30102

Find dydx\frac{d y}{d x}dxdy​ for y=9x+9x6y=9 x+\frac{9}{x^{6}}y=9x+x69​ at x=1x=1x=1. What is the simplified value?

Problem 30103

Find the derivative f′(2)f^{\prime}(2)f′(2) for the function f(x)=2x2−6x+9f(x)=2 x^{2}-6 x+9f(x)=2x2−6x+9. Simplify your answer.

Problem 30104

Find the intervals where the function f(x)=excos⁡(x)f(x)=e^{x} \cos (x)f(x)=excos(x) is increasing or decreasing.

Problem 30105

Problem 30106

Find where the tangent line is horizontal for the function y=x2−5y=x^{2}-5y=x2−5. A. The point(s) is(are) □\square□. B. The tangent line is horizontal at all points. C. No points have a horizontal tangent line.

Problem 30107

Aufgabe: Analysiere die Stückkosten k(x)k(x)k(x) und h(x)h(x)h(x) für große xxx und finde die Stückzahl, bei der h(x)<k(300)h(x) < k(300)h(x)<k(300).

Problem 30108

Gegeben ist die Funktion f(x)=−0,5x2+4xf(x)=-0,5 x^{2}+4 xf(x)=−0,5x2+4x. Zeichne den Graphen für x∈[0;5]x \in [0; 5]x∈[0;5] und bestimme die Änderungsrate für x=3x=3x=3.

Problem 30109

Find the points of discontinuity for f(x)=x2+11−x2f(x)=\frac{x^{2}+1}{1-x^{2}}f(x)=1−x2x2+1​ and their one-sided limits.

Problem 30110

Find where the tangent line to the graph of y=x2+2y=x^{2}+2y=x2+2 is horizontal. A. The point(s) is(are) □\square□. B. The tangent line is horizontal at all points. C. No points have a horizontal tangent line.

Problem 30111

Find the point on the curve y=3x2−2xy=3x^{2}-2xy=3x2−2x where the tangent slope is -26. The point is □\square□.

Problem 30112

Plot the line segment on f(x)=x3+11x2+23x−35f(x)=x^{3}+11x^{2}+23x-35f(x)=x3+11x2+23x−35 between x=−8x=-8x=−8 and x=0x=0x=0. Find all ccc for the Mean Value Theorem on [−8,0][-8, 0][−8,0].

Problem 30113

Find the limit: lim⁡x→−32x2−x−21x+3\lim _{x \rightarrow-3} \frac{2 x^{2}-x-21}{x+3}limx→−3​x+32x2−x−21​.

Problem 30114

Find where the tangent line of f(x)=5x2−4x+4f(x)=5 x^{2}-4 x+4f(x)=5x2−4x+4 is horizontal. State if none exist.

Problem 30115

Plot the line segment between f(−5)f(-5)f(−5) and f(0)f(0)f(0), then find all ccc satisfying the Mean Value Theorem on [−5,0][-5, 0][−5,0].

Problem 30116

Find the weight change rate for a boy aged ttt months using w(t)=8.71+1.69t−0.0095t2+0.000035t3w(t)=8.71+1.69t-0.0095t^2+0.000035t^3w(t)=8.71+1.69t−0.0095t2+0.000035t3. Calculate w′(t)w'(t)w′(t).

Problem 30117

Problem 30118

Graph the function f(x)=x3−13x2+44x−42f(x)=x^{3}-13x^{2}+44x-42f(x)=x3−13x2+44x−42. Find ccc values for the Mean Value Theorem on [0,8][0, 8][0,8].

Problem 30119

A city’s population grows from 800,000 to P(t)=800,000+1000t2P(t)=800,000+1000 t^{2}P(t)=800,000+1000t2 in ttt years. Find: a) dPdt\frac{d P}{d t}dtdP​ b) P(10)P(10)P(10) c) dPdt\frac{d P}{d t}dtdP​ at t=10t=10t=10 d) Meaning of part (c): A, B, C, or D?

Problem 30120

Find the derivative of f(x)=(x−5)(5x+2)f(x)=(x-5)(5x+2)f(x)=(x−5)(5x+2) using the Product Rule and by expanding the product.

Problem 30121

Find all values of ccc satisfying the Mean Value Theorem for f(x)=x3−10x2+13x+24f(x)=x^{3}-10x^{2}+13x+24f(x)=x3−10x2+13x+24 on [−1,7][-1, 7][−1,7].

Problem 30122

Plot the line segment between f(0)f(0)f(0) and f(8)f(8)f(8) for f(x)=x3−13x2+44x−42f(x)=x^{3}-13x^{2}+44x-42f(x)=x3−13x2+44x−42. Find all ccc satisfying the Mean Value Theorem on [0,8][0, 8][0,8].

Problem 30123

Find where the function f(x)=excos⁡(x)f(x)=e^{x} \cos (x)f(x)=excos(x) is increasing or decreasing. Also, find the edge length of a square-based container with volume 0.5 m³ that minimizes material use.

Problem 30124

Find the derivative of y=x4⋅x6y=x^{4} \cdot x^{6}y=x4⋅x6 using the Product Rule and by multiplying first. Verify both results.

Problem 30125

Plot the line segment on f(x)=x3+2x2−11x−22f(x)=x^{3}+2 x^{2}-11 x-22f(x)=x3+2x2−11x−22 between x=−5x=-5x=−5 and x=3x=3x=3. Find all ccc for the Mean Value Theorem on [−5,3][-5, 3][−5,3].

Problem 30126

Problem 30127

Plot a line segment on f(x)=x3+8x2+9x−28f(x)=x^{3}+8x^{2}+9x-28f(x)=x3+8x2+9x−28 between x=−7x=-7x=−7 and x=1x=1x=1. Find all ccc for the Mean Value Theorem.

Problem 30128

Calculate the integral: ∫6x2sin⁡3x dx\int 6 x^{2} \sin 3 x \, dx∫6x2sin3xdx

Problem 30129

Find the second derivative y′′y^{\prime \prime}y′′ for the function y=3x−42x+5y=\frac{3 x-4}{2 x+5}y=2x+53x−4​.

Problem 30130

Find ccc in the interval [−6,−2][-6, -2][−6,−2] using the Mean Value Theorem for the function y=(4x+24)12y=(4x+24)^{\frac{1}{2}}y=(4x+24)21​.

Problem 30131

Problem 30132

Find the derivative of h(z)=(4−z2)(z3−3z+1)h(z)=(4-z^{2})(z^{3}-3z+1)h(z)=(4−z2)(z3−3z+1) using the product rule and by expanding the product.

Problem 30133

Scrieți diferențialele de ordinul întâi și doi pentru funcțiile date și calculați-le în punctele specificate.

Problem 30134

Find the second derivative f′′(x)f^{\prime \prime}(x)f′′(x) for f(x)=(x2+9)34f(x)=\sqrt[4]{(x^{2}+9)^{3}}f(x)=4(x2+9)3​.

Problem 30135

Find the derivative of y=x9x5y=\frac{x^{9}}{x^{5}}y=x5x9​ using the Quotient Rule and by simplifying first.

Problem 30136

Find the fourth derivative of the function y=7x5y=7 x^{5}y=7x5. What is d4ydx4\frac{d^{4} y}{d x^{4}}dx4d4y​?

Problem 30137

Calculaţi derivatele parţiale pentru funcţiile: (a) f(x,y)=xy3eyf(x, y)=x y^{3} e^{y}f(x,y)=xy3ey, (b) f(p,q)=3p3q2f(p, q)=3 p^{3} q^{2}f(p,q)=3p3q2, (c) f(k,l)=5kl3f(k, l)=5 \sqrt{k} l^{3}f(k,l)=5k​l3.

Problem 30138

Find the second derivative of y=2x+6y=2x+6y=2x+6. What is d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​?

Problem 30139

Find the fourth derivative t(4)(n)t^{(4)}(n)t(4)(n) of t(n)=4n−1/4+5n5/4t(n)=4 n^{-1/4}+5 n^{5/4}t(n)=4n−1/4+5n5/4. Simplify your answer.

Problem 30140

Find the derivative of F(x)=8x4(x2−5x)F(x)=8 x^{4}(x^{2}-5 x)F(x)=8x4(x2−5x) using the Product Rule and by multiplying first. Check if both methods give the same result.

Problem 30141

Problem 30142

Find the derivative of y=x9x4y=\frac{x^{9}}{x^{4}}y=x4x9​ using the Quotient Rule or by simplifying first. Choose the correct option.

Find the derivative $\frac{d y}{d x}$ of $y=\frac{1}{x^{3}}$ at $x=-2$ . What is the value?

Find $\frac{d y}{d x}$ for $y=9 x+\frac{9}{x^{6}}$ at $x=1$ . What is the simplified value?

Find the derivative $f^{\prime}(2)$ for the function $f(x)=2 x^{2}-6 x+9$ . Simplify your answer.

Find the intervals where the function $f(x)=e^{x} \cos (x)$ is increasing or decreasing.

Find where the tangent line is horizontal for the function $y=x^{2}-5$ .
A. The point(s) is(are) $\square$ . B. The tangent line is horizontal at all points. C. No points have a horizontal tangent line.

Aufgabe: Analysiere die Stückkosten $k(x)$ und $h(x)$ für große $x$ und finde die Stückzahl, bei der $h(x) < k(300)$ .

Gegeben ist die Funktion $f(x)=-0,5 x^{2}+4 x$ . Zeichne den Graphen für $x \in [0; 5]$ und bestimme die Änderungsrate für $x=3$ .

Find the points of discontinuity for $f(x)=\frac{x^{2}+1}{1-x^{2}}$ and their one-sided limits.

Find where the tangent line to the graph of $y=x^{2}+2$ is horizontal.
A. The point(s) is(are) $\square$ . B. The tangent line is horizontal at all points. C. No points have a horizontal tangent line.

Find the point on the curve $y=3x^{2}-2x$ where the tangent slope is -26. The point is $\square$ .

Plot the line segment on $f(x)=x^{3}+11x^{2}+23x-35$ between $x=-8$ and $x=0$ . Find all $c$ for the Mean Value Theorem on $[-8, 0]$ .

Find the limit: $\lim _{x \rightarrow-3} \frac{2 x^{2}-x-21}{x+3}$ .

Find where the tangent line of $f(x)=5 x^{2}-4 x+4$ is horizontal. State if none exist.

Plot the line segment between $f(-5)$ and $f(0)$ , then find all $c$ satisfying the Mean Value Theorem on $[-5, 0]$ .

Find the weight change rate for a boy aged $t$ months using $w(t)=8.71+1.69t-0.0095t^2+0.000035t^3$ . Calculate $w'(t)$ .

Graph the function $f(x)=x^{3}-13x^{2}+44x-42$ . Find $c$ values for the Mean Value Theorem on $[0, 8]$ .

A city’s population grows from 800,000 to $P(t)=800,000+1000 t^{2}$ in $t$ years. Find: a) $\frac{d P}{d t}$ b) $P(10)$ c) $\frac{d P}{d t}$ at $t=10$ d) Meaning of part (c): A, B, C, or D?

Find the derivative of $f(x)=(x-5)(5x+2)$ using the Product Rule and by expanding the product.

Find all values of $c$ satisfying the Mean Value Theorem for $f(x)=x^{3}-10x^{2}+13x+24$ on $[-1, 7]$ .

Plot the line segment between $f(0)$ and $f(8)$ for $f(x)=x^{3}-13x^{2}+44x-42$ . Find all $c$ satisfying the Mean Value Theorem on $[0, 8]$ .

Find where the function $f(x)=e^{x} \cos (x)$ is increasing or decreasing. Also, find the edge length of a square-based container with volume 0.5 m³ that minimizes material use.

Find the derivative of $y=x^{4} \cdot x^{6}$ using the Product Rule and by multiplying first. Verify both results.

Plot the line segment on $f(x)=x^{3}+2 x^{2}-11 x-22$ between $x=-5$ and $x=3$ . Find all $c$ for the Mean Value Theorem on $[-5, 3]$ .

Plot a line segment on $f(x)=x^{3}+8x^{2}+9x-28$ between $x=-7$ and $x=1$ . Find all $c$ for the Mean Value Theorem.

Calculate the integral: $\int 6 x^{2} \sin 3 x \, dx$

Find the second derivative $y^{\prime \prime}$ for the function $y=\frac{3 x-4}{2 x+5}$ .

Find $c$ in the interval $[-6, -2]$ using the Mean Value Theorem for the function $y=(4x+24)^{\frac{1}{2}}$ .

Find the derivative of $h(z)=(4-z^{2})(z^{3}-3z+1)$ using the product rule and by expanding the product.

Find the second derivative $f^{\prime \prime}(x)$ for $f(x)=\sqrt[4]{(x^{2}+9)^{3}}$ .

Find the derivative of $y=\frac{x^{9}}{x^{5}}$ using the Quotient Rule and by simplifying first.

Find the fourth derivative of the function $y=7 x^{5}$ . What is $\frac{d^{4} y}{d x^{4}}$ ?

Calculaţi derivatele parţiale pentru funcţiile: (a) $f(x, y)=x y^{3} e^{y}$ , (b) $f(p, q)=3 p^{3} q^{2}$ , (c) $f(k, l)=5 \sqrt{k} l^{3}$ .

Find the second derivative of $y=2x+6$ . What is $\frac{d^{2} y}{d x^{2}}$ ?

Find the fourth derivative $t^{(4)}(n)$ of $t(n)=4 n^{-1/4}+5 n^{5/4}$ . Simplify your answer.

Find the derivative of $F(x)=8 x^{4}(x^{2}-5 x)$ using the Product Rule and by multiplying first. Check if both methods give the same result.

Find the derivative of $y=\frac{x^{9}}{x^{4}}$ using the Quotient Rule or by simplifying first. Choose the correct option.

Evaluate the integral $\int_{0}^{a} x^{2} \sin ^{2}\left(\frac{n \pi x}{a}\right) d x$ for constants $n$ , $\pi$ , and $a$ .

Find the rate of change of average cost when 174 belts are produced, given $C(x)=770+32x-0.067x^2$ . Calculate $\overline{C}'(174)$ .

Divide $x^{9}$ by $x^{4}$ , simplify to get $x^{5}$ , then find $\frac{d y}{d x}$ and verify both results equal.

Find the rate of change of the particle's position for $s=\sqrt{37+4t}$ at $t=3$ sec. Answer: $\square \mathrm{m} / \mathrm{sec}$ .

Given the temperature function $T(t)=\frac{8 t}{t^{2}+2}+986$ , find:
(a) $T^{\prime}(t)$ , (b) $T(1)$ , (c) $T^{\prime}(1)$ .

Evaluate the integral $f(x)=\int \frac{d x}{(2 x+1) \sqrt{4 x+4 x^{2}}}$ .

Differentiate the function $y=(2x^{2}-6)^{-10}$ . Find $\frac{dy}{dx}=\square$ .

Differentiate the function $y=\frac{1}{(3 x+7)^{2}}$ . Find $\frac{d y}{d x}$ .

Differentiate the function $y=(7 x-2)^{3}$ . Find $\frac{d y}{d x}$ .

Differentiate the function $f(x)=\left(\frac{x+3}{x-6}\right)^{8}$ . Find $f^{\prime}(x)=$ .

Differentiate the function $y=(18-x)^{65}$ . Find $\frac{d y}{d x}=\square$ .

If $f^{\prime \prime}(x)>0$ , what can we conclude about $f(x)$ ? Choose the correct option.

Given $s(t)=4t+8$ , find: a) $v(t)$ , b) $a(t)$ , c) values at $t=2$ hr, d) meaning of uniform motion.

A company's sales function is $S(t)=2t^{3}-45t^{2}+270t+180$ . Find $S'(2)$ , $S'(3)$ , $S'(5)$ and $S''(2)$ , $S''(3)$ , $S''(5)$ .

Find the mean of the density function $p(t)=0.2 e^{-0.2 t}$ for $0 \leq t \leq 30$ , rounded to 2 decimal places.

Given $s(t)=t^{2}+t$ , find the velocity $v(t)$ , acceleration $a(t)$ , and their values at $t=4$ sec.

Differentiate $G(x)=(4 x^{2}+5)(2 x+\sqrt{x})$ . Find $G^{\prime}(x)$ .

Find the derivative of $f(s)=\frac{\sqrt{s}-5}{\sqrt{s}+5}$ . Choose the correct answer: A, B, C, or D.

Differentiate the function $g(x)=\frac{9 x-7}{7 x+6}+x^{3}$ . Find $g^{\prime}(x)=$ .

An object falls on a planet with $s(t)=12 t^{2}$ . Find (a) distance in 3 sec, (b) speed in 3 sec, (c) acceleration after 3 sec.

An object falls a distance of $s(t)=12t^{2}$ feet. Find (a) distance in 3 sec, (b) speed in 3 sec, (c) acceleration after 3 sec.

Find the derivative using the quotient rule for $y=\frac{x^{2}-3 x+7}{x^{2}+9}$ . What is $y^{\prime}=$ ?

Find the derivative using the quotient rule for $y=\frac{x^{2}-3 x+7}{x^{2}+9}$ . What is $y^{\prime}=\square$ ?

Differentiate the function $g(x)=5 x^{-3}(x^{4}-5 x^{3}+12 x-7)$ . Find $g^{\prime}(x)$ .

Find the horizontal asymptote of $f(x)=\frac{2 x^{4}+5 x^{3}+3 x^{2}+2 x-50}{x^{4}+3}$ as $x \to +\infty$ .

What is the behavior of $f'(x)$ if $f''(x) > 0$ ? a) Decreasing b) Increasing c) Stationary d) More info needed.

What are the possible values for $f^{\prime}(x)$ if $f(x_1) > f(x_2)$ for every $x_1 < x_2$ ?

Find the rate of change of average cost when 172 belts are produced, given $C(x)=780+40x-0.064x^2$ . $\overline{C}'(x)=\square$