Exam 12: Limits, Derivatives, and Definite Integrals

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow \infty} f(x)$ $find the value of the given limit if it exists. If the function approaches \infty or -\infty , say so. - \lim _{x \rightarrow \infty} f(x)$

(Short Answer)

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Question 41

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow \infty} \frac{1+5 x}{3 x+9}$

(Short Answer)

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Question 42

$\lim _{x \rightarrow 0^{+}} \sqrt{2 x}$

(Short Answer)

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Question 43

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 3^{+}} f(x)$ where $f(x)= \begin{cases}x^{2}-2 x & \text { if } x<3 \\ 9-3 x & \text { if } x \geq 3\end{cases}$

(Short Answer)

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Question 44

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow-4} \frac{x+4}{x^{2}+4 x}$

(Short Answer)

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Question 45

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow-\infty} \frac{8 x-5}{9 x+5}$

(Short Answer)

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Question 46

$\lim _{x \rightarrow 2^{-}} \frac{|x-2|}{x-2}$

(Short Answer)

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Question 47

Approximate the area under the graph of $f(x)=(x-4)^{2}+2$ from $x=0$ to $x=8$ and above the $x$ -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

(Short Answer)

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Question 48

$\lim _{x \rightarrow 2^{-}}\left(3 x^{2}+2 x\right)$

(Short Answer)

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Question 49

$\lim _{x \rightarrow 3^{+}} \sqrt{3-x}$

(Short Answer)

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Question 50

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 3^{+}} f(x)$ $find the value of the given limit if it exists. If the function approaches \infty or -\infty , say so. - \lim _{x \rightarrow 3^{+}} f(x)$

(Short Answer)

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Question 51

The population of bacteria in a culture follows the model $f(t)=6500\left(1-e^{-.35 t}\right)$ , where $f(t)$ is the number of bacteria after $t$ days. Use a calculator to graph $y=f(t)$ for $0 \leq t \leq 10$ , and determine $f^{\prime}(6)$ . What does $f^{\prime}(6)$ tell about the situation?

(Essay)

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Question 52

Five grams of a radioactive substance is placed in a vault. The function defined by $f(t)$ shown in the graph gives the amount remaining after $t$ years. At what rate (in grams per year) is the substance disintegrating after 20 years?

(Short Answer)

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Question 53

The total number of people who have seen a newly released movie at a particular theater chain follows the model $f(t)=10,000\left(1-e^{-.3 t}\right)$ , where $f(t)$ is the number of people who have seen the movie after $t$ days. Use a calculator to graph $y=f(t)$ for $0 \leq t \leq 16$ , and determine $f^{\prime}(5)$ . What does $f^{\prime}(5)$ tell about the situation?

(Essay)

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Question 54

Approximate the area under the graph of $f(x)=(x+1)^{2}+1$ from $x=-2$ to $x=2$ and above the $x$ -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

(Short Answer)

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Question 55

Use a calculator to estimate $f^{\prime}(2)$ , where $f(x)=\frac{2-3 e^{x}}{x}$ .

(Short Answer)

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Question 56

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 3} \frac{x^{2}+x+4}{x^{2}-1}$

(Short Answer)

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Question 57

$\lim _{x \rightarrow 1^{+}} \frac{|1-x|}{1-x}$

(Short Answer)

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Question 58

The total number of people who have seen a newly released movie at a particular theater chain follows the model $f(t)=5000\left(1-e^{-.2 t}\right)$ , where $f(t)$ is the number of people who have seen the movie after $t$ days. Use a calculator to graph $y=f(t)$ for $0 \leq t \leq 20$ , and determine $f^{\prime}(5)$ . What does $f^{\prime}(5)$ tell about the situation?

(Essay)

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Question 59

Find $\int_{-3}^{4}(x+3) d x$ by using the formula for the area of a triangle.

(Short Answer)

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Question 60

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow \infty} f(x)$ $find the value of the given limit if it exists. If the function approaches \infty or -\infty , say so. - \lim _{x \rightarrow \infty} f(x)$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow \infty} \frac{1+5 x}{3 x+9}$

$\lim _{x \rightarrow 0^{+}} \sqrt{2 x}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 3^{+}} f(x)$ where $f(x)= \begin{cases}x^{2}-2 x & \text { if } x<3 \\ 9-3 x & \text { if } x \geq 3\end{cases}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow-4} \frac{x+4}{x^{2}+4 x}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow-\infty} \frac{8 x-5}{9 x+5}$

$\lim _{x \rightarrow 2^{-}} \frac{|x-2|}{x-2}$

Approximate the area under the graph of $f(x)=(x-4)^{2}+2$ from $x=0$ to $x=8$ and above the $x$ -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

$\lim _{x \rightarrow 2^{-}}\left(3 x^{2}+2 x\right)$

$\lim _{x \rightarrow 3^{+}} \sqrt{3-x}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 3^{+}} f(x)$ $find the value of the given limit if it exists. If the function approaches \infty or -\infty , say so. - \lim _{x \rightarrow 3^{+}} f(x)$

The population of bacteria in a culture follows the model $f(t)=6500\left(1-e^{-.35 t}\right)$ , where $f(t)$ is the number of bacteria after $t$ days. Use a calculator to graph $y=f(t)$ for $0 \leq t \leq 10$ , and determine $f^{\prime}(6)$ . What does $f^{\prime}(6)$ tell about the situation?

Five grams of a radioactive substance is placed in a vault. The function defined by $f(t)$ shown in the graph gives the amount remaining after $t$ years. At what rate (in grams per year) is the substance disintegrating after 20 years?

Approximate the area under the graph of $f(x)=(x+1)^{2}+1$ from $x=-2$ to $x=2$ and above the $x$ -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

Use a calculator to estimate $f^{\prime}(2)$ , where $f(x)=\frac{2-3 e^{x}}{x}$ .

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 3} \frac{x^{2}+x+4}{x^{2}-1}$

$\lim _{x \rightarrow 1^{+}} \frac{|1-x|}{1-x}$

Find $\int_{-3}^{4}(x+3) d x$ by using the formula for the area of a triangle.

Linear Functions, Equations, and Inequalities

Analysis of Graphs of Functions

Polynomial Functions

Rational, Power, and Root Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Analytic Geometry and Nonlinear Systems

The Unit Circle and Functions of Trigonometry

Trigonometric Identities and Equations

Applications of Trigonometry and Vectors

Further Topics in Algebra

Reference: Basic Algebraic Concepts

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Exam 12: Limits, Derivatives, and Definite Integrals

find the value of the given limit if it exists. If the function approaches ∞\infty∞ or −∞-\infty−∞ , say so. - lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x)limx→∞​f(x)

find the value of the given limit if it exists. If the function approaches ∞\infty∞ or −∞-\infty−∞ , say so. - lim⁡x→∞1+5x3x+9\lim _{x \rightarrow \infty} \frac{1+5 x}{3 x+9}limx→∞​3x+91+5x​

lim⁡x→0+2x\lim _{x \rightarrow 0^{+}} \sqrt{2 x}limx→0+​2x​

find the value of the given limit if it exists. If the function approaches ∞\infty∞ or −∞-\infty−∞ , say so. - lim⁡x→−4x+4x2+4x\lim _{x \rightarrow-4} \frac{x+4}{x^{2}+4 x}limx→−4​x2+4xx+4​

find the value of the given limit if it exists. If the function approaches ∞\infty∞ or −∞-\infty−∞ , say so. - lim⁡x→−∞8x−59x+5\lim _{x \rightarrow-\infty} \frac{8 x-5}{9 x+5}limx→−∞​9x+58x−5​

lim⁡x→2−∣x−2∣x−2\lim _{x \rightarrow 2^{-}} \frac{|x-2|}{x-2}limx→2−​x−2∣x−2∣​

Approximate the area under the graph of f(x)=(x−4)2+2f(x)=(x-4)^{2}+2f(x)=(x−4)2+2 from x=0x=0x=0 to x=8x=8x=8 and above the xxx -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

lim⁡x→2−(3x2+2x)\lim _{x \rightarrow 2^{-}}\left(3 x^{2}+2 x\right)limx→2−​(3x2+2x)

lim⁡x→3+3−x\lim _{x \rightarrow 3^{+}} \sqrt{3-x}limx→3+​3−x​

find the value of the given limit if it exists. If the function approaches ∞\infty∞ or −∞-\infty−∞ , say so. - lim⁡x→3+f(x)\lim _{x \rightarrow 3^{+}} f(x)limx→3+​f(x)

Five grams of a radioactive substance is placed in a vault. The function defined by f(t)f(t)f(t) shown in the graph gives the amount remaining after ttt years. At what rate (in grams per year) is the substance disintegrating after 20 years?

Approximate the area under the graph of f(x)=(x+1)2+1f(x)=(x+1)^{2}+1f(x)=(x+1)2+1 from x=−2x=-2x=−2 to x=2x=2x=2 and above the xxx -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

Use a calculator to estimate f′(2)f^{\prime}(2)f′(2) , where f(x)=2−3exxf(x)=\frac{2-3 e^{x}}{x}f(x)=x2−3ex​ .

find the value of the given limit if it exists. If the function approaches ∞\infty∞ or −∞-\infty−∞ , say so. - lim⁡x→3x2+x+4x2−1\lim _{x \rightarrow 3} \frac{x^{2}+x+4}{x^{2}-1}limx→3​x2−1x2+x+4​

lim⁡x→1+∣1−x∣1−x\lim _{x \rightarrow 1^{+}} \frac{|1-x|}{1-x}limx→1+​1−x∣1−x∣​

Find ∫−34(x+3)dx\int_{-3}^{4}(x+3) d x∫−34​(x+3)dx by using the formula for the area of a triangle.

Linear Functions, Equations, and Inequalities

Analysis of Graphs of Functions

Polynomial Functions

Rational, Power, and Root Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Analytic Geometry and Nonlinear Systems

The Unit Circle and Functions of Trigonometry

Trigonometric Identities and Equations

Applications of Trigonometry and Vectors

Further Topics in Algebra

Reference: Basic Algebraic Concepts

Filters

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow \infty} f(x)$ $find the value of the given limit if it exists. If the function approaches \infty or -\infty , say so. - \lim _{x \rightarrow \infty} f(x)$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow \infty} \frac{1+5 x}{3 x+9}$

$\lim _{x \rightarrow 0^{+}} \sqrt{2 x}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow-4} \frac{x+4}{x^{2}+4 x}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow-\infty} \frac{8 x-5}{9 x+5}$

$\lim _{x \rightarrow 2^{-}} \frac{|x-2|}{x-2}$

Approximate the area under the graph of $f(x)=(x-4)^{2}+2$ from $x=0$ to $x=8$ and above the $x$ -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

$\lim _{x \rightarrow 2^{-}}\left(3 x^{2}+2 x\right)$

$\lim _{x \rightarrow 3^{+}} \sqrt{3-x}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 3^{+}} f(x)$ $find the value of the given limit if it exists. If the function approaches \infty or -\infty , say so. - \lim _{x \rightarrow 3^{+}} f(x)$

Five grams of a radioactive substance is placed in a vault. The function defined by $f(t)$ shown in the graph gives the amount remaining after $t$ years. At what rate (in grams per year) is the substance disintegrating after 20 years?

Approximate the area under the graph of $f(x)=(x+1)^{2}+1$ from $x=-2$ to $x=2$ and above the $x$ -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

Use a calculator to estimate $f^{\prime}(2)$ , where $f(x)=\frac{2-3 e^{x}}{x}$ .

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 3} \frac{x^{2}+x+4}{x^{2}-1}$

$\lim _{x \rightarrow 1^{+}} \frac{|1-x|}{1-x}$

Find $\int_{-3}^{4}(x+3) d x$ by using the formula for the area of a triangle.