Stewart Precalc 6e Chapter 13 Review: Concept Check

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Stewart Precalc 6e Chapter 13 Review: Concept Check 0/48
1 Concept - Definition of Limit · Level 1
Explain in your own words what is meant by the equation \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x) = 5\) Is it possible for this statement to be true and yet \(f(2) = 3\)? Explain.
2 Concept - One-Sided Limits · Level 1
Explain what it means to say that \(\operatorname*{lim}\limits_{x \rightarrow 1^-} f(x) = 3\) and \(\operatorname*{lim}\limits_{x \rightarrow 1^+} f(x) = 7\) In this situation is it possible that \(\operatorname*{lim}\limits_{x \rightarrow 1} f(x)\) exists? Explain.
3 Concept - Limit Failures · Level 1
Describe several ways in which a limit can fail to exist. Illustrate with sketches.
4 Concept - Limit Laws · Level 1
State the following Limit Laws.
(a) Sum Law
(b) Difference Law
(c) Constant Multiple Law
(d) Product Law
(e) Quotient Law
(f) Power Law (g) Root Law

Enter your answer directly below each part above.

5 Concept - Slope of Tangent Line · Level 1
Write an expression for the slope of the tangent line to the curve \(y = f(x)\) at the point \((a, f(a))\).
6 Concept - Derivative · Level 1
Define the derivative \(f'(a)\). Discuss two ways of interpreting this number.
7 Concept - Rate of Change · Level 1
If \(y = f(x)\), write expressions for the following.
(a) The average rate of change of \(y\) with respect to \(x\) between the numbers \(a\) and \(x\).
(b) The instantaneous rate of change of \(y\) with respect to \(x\) at \(x = a\).

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8 Concept - Limit at Infinity · Level 1
Explain the meaning of the equation \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 2\) Draw sketches to illustrate the various possibilities.
9 Concept - Horizontal Asymptote · Level 1
(a) What does it mean to say that the line \(y = L\) is a horizontal asymptote of the curve \(y = f(x)\)? Draw curves to illustrate the various possibilities.
(b) Which of the following curves have horizontal asymptotes? (i) \(y = x^2\) (ii) \(y = \dfrac{1}{x}\) (iii) \(y = \sin x\) (iv) \(y = \arctan x\) (v) \(y = e^x\)

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10 Concept - Convergent Sequence · Level 1
(a) What is a convergent sequence?
(b) What does \(\operatorname*{lim}\limits_{n \rightarrow \infty} a_n = 3\) mean?

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11 Concept - Area Under Graph · Level 1
Suppose \(S\) is the region that lies under the graph of \(y = f(x)\), \(a \leq x \leq b\).
(a) Explain how this area is approximated using rectangles.
(b) Write an expression for the area of \(S\) as a limit of sums.

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12 Limit - Factoring · Level 2
Use the Limit Laws to evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{x \rightarrow -2} \dfrac{x^2 - 4}{x^2 + x - 2}\)
13 Limit - Algebraic Simplification · Level 2
Use the Limit Laws to evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{u \rightarrow 0} ((u + 1)^2 - 1)/u\)
14 Limit - Rationalize · Level 2
Use the Limit Laws to evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{z \rightarrow 9} \dfrac{\sqrt{z} - 3}{z - 9}\)
15 Limit - One-Sided with Absolute Value · Level 2
Use the Limit Laws to evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{x \rightarrow 3^-} (x - 3)/|x - 3|\)
16 Limit - Algebraic Simplification · Level 2
Use the Limit Laws to evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{x \rightarrow 0} (\dfrac{1}{x} + 2/(x^2 - 2 x))\)
17 Limit - At Infinity · Level 2
Use the Limit Laws to evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{2 x}{x - 4}\)
18 Limit - At Infinity · Level 2
Use the Limit Laws to evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{x \rightarrow \infty} \dfrac{x^2 + 1}{x^4 - 3 x + 6}\)
19 Limit - Oscillating · Level 2
Use the Limit Laws to evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{x \rightarrow \infty} \cos^2 x\)
20 Limit - At Negative Infinity · Level 2
Use the Limit Laws to evaluate the limit, if it exists. \(\operatorname*{lim}\limits_{t \rightarrow -\infty} t^4/(t^3 - 1)\)
21 Derivative - At a Number · Level 2
Find the derivative of the function at the given number. \(f(x) = 3 x - 5\), at \(4\)
22 Derivative - At a Number · Level 2
Find the derivative of the function at the given number. \(g(x) = 2 x^2 - 1\), at \(-1\)
23 Derivative - At a Number · Level 2
Find the derivative of the function at the given number. \(f(x) = \sqrt{x}\), at \(16\)
24 Derivative - At a Number · Level 2
Find the derivative of the function at the given number. \(f(x) = x/(x + 1)\), at \(1\)
25 Derivative - General · Level 2
(a) Find \(f'(a)\). (b) Find \(f'(2)\) and \(f'(-2)\). \(f(x) = 6 - 2 x\)
26 Derivative - General · Level 2
(a) Find \(f'(a)\). (b) Find \(f'(2)\) and \(f'(-2)\). \(f(x) = x^2 - 3 x\)
27 Derivative - General · Level 3
(a) Find \(f'(a)\). (b) Find \(f'(2)\) and \(f'(-2)\). \(f(x) = \sqrt{x + 6}\)
28 Derivative - General · Level 3
(a) Find \(f'(a)\). (b) Find \(f'(2)\) and \(f'(-2)\). \(f(x) = \dfrac{4}{x}\)
29 Tangent Line - From Figure · Level 2
Find an equation of the tangent line shown in the figure.
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30 Tangent Line - From Figure · Level 2
Find an equation of the tangent line shown in the figure.
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31 Tangent Line - At a Point · Level 2
Find an equation of the line tangent to the graph of \(f\) at the given point. \(f(x) = 2 x\), at \((3, 6)\)
32 Tangent Line - At a Point · Level 2
Find an equation of the line tangent to the graph of \(f\) at the given point. \(f(x) = x^2 - 3\), at \((2, 1)\)
33 Tangent Line - At a Point · Level 2
Find an equation of the line tangent to the graph of \(f\) at the given point. \(f(x) = \dfrac{1}{x}\), at \(\left(2, \dfrac{1}{2}\right)\)
34 Tangent Line - At a Point · Level 2
Find an equation of the line tangent to the graph of \(f\) at the given point. \(f(x) = \sqrt{x + 1}\), at \((3, 2)\)
35 Application - Velocity · Level 3
A stone is dropped from the roof of a building 640 ft above the ground. Its height (in feet) after \(t\) seconds is given by \(h(t) = 640 - 16 t^2\).
(a) Find the velocity of the stone when \(t = 2\).
(b) Find the velocity of the stone when \(t = a\).
(c) At what time \(t\) will the stone hit the ground?
(d) With what velocity will the stone hit the ground?

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36 Application - Boyle's Law · Level 3
If a gas is confined in a fixed volume, then according to Boyle's Law the product of the pressure \(P\) and the temperature \(T\) is a constant. For a certain gas, \(P T = 100\), where \(P\) is measured in lb/in² and \(T\) is measured in kelvins (K).
(a) Express \(P\) as a function of \(T\).
(b) Find the instantaneous rate of change of \(P\) with respect to \(T\) when \(T = 300\) K.

Enter your answer directly below each part above.

37 Sequence - Limit · Level 2
If the sequence is convergent, find its limit. If it is divergent, explain why. \(a_n = n/(5 n + 1)\)
38 Sequence - Limit · Level 2
If the sequence is convergent, find its limit. If it is divergent, explain why. \(a_n = n^3/(n^3 + 1)\)
39 Sequence - Limit · Level 2
If the sequence is convergent, find its limit. If it is divergent, explain why. \(a_n = \dfrac{n(n + 1)}{2 n^2}\)
40 Sequence - Divergent · Level 2
If the sequence is convergent, find its limit. If it is divergent, explain why. \(a_n = n^3/(2 n + 6)\)
41 Sequence - Divergent · Level 2
If the sequence is convergent, find its limit. If it is divergent, explain why. \(a_n = \cos\left(n \dfrac{\pi}{2}\right)\)
42 Sequence - Limit · Level 2
If the sequence is convergent, find its limit. If it is divergent, explain why. \(a_n = 10/3^n\)
43 Approximate Area · Level 2
Approximate the area of the shaded region under the graph of the given function by using the indicated rectangles. (The rectangles have equal width.) \(f(x) = \sqrt{x}\)
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44 Approximate Area · Level 2
Approximate the area of the shaded region under the graph of the given function by using the indicated rectangles. (The rectangles have equal width.) \(f(x) = 4 x - x^2\)
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45 Area as Limit · Level 3
Use the limit definition of area to find the area of the region that lies under the graph of \(f\) over the given interval. \(f(x) = 2 x + 3\), \(0 \leq x \leq 2\)
46 Area as Limit · Level 3
Use the limit definition of area to find the area of the region that lies under the graph of \(f\) over the given interval. \(f(x) = x^2 + 1\), \(0 \leq x \leq 3\)
47 Area as Limit · Level 3
Use the limit definition of area to find the area of the region that lies under the graph of \(f\) over the given interval. \(f(x) = x^2 - x\), \(1 \leq x \leq 2\)
48 Area as Limit · Level 3
Use the limit definition of area to find the area of the region that lies under the graph of \(f\) over the given interval. \(f(x) = x^3\), \(1 \leq x \leq 2\)

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