Calculus

Problem 8001

Find the tangent line equation for $f(x)=x^{3}-3x+1$ at the point $(4,53)$ . What is $y=$ ?

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Problem 8002

Find the slope and equation of the tangent line to the curve $f(x)=x-x^{3}$ at the point $(1,0)$ . $y=$

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Problem 8003

Find the tangent line equation at $(1,-1)$ for $f(x)=5x^3-7x^2+1$ . Use $y - y_1 = m(x - x_1)$ .

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Problem 8004

Find the derivative $f^{\prime}(a)$ for the function $f(t)=\frac{4t+8}{t+5}$ .

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Problem 8005

Find intervals of increase and decrease for $F(x)=x \sqrt{6-x}$ . Also, find local min and max values.

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Problem 8006

Find the tangent line equation for $f(x)=\sqrt{x}$ at the point $(1,1)$ . What is $y=$ ?

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Problem 8007

Find the derivative $f^{\prime}(9)$ for the function $f(x)=10 x^{\ln (x)}$ .

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Problem 8008

Find the derivative $f^{\prime}(a)$ for the function $f(x)=\sqrt{3-6x}$ .

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Problem 8009

Find the tangent line equation at $(2,20)$ for $f(x)=5 x^{3}-7 x^{2}+8$ . Use $y - y_1 = m(x - x_1)$ .

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Problem 8010

A ball is thrown with a velocity of $34 \mathrm{ft} / \mathrm{s}$ . Its height is $s(t)=34t-16t^{2}$ . Find the velocity at $t=1 \mathrm{s}$ .

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Problem 8011

A projectile is launched at 64 ft/s from a 45-ft platform. Find its maximum height.

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Problem 8012

A projectile is launched at 64 ft/s from a 45-ft platform. Find its maximum height. Use $h(t) = -16t^2 + 64t + 45$ .

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Problem 8013

Evaluate the integral: $\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}$ . What is the result?

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Problem 8014

Given the cost function $C(x)=44100+400 x+x^{2}$ , find: a) Cost at $x=1550$ b) Average cost at $x=1550$ c) Marginal cost at $x=1550$ d) Production level minimizing average cost e) Minimal average cost

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Problem 8015

Find $f'(x)$ and $f'(6)$ for $f(x)=2 \ln (x)$ . Also, find $f''(x)$ and $f''(6)$ .

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Problem 8016

Find the area between the velocity-time graph and the time axis from $t_{1} = 7.2s$ to $t_{2} = 14.4s$ for displacement.

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Problem 8017

Find the limit: $\lim _{x \rightarrow 0} \frac{7 x^{4}+5 x+12}{3 x^{4}+4}$ .

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Problem 8018

Find the growth rate of a snake at 3 weeks using the model $L(t)=19\left(1-\left(1+0.553 t+0.014 t^{2}\right)^{-1}\right)$ . Round to 2 decimal places.

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Problem 8019

Find the second derivative $y^{\prime \prime}$ for $y=\frac{\csc x}{2}$ .

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Problem 8020

Write the sigma notation for the sum of $f(x)=x^{2}$ from $x=1$ to $x=2$ with $n=30$ .

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Problem 8021

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\cos x + 3 e^{x}}{2 e^{x}}$ .

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Problem 8022

Find the derivative $y^{\prime}$ of the function $y=\ln \sqrt{x^{2}-4 x-7}$ .

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Problem 8023

Find the derivative $h^{\prime}(x)$ for $h(x)=\ln \frac{x(x-1)^{3}}{\sqrt{x-2}}$ . No need to simplify.

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Problem 8024

Find the derivative $y^{\prime}$ of the function $y=3^{5x}$ .

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Problem 8025

A company's revenue is $R(q)=-q^{3}+230 q^{2}$ and cost is $C(q)=480+10 q$ .
A) Find the marginal profit function $M P(q)$ . B) Determine the number of hundreds of units to sell for maximum profit.

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Problem 8026

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin x-1}{\cos ^{2} x}$ .

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Problem 8027

Find how many positive values of $b$ make $\lim _{x \rightarrow b} f(x)=2$ for $f(x)=0.1 x^{4}-0.5 x^{3}-3.3 x^{2}+7.7 x-1.99$ .

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Problem 8028

1. In a piecewise hazard rate model with intervals $[\tau_{i-1}, \tau_{i})$ , find the survival function and mean residual-life function.

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Problem 8029

Find the tangent line equation at $x=1$ for $y=x^{4}-3x^{2}-5x-4$ and graph both.

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Problem 8030

Find values of $x$ where the tangent line of $f(x)=\sqrt{x^{3}-9x^{2}+24x+1}$ is horizontal.

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Problem 8031

Analyze the sequence $a_{n}=\frac{e^{n}}{4^{n}}$ for convergence or divergence and find the limit if it converges.

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Problem 8032

Find the number of apartments filled, $x$ , that maximizes revenue given by $R(x)=600x-0.5x^{2}$ .

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Problem 8033

Find $f^{\prime}(x)$ and where the tangent line of $f(x)=\frac{x}{(2 x-5)^{3}}$ is horizontal.

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Problem 8034

Given the function $f(x)=\frac{4}{x^{2}-25}$ , which statement is FALSE?
1. Local max at $(0, f(0))$
2. Horizontal asymptote $y=0$
3. Increasing on $(0,5)$ and $(5, \infty)$
4. No points of inflection
5. Concave down on $(-5,5)$
6. None of the above

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Problem 8035

Which statements about the graph of $f(x)=27 x^{2}+\frac{2}{x}-2$ are true?

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Problem 8036

Find the tangent line equation at $x=2$ for $y=x^{4}-3x^{3}+2x+4$ . Graph both on the same axes.

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Problem 8037

Find the number of boats $x$ that minimizes the cost $C(x)=17000-40 x+0.04 x^{2}$ .

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Problem 8038

Identify the FALSE statement about the function $f(x)=\frac{3 x}{(x-5)^{2}}$ and its derivatives.

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Problem 8039

Find values of $x$ where the tangent line of $f(x)=\sqrt{x^{3}-9 x^{2}+24 x+3}$ is horizontal.

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Problem 8040

Check if the graph of $f(x)=3 x \sqrt[3]{x-6}$ has a vertical tangent or cusp at $x=6$ . Options: cusp, tangent, neither, both.

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Problem 8041

Consider the function $f(x)=(x^{2}-4x)^{\frac{1}{5}}$ . Which statement about $f(x)$ is FALSE?

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Problem 8042

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

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Problem 8043

Find the derivative of $y=\ln \cos(2x)$ .

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Problem 8044

A golfer hits a ball at $42 \mathrm{~m/s}$ and $32^{\circ}$ . Find the maximum height reached, ignoring air resistance.

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Problem 8045

Differentiate $g(t)=\ln \left(\frac{t(t^{2}+1)^{4}}{\sqrt[7]{6 t-1}}\right)$ .

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Problem 8046

Identify the FALSE statement about the function $f(x)=\frac{2}{x^{2}-9}$ and its derivatives.

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Problem 8047

Identify the FALSE statement about the function $f(x)=\left(5 x^{2}-2 x\right)^{\frac{1}{3}}$ .

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Problem 8048

Find the population change in 2013 for the model $P(t)=200(1.052)^{t}$ (in thousands). Choices are given.

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Problem 8049

Check if the graph of $f(x)=2(x-3)^{4/5}$ has a vertical tangent or cusp at $x=3$ . Options: neither, vertical cusp, both, vertical tangent.

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Problem 8050

Which statements about the graph of $f(x)=x+\sin(2x)+4$ on $[0, \pi]$ are true?

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Problem 8051

Evaluate $f(x)=3 x^{2}+12 x+5$ at the endpoints $-7$ and $3$ . Does Rolle's Theorem apply? Find $c$ or enter "DNE".

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Problem 8052

A kicker must kick a ball from $34 \mathrm{~m}$ to clear a $3.05 \mathrm{~m}$ crossbar. The ball's speed is $20 \mathrm{~m/s}$ at $42.2^{\circ}$ . How much higher does it go than the crossbar? Answer in $\mathrm{m}$ .

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Problem 8053

Calculate the derivative value $f^{\prime}(1)=(-1+2)e^{3(1)^{2}}$ .

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Problem 8054

A pendulum of mass $m$ is released from horizontal. Find speed, tension, max height, and largest $r$ before breaking.

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Problem 8055

Evaluate $f(x)=\cos(3 \pi x)$ at $x=-\frac{5}{3}$ and $x=-1$ . Does Rolle's Theorem apply? Find $c$ where $f'(c)=0$ .

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Problem 8056

Find all numbers that satisfy the mean value theorem for the function $f(x)=7x-\frac{4}{x}$ on the interval $[3,10]$ .

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Problem 8057

Differentiate the following with respect to $x$ :
1. $\frac{\sin 2 x}{2 x+5}$
2. $\frac{(3 x+1) \cos 2 x}{e^{2 x}}$
3. $\tan x$

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Problem 8058

Which statements about the graph of $f(x)=x+\sin (2 x)+4$ on $[0, \pi]$ are true?

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Problem 8059

Evaluate the integral $\int_{0}^{1} \frac{\ln (x+1)}{x^{2}+1} dx$ .

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Problem 8060

Find the slope of the secant line for $f(x)=-4 x^{2}-5 x+6$ on $[-3,-1]$ . Then find $c$ in $(-3,-1)$ where $m=f^{\prime}(c)$ .

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Problem 8061

Calculate the area under the curve of $3x^2 - x + 7$ from $0$ to $1$ using Riemann sums: $\int_{0}^{1}(3x^2 - x + 7)dx$ .

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Problem 8062

Find the value(s) of $c$ that satisfy the Mean Value Theorem for $f(x)=\ln(x^{7})$ on the interval $[1,7]$ .

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Problem 8063

Find the derivatives of these functions: (1) $y=e^{\sin 2 x}$ (2) $y=\sin^{2} x$ (3) $y=\ln \cos 3 x$ (4) $y=\cos^{3}(3 x)$ (5) $y=\log_{10}(2 x-1)$ .

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Problem 8064

Quiz 3: Provide examples or explanations for these functions: a. Continuous at $a=2$ , not differentiable there. b. Differentiable at $a=3$ , no limit at $a=3$ . c. Limit exists at $a=-2$ , defined there, not continuous. d. Function $p$ with: i. $p(-1)=3$ , $\lim_{x \to -1} p(x)=2$ ; ii. $p(0)=1$ , $p'(0)=0$ ; iii. $\lim_{x \to 1} p(x)=p(1)$ , $p'(1)$ doesn't exist.

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Problem 8065

Find the slope of the secant line for $f(x)=2 x^{3}-5 x$ on $[1,4]$ . Then, find $c$ in $(1,4)$ where $m=f^{\prime}(c)$ .

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Problem 8066

Find the slope of the secant line for $f(x)=-3 x^{2}-5 x+1$ on $[-3,1]$ . Then, find $c$ in $(-3,1)$ where $m=f^{\prime}(c)$ .

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Problem 8067

How much faster is a person on an $18.6 \mathrm{~m}$ building than on the ground? Use a day period and Earth's radius $6400 \mathrm{~km}$ .

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Problem 8068

Find the mean slope of $f(x)=3-2x^{2}$ on $[-5,4]$ and the value of $c$ where $f'(c)$ equals this slope.

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Problem 8069

Given $f(x)=\frac{1}{x^{2}}$ on $[-2,2]$ , answer: a. Is $f(-2)=f(2)$ ? b. Is $f$ continuous on $[-2,2]$ ? c. Is $f$ differentiable on $(-2,2)$ ? d. Can Rolle's or Mean Value Theorem apply? e. If applicable, find $c$ . $c=$

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Problem 8070

Find the derivative of the function $f(x)=2 \tan ^{-1}(6 \sin (2 x))$ , i.e., compute $f^{\prime}(x)$ .

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Problem 8071

Find the average slope of the function $f(x)=5 x^{3}-4 x$ on $[-2,2]$ and determine the two values of $c$ where $f^{\prime}(c)$ equals this slope.

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Problem 8072

Find the limit: $\lim _{h \rightarrow 0} \frac{(16+h)^{\frac{1}{4}}-2}{h}$ . What is its value?

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Problem 8073

Find the derivative of the function $f(x)=6 \tan ^{-1}(8 \sin (3 x))$ , denoted as $f^{\prime}(x)$ .

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Problem 8074

Choose the correct answer for the given conditions on the function $f(x)$ and its derivatives.
2. If $f^{\prime \prime}(x)>0$ , then $f(x)$ is: a. Concave up b. Concave down c. Increasing d. Decreasing e. Neither
3. If $f^{\prime \prime}(x)<0$ , then $f(x)$ is: a. Concave up b. Concave down c. Increasing d. Decreasing e. Neither
4. If $f^{\prime}(x)>0$ , then $f(x)$ is: a. Concave up b. Concave down c. Increasing d. Decreasing e. Neither
5. If $f^{\prime}(x)=0$ , then $f(x)$ is: a. Concave up b. Concave down c. Increasing d. Decreasing e. Neither

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Problem 8075

Jamie needs to find the optimal landing point $L=(x,0)$ to minimize travel time $T(x)$ from $J=(0,a)$ to $D=(c,-b)$ .
a. Derive the formula for total travel time $T(x)$ . b. Show that $T'(x)=\frac{\sin (LJO)}{p}-\frac{\sin (LDN)}{q}$ at the minimum travel time. c. Prove that the critical $x$ produces a local minimum for $T(x)$ .

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Problem 8076

Choose the correct option for each statement about the function $f(x)$ :
2. If $f^{\prime \prime}(x)>0$ , then $f(x)$ is: a. Concave up b. Concave down c. Increasing d. Decreasing e. Neither
3. If $f^{\prime \prime}(x)<0$ , then $f(x)$ is: a. Concave up b. Concave down c. Increasing d. Decreasing e. Neither
4. If $f^{\prime}(x)>0$ , then $f(x)$ is: a. Concave up b. Concave down c. Increasing d. Decreasing e. Neither
5. If $f^{\prime}(x)=0$ , then $f(x)$ is: a. Concave up b. Concave down c. Increasing d. Decreasing e. Neither

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Problem 8077

Find the growth rate of the epidemic given by $P=\frac{700}{1+20,000 e^{-0.549 t}}$ after 10, 20, and 30 years.

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Problem 8078

A toy rocket launched from a 5-ft pad at 160 ft/s reaches max height. Find time to max height and the max height. Use $h(t)=-16t^2+160t+5$ .

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Problem 8079

Berechnen Sie die Ableitung $f'(x)$ der Funktion $f(x)=x^{3}-6 x^{2}+20$ und den Wert von $f'(-1)$ .

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Problem 8080

Zeigen Sie, dass die Funktion $f(x)=x^{3}-6x^{2}+20$ bei $x=2$ einen Wendepunkt hat.

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Problem 8081

Bestimme die Ableitung von $f(x)=7 \cdot \sqrt{2.5 x}-\frac{4}{3} x^{2}-7$ .

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Problem 8082

Find the limit: $\lim _{n \rightarrow \infty} \sqrt{4 n+5}(\sqrt{8 n+3}-\sqrt{8 n+1})$ .

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Problem 8083

Berechne $f^{\prime}(12)$ und den Durchschnitt $\frac{f(18)-f(12)}{18-12}$ für $f(x)=0,1 x t^3+4,5 x t^2-66,3 x +322,7$ .

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Problem 8084

Find the limit: $\lim _{n \rightarrow \infty} \frac{\sqrt{81 n^{4}+6 n^{3}+2}-\sqrt{81 n^{4}-n-5}}{n+1}$ .

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Problem 8085

Find the extrema of $f_{t}(x)=x^{3}-t \cdot x^{2}+9 t$ and use the first derivative test.

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Problem 8086

Berechne die Fläche zwischen dem Graphen von $f$ und der $x$ -Achse für: a) $f(x)=3 x^{2}-3$ , b) $f(x)=4-x^{2}$ , c) $f(x)=5 x^{4}-80$ , d) $f(x)=x^{3}-3 x$ .

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Problem 8087

Gegeben sind die Funktionen $p(x)=-x^{2}-x+1$ und $q(x)=\mathrm{e}^{-x}$ .
(1) Wie entsteht der Graph von $q'$ aus $q$ ? (2) Zeigen Sie, dass $p$ und $q$ im gemeinsamen Punkt eine Tangente haben und geben Sie deren Gleichung an. (3) Berechnen Sie $\int_{0}^{2}(q(x)-p(x)) \mathrm{d} x$ und interpretieren Sie den Wert geometrisch.

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Problem 8088

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

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Problem 8089

Find $\lim _{x \rightarrow 2} f(x)$ given that $\lim _{x \rightarrow 2} \frac{2 f(x)+\pi}{x^{2}-3 x+2}=e$ .

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Problem 8090

When marginal revenue equals marginal cost, what does it imply for the company? a. max profit b. min profit c. expand output d. shutdown point

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Problem 8091

Bestimme den Schnittwinkel der Funktionen $f(x)=2 x^{2}+5 x-2$ und $g(x)=2 x^{2}-5 x+2$ an ihrem Schnittpunkt.

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Problem 8092

Bestimmen Sie die Ableitung der Funktion $f(x) = \sqrt[3]{2}$ , also $f'(x)$ .

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Problem 8093

Find the limit: $\lim _{x \rightarrow \infty} \frac{103 x^{2}-8}{4 x^{2}+99 x-3}$ .

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Problem 8094

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

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Problem 8095

Determine the horizontal asymptote of $g(x)=\frac{(x-2)^{3}(x+4)}{(x+5)(x-2)}$ .

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Problem 8096

Find the rate of change of the cost function $C(Q)=\frac{a}{2} Q^{3}+b$ when production changes from $Q$ to $Q+h$ .

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Problem 8097

Find the rate of change of the revenue function $R(Q)=100 Q^{2}+500$ as $\Delta R\left(Q_{0}\right) / \Delta Q$ .

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Problem 8098

Prove that the series $\sum_{n=1}^{\infty} a_{n}$ , where $a_{n} = 1/k$ for odd $n$ and $a_{n} = -1/k$ for even $n$ , is conditionally convergent.

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Problem 8099

Finde die x-Koordinaten und zeichne die Graphen von $f(x) = 0,5x^{3}$ und $g(x) = 0,25x^{5}$ . Bestimme Nullstellen, Extrema und Wendepunkte.

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Problem 8100

Find the limit as h approaches 0 for $\frac{\tan (x)+\tan (h)-\tan (x)(1-\tan (x) \tan (h))}{h(1-\tan (x) \tan (h))}$ and show it simplifies to $\sec^2(x)$ .

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Calculus

Problem 8001

Find the tangent line equation for f(x)=x3−3x+1f(x)=x^{3}-3x+1f(x)=x3−3x+1 at the point (4,53)(4,53)(4,53). What is y=y=y=?

Problem 8002

Find the slope and equation of the tangent line to the curve f(x)=x−x3f(x)=x-x^{3}f(x)=x−x3 at the point (1,0)(1,0)(1,0). y=y=y=

Problem 8003

Find the tangent line equation at (1,−1)(1,-1)(1,−1) for f(x)=5x3−7x2+1f(x)=5x^3-7x^2+1f(x)=5x3−7x2+1. Use y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​).

Problem 8004

Find the derivative f′(a)f^{\prime}(a)f′(a) for the function f(t)=4t+8t+5f(t)=\frac{4t+8}{t+5}f(t)=t+54t+8​.

Problem 8005

Find intervals of increase and decrease for F(x)=x6−xF(x)=x \sqrt{6-x}F(x)=x6−x​. Also, find local min and max values.

Problem 8006

Find the tangent line equation for f(x)=xf(x)=\sqrt{x}f(x)=x​ at the point (1,1)(1,1)(1,1). What is y=y=y=?

Problem 8007

Find the derivative f′(9)f^{\prime}(9)f′(9) for the function f(x)=10xln⁡(x)f(x)=10 x^{\ln (x)}f(x)=10xln(x).

Problem 8008

Find the derivative f′(a)f^{\prime}(a)f′(a) for the function f(x)=3−6xf(x)=\sqrt{3-6x}f(x)=3−6x​.

Problem 8009

Find the tangent line equation at (2,20)(2,20)(2,20) for f(x)=5x3−7x2+8f(x)=5 x^{3}-7 x^{2}+8f(x)=5x3−7x2+8. Use y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​).

Problem 8010

A ball is thrown with a velocity of 34ft/s34 \mathrm{ft} / \mathrm{s}34ft/s. Its height is s(t)=34t−16t2s(t)=34t-16t^{2}s(t)=34t−16t2. Find the velocity at t=1st=1 \mathrm{s}t=1s.

Problem 8011

A projectile is launched at 64 ft/s from a 45-ft platform. Find its maximum height.

Problem 8012

A projectile is launched at 64 ft/s from a 45-ft platform. Find its maximum height. Use h(t)=−16t2+64t+45h(t) = -16t^2 + 64t + 45h(t)=−16t2+64t+45.

Problem 8013

Evaluate the integral: ∫x3dx9−x2\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}∫9−x2​x3dx​. What is the result?

Problem 8014

Given the cost function C(x)=44100+400x+x2C(x)=44100+400 x+x^{2}C(x)=44100+400x+x2, find: a) Cost at x=1550x=1550x=1550 b) Average cost at x=1550x=1550x=1550 c) Marginal cost at x=1550x=1550x=1550 d) Production level minimizing average cost e) Minimal average cost

Problem 8015

Find f′(x)f'(x)f′(x) and f′(6)f'(6)f′(6) for f(x)=2ln⁡(x)f(x)=2 \ln (x)f(x)=2ln(x). Also, find f′′(x)f''(x)f′′(x) and f′′(6)f''(6)f′′(6).

Problem 8016

Find the area between the velocity-time graph and the time axis from t1=7.2st_{1} = 7.2st1​=7.2s to t2=14.4st_{2} = 14.4st2​=14.4s for displacement.

Problem 8017

Find the limit: lim⁡x→07x4+5x+123x4+4\lim _{x \rightarrow 0} \frac{7 x^{4}+5 x+12}{3 x^{4}+4}limx→0​3x4+47x4+5x+12​.

Problem 8018

Find the growth rate of a snake at 3 weeks using the model L(t)=19(1−(1+0.553t+0.014t2)−1)L(t)=19\left(1-\left(1+0.553 t+0.014 t^{2}\right)^{-1}\right)L(t)=19(1−(1+0.553t+0.014t2)−1). Round to 2 decimal places.

Problem 8019

Find the second derivative y′′y^{\prime \prime}y′′ for y=csc⁡x2y=\frac{\csc x}{2}y=2cscx​.

Problem 8020

Write the sigma notation for the sum of f(x)=x2f(x)=x^{2}f(x)=x2 from x=1x=1x=1 to x=2x=2x=2 with n=30n=30n=30.

Problem 8021

Calculate the limit: lim⁡x→0cos⁡x+3ex2ex\lim _{x \rightarrow 0} \frac{\cos x + 3 e^{x}}{2 e^{x}}limx→0​2excosx+3ex​.

Problem 8022

Find the derivative y′y^{\prime}y′ of the function y=ln⁡x2−4x−7y=\ln \sqrt{x^{2}-4 x-7}y=lnx2−4x−7​.

Problem 8023

Find the derivative h′(x)h^{\prime}(x)h′(x) for h(x)=ln⁡x(x−1)3x−2h(x)=\ln \frac{x(x-1)^{3}}{\sqrt{x-2}}h(x)=lnx−2​x(x−1)3​. No need to simplify.

Problem 8024

Find the derivative y′y^{\prime}y′ of the function y=35xy=3^{5x}y=35x.

Problem 8025

A company's revenue is R(q)=−q3+230q2R(q)=-q^{3}+230 q^{2}R(q)=−q3+230q2 and cost is C(q)=480+10qC(q)=480+10 qC(q)=480+10q. A) Find the marginal profit function MP(q)M P(q)MP(q). B) Determine the number of hundreds of units to sell for maximum profit.

Problem 8026

Find the limit: lim⁡x→π2sin⁡x−1cos⁡2x\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin x-1}{\cos ^{2} x}limx→2π​​cos2xsinx−1​.

Problem 8027

Find how many positive values of bbb make lim⁡x→bf(x)=2\lim _{x \rightarrow b} f(x)=2limx→b​f(x)=2 for f(x)=0.1x4−0.5x3−3.3x2+7.7x−1.99f(x)=0.1 x^{4}-0.5 x^{3}-3.3 x^{2}+7.7 x-1.99f(x)=0.1x4−0.5x3−3.3x2+7.7x−1.99.

Problem 8028

1. In a piecewise hazard rate model with intervals [τi−1,τi)[\tau_{i-1}, \tau_{i})[τi−1​,τi​), find the survival function and mean residual-life function.

Problem 8029

Find the tangent line equation at x=1x=1x=1 for y=x4−3x2−5x−4y=x^{4}-3x^{2}-5x-4y=x4−3x2−5x−4 and graph both.

Problem 8030

Find values of xxx where the tangent line of f(x)=x3−9x2+24x+1f(x)=\sqrt{x^{3}-9x^{2}+24x+1}f(x)=x3−9x2+24x+1​ is horizontal.

Problem 8031

Analyze the sequence an=en4na_{n}=\frac{e^{n}}{4^{n}}an​=4nen​ for convergence or divergence and find the limit if it converges.

Problem 8032

Find the number of apartments filled, xxx, that maximizes revenue given by R(x)=600x−0.5x2R(x)=600x-0.5x^{2}R(x)=600x−0.5x2.

Problem 8033

Find f′(x)f^{\prime}(x)f′(x) and where the tangent line of f(x)=x(2x−5)3f(x)=\frac{x}{(2 x-5)^{3}}f(x)=(2x−5)3x​ is horizontal.

Problem 8034

Problem 8035

Which statements about the graph of f(x)=27x2+2x−2f(x)=27 x^{2}+\frac{2}{x}-2f(x)=27x2+x2​−2 are true?

Problem 8036

Find the tangent line equation at x=2x=2x=2 for y=x4−3x3+2x+4y=x^{4}-3x^{3}+2x+4y=x4−3x3+2x+4. Graph both on the same axes.

Problem 8037

Find the number of boats xxx that minimizes the cost C(x)=17000−40x+0.04x2C(x)=17000-40 x+0.04 x^{2}C(x)=17000−40x+0.04x2.

Problem 8038

Identify the FALSE statement about the function f(x)=3x(x−5)2f(x)=\frac{3 x}{(x-5)^{2}}f(x)=(x−5)23x​ and its derivatives.

Problem 8039

Find values of xxx where the tangent line of f(x)=x3−9x2+24x+3f(x)=\sqrt{x^{3}-9 x^{2}+24 x+3}f(x)=x3−9x2+24x+3​ is horizontal.

Problem 8040

Check if the graph of f(x)=3xx−63f(x)=3 x \sqrt[3]{x-6}f(x)=3x3x−6​ has a vertical tangent or cusp at x=6x=6x=6. Options: cusp, tangent, neither, both.

Find the tangent line equation for $f(x)=x^{3}-3x+1$ at the point $(4,53)$ . What is $y=$ ?

Find the slope and equation of the tangent line to the curve $f(x)=x-x^{3}$ at the point $(1,0)$ . $y=$

Find the tangent line equation at $(1,-1)$ for $f(x)=5x^3-7x^2+1$ . Use $y - y_1 = m(x - x_1)$ .

Find the derivative $f^{\prime}(a)$ for the function $f(t)=\frac{4t+8}{t+5}$ .

Find intervals of increase and decrease for $F(x)=x \sqrt{6-x}$ . Also, find local min and max values.

Find the tangent line equation for $f(x)=\sqrt{x}$ at the point $(1,1)$ . What is $y=$ ?

Find the derivative $f^{\prime}(9)$ for the function $f(x)=10 x^{\ln (x)}$ .

Find the derivative $f^{\prime}(a)$ for the function $f(x)=\sqrt{3-6x}$ .

Find the tangent line equation at $(2,20)$ for $f(x)=5 x^{3}-7 x^{2}+8$ . Use $y - y_1 = m(x - x_1)$ .

A ball is thrown with a velocity of $34 \mathrm{ft} / \mathrm{s}$ . Its height is $s(t)=34t-16t^{2}$ . Find the velocity at $t=1 \mathrm{s}$ .

A projectile is launched at 64 ft/s from a 45-ft platform. Find its maximum height. Use $h(t) = -16t^2 + 64t + 45$ .

Evaluate the integral: $\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}$ . What is the result?

Given the cost function $C(x)=44100+400 x+x^{2}$ , find: a) Cost at $x=1550$ b) Average cost at $x=1550$ c) Marginal cost at $x=1550$ d) Production level minimizing average cost e) Minimal average cost

Find $f'(x)$ and $f'(6)$ for $f(x)=2 \ln (x)$ . Also, find $f''(x)$ and $f''(6)$ .

Find the area between the velocity-time graph and the time axis from $t_{1} = 7.2s$ to $t_{2} = 14.4s$ for displacement.

Find the limit: $\lim _{x \rightarrow 0} \frac{7 x^{4}+5 x+12}{3 x^{4}+4}$ .

Find the growth rate of a snake at 3 weeks using the model $L(t)=19\left(1-\left(1+0.553 t+0.014 t^{2}\right)^{-1}\right)$ . Round to 2 decimal places.

Find the second derivative $y^{\prime \prime}$ for $y=\frac{\csc x}{2}$ .

Write the sigma notation for the sum of $f(x)=x^{2}$ from $x=1$ to $x=2$ with $n=30$ .

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\cos x + 3 e^{x}}{2 e^{x}}$ .

Find the derivative $y^{\prime}$ of the function $y=\ln \sqrt{x^{2}-4 x-7}$ .

Find the derivative $h^{\prime}(x)$ for $h(x)=\ln \frac{x(x-1)^{3}}{\sqrt{x-2}}$ . No need to simplify.

Find the derivative $y^{\prime}$ of the function $y=3^{5x}$ .

A company's revenue is $R(q)=-q^{3}+230 q^{2}$ and cost is $C(q)=480+10 q$ .
A) Find the marginal profit function $M P(q)$ . B) Determine the number of hundreds of units to sell for maximum profit.

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin x-1}{\cos ^{2} x}$ .

Find how many positive values of $b$ make $\lim _{x \rightarrow b} f(x)=2$ for $f(x)=0.1 x^{4}-0.5 x^{3}-3.3 x^{2}+7.7 x-1.99$ .

1. In a piecewise hazard rate model with intervals $[\tau_{i-1}, \tau_{i})$ , find the survival function and mean residual-life function.

Find the tangent line equation at $x=1$ for $y=x^{4}-3x^{2}-5x-4$ and graph both.

Find values of $x$ where the tangent line of $f(x)=\sqrt{x^{3}-9x^{2}+24x+1}$ is horizontal.

Analyze the sequence $a_{n}=\frac{e^{n}}{4^{n}}$ for convergence or divergence and find the limit if it converges.

Find the number of apartments filled, $x$ , that maximizes revenue given by $R(x)=600x-0.5x^{2}$ .

Find $f^{\prime}(x)$ and where the tangent line of $f(x)=\frac{x}{(2 x-5)^{3}}$ is horizontal.

Which statements about the graph of $f(x)=27 x^{2}+\frac{2}{x}-2$ are true?

Find the tangent line equation at $x=2$ for $y=x^{4}-3x^{3}+2x+4$ . Graph both on the same axes.

Find the number of boats $x$ that minimizes the cost $C(x)=17000-40 x+0.04 x^{2}$ .

Identify the FALSE statement about the function $f(x)=\frac{3 x}{(x-5)^{2}}$ and its derivatives.

Find values of $x$ where the tangent line of $f(x)=\sqrt{x^{3}-9 x^{2}+24 x+3}$ is horizontal.

Check if the graph of $f(x)=3 x \sqrt[3]{x-6}$ has a vertical tangent or cusp at $x=6$ . Options: cusp, tangent, neither, both.

Consider the function $f(x)=(x^{2}-4x)^{\frac{1}{5}}$ . Which statement about $f(x)$ is FALSE?

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

Find the derivative of $y=\ln \cos(2x)$ .

A golfer hits a ball at $42 \mathrm{~m/s}$ and $32^{\circ}$ . Find the maximum height reached, ignoring air resistance.

Differentiate $g(t)=\ln \left(\frac{t(t^{2}+1)^{4}}{\sqrt[7]{6 t-1}}\right)$ .

Identify the FALSE statement about the function $f(x)=\frac{2}{x^{2}-9}$ and its derivatives.

Identify the FALSE statement about the function $f(x)=\left(5 x^{2}-2 x\right)^{\frac{1}{3}}$ .

Find the population change in 2013 for the model $P(t)=200(1.052)^{t}$ (in thousands). Choices are given.

Check if the graph of $f(x)=2(x-3)^{4/5}$ has a vertical tangent or cusp at $x=3$ . Options: neither, vertical cusp, both, vertical tangent.

Which statements about the graph of $f(x)=x+\sin(2x)+4$ on $[0, \pi]$ are true?

Evaluate $f(x)=3 x^{2}+12 x+5$ at the endpoints $-7$ and $3$ . Does Rolle's Theorem apply? Find $c$ or enter "DNE".

A kicker must kick a ball from $34 \mathrm{~m}$ to clear a $3.05 \mathrm{~m}$ crossbar. The ball's speed is $20 \mathrm{~m/s}$ at $42.2^{\circ}$ . How much higher does it go than the crossbar? Answer in $\mathrm{m}$ .

Calculate the derivative value $f^{\prime}(1)=(-1+2)e^{3(1)^{2}}$ .

A pendulum of mass $m$ is released from horizontal. Find speed, tension, max height, and largest $r$ before breaking.

Evaluate $f(x)=\cos(3 \pi x)$ at $x=-\frac{5}{3}$ and $x=-1$ . Does Rolle's Theorem apply? Find $c$ where $f'(c)=0$ .

Find all numbers that satisfy the mean value theorem for the function $f(x)=7x-\frac{4}{x}$ on the interval $[3,10]$ .

Differentiate the following with respect to $x$ :
1. $\frac{\sin 2 x}{2 x+5}$
2. $\frac{(3 x+1) \cos 2 x}{e^{2 x}}$
3. $\tan x$

Which statements about the graph of $f(x)=x+\sin (2 x)+4$ on $[0, \pi]$ are true?

Evaluate the integral $\int_{0}^{1} \frac{\ln (x+1)}{x^{2}+1} dx$ .

Find the slope of the secant line for $f(x)=-4 x^{2}-5 x+6$ on $[-3,-1]$ . Then find $c$ in $(-3,-1)$ where $m=f^{\prime}(c)$ .

Calculate the area under the curve of $3x^2 - x + 7$ from $0$ to $1$ using Riemann sums: $\int_{0}^{1}(3x^2 - x + 7)dx$ .

Find the value(s) of $c$ that satisfy the Mean Value Theorem for $f(x)=\ln(x^{7})$ on the interval $[1,7]$ .

Find the derivatives of these functions: (1) $y=e^{\sin 2 x}$ (2) $y=\sin^{2} x$ (3) $y=\ln \cos 3 x$ (4) $y=\cos^{3}(3 x)$ (5) $y=\log_{10}(2 x-1)$ .

Find the slope of the secant line for $f(x)=2 x^{3}-5 x$ on $[1,4]$ . Then, find $c$ in $(1,4)$ where $m=f^{\prime}(c)$ .

Find the slope of the secant line for $f(x)=-3 x^{2}-5 x+1$ on $[-3,1]$ . Then, find $c$ in $(-3,1)$ where $m=f^{\prime}(c)$ .

How much faster is a person on an $18.6 \mathrm{~m}$ building than on the ground? Use a day period and Earth's radius $6400 \mathrm{~km}$ .

Find the mean slope of $f(x)=3-2x^{2}$ on $[-5,4]$ and the value of $c$ where $f'(c)$ equals this slope.

Find the derivative of the function $f(x)=2 \tan ^{-1}(6 \sin (2 x))$ , i.e., compute $f^{\prime}(x)$ .

Find the average slope of the function $f(x)=5 x^{3}-4 x$ on $[-2,2]$ and determine the two values of $c$ where $f^{\prime}(c)$ equals this slope.

Find the limit: $\lim _{h \rightarrow 0} \frac{(16+h)^{\frac{1}{4}}-2}{h}$ . What is its value?

Find the derivative of the function $f(x)=6 \tan ^{-1}(8 \sin (3 x))$ , denoted as $f^{\prime}(x)$ .