DP Maths AA HL - Functions - Modeling with Functions

DP Maths AA HL - Functions - Modeling with Functions - Full notes

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DP Maths AA HL | Modeling with Functions

Modeling with Functions

This topic contains 1 note:

2.8 Rational, Exponential and Log Functions

DP MATHEMATICS AA · TOPIC 2: FUNCTIONS

2.8

Rational, Exponential and Logarithmic Functions

Full notes for HL · Syllabus SL 2.8, 2.9, 2.10

Contents

1	The reciprocal function
2	Rational functions of the form (ax + b)/(cx + d)
3	Exponential functions
4	The number e and natural growth
5	Logarithmic functions
6	Transformed exponential and logarithmic graphs
7	Solving equations analytically and graphically
8	Modelling with exponential functions
9	The reciprocal function and its transformations
10	Rational functions (ax + b)/(cx + d)
11	Exponential functions and their features
12	Logarithmic functions and their features
13	Modelling with exponential functions
14	Equations with exponentials and logarithms
15	Comparing growth rates
16	Mixed review: worked problems
17	Additional exam-style practice
18	Graphs of y = f(x) combined with exponentials
19	Exam-style questions with markscheme
20	Common misconceptions and links
21	Check your understanding
22	Summary and topic map
23	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the Quick check.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
sketch y = 1/x and recognize it as self-inverse
find asymptotes and intercepts of (ax + b)/(cx + d) and sketch its graph
sketch exponential functions ax and ex, identifying intercepts and asymptotes
explain the special role of e and write ax as ex ln a
sketch logarithmic functions as inverses of exponentials
find asymptotes, domains and ranges of transformed exponential and log functions
solve equations analytically and graphically, including with technology
model growth and decay, including half-life and Newton's law of cooling

This booklet covers SL 2.8, 2.9, 2.10, which is common to SL and HL. HL students extend this topic in booklet 2.13.

Guiding questions

How do asymptotes shape the graphs of rational, exponential and logarithmic functions? Why is e the natural base for growth?

How this topic is examined

These functions appear on both papers. Paper 1 asks for sketches with asymptotes and intercepts, domains and ranges, and exact solutions; Paper 2 asks for modelling questions and graphical solutions.

Always write asymptotes as equations (x = 2, y = 3), not just numbers, and label them on sketches with dashed lines.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working
Solve	find the values of the unknowns, using algebra or technology as allowed
Show that	obtain the given result; every step must be shown and the result must not be assumed
Hence / Hence or otherwise	use the previous result (or another valid method)
Sketch	a diagram showing the general shape and key features, labelled
Prove	a formal, rigorous argument from accepted facts
Justify / Explain	give valid reasons or evidence

How to use these notes

Each section teaches one part of the syllabus. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or proof of a key result, so you understand where formulas come from
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently on Paper 2
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given); examples are graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet
Quick check	short questions to check the key ideas

1 The reciprocal function

y = 1/x.

Think about it

If you double the speed of a journey, the time halves. If you triple it, the time is a third. What shape does a graph of time against speed have, and what happens at very low speeds?

The reciprocal function f(x) = 1/x, x ≠ 0, describes quantities that are inversely proportional. Its graph is a rectangular hyperbola in the first and third quadrants. As x → 0, |f(x)| grows without bound: the vertical asymptote is x = 0. As x → ±∞, f(x) → 0: the horizontal asymptote is y = 0.

1/x is self-inverse: f−1(x) = 1/x, so its graph is symmetric about y = x (and also about y = −x). It is an odd function: f(−x) = −f(x).

Figure 1. Reciprocal functions: y = k/x is a vertical stretch of y = 1/x.

2 Rational functions of the form (ax + b)/(cx + d)

Hyperbolas with shifted asymptotes.

For f(x) = (ax + b)/(cx + d), c ≠ 0:

Vertical asymptote: where the denominator is zero, x = −d/c (provided the numerator is not also zero there).

Horizontal asymptote: y = a/c. For large |x|, the constants b and d become negligible, so f(x) ≈ ax/(cx) = a/c.

Intercepts: x-intercept where the numerator is zero, x = −b/a; y-intercept f(0) = b/d.

Writing f(x) in the form a/c + k/(cx + d) by division shows it is a transformation of 1/x (2.11).

WORKED EXAMPLE 1 Sketching a rational function (standard)

For f(x) = (2x − 1)/(x + 3), find the asymptotes and intercepts, and sketch the graph.

Solution

Vertical asymptote: x = −3A1

Horizontal asymptote: y = 2A1

x-intercept (½, 0); y-intercept (0, −1/3)A1

Figure 2. Asymptotes x = −3 and y = 2.

WORKED EXAMPLE 2 Domain, range and inverse (exam level)

f(x) = (3x + 1)/(x − 2). State the domain and range of f, and find f−1(x).

Solution

Domain x ≠ 2; range f(x) ≠ 3A1 A1

x(y − 2) = 3y + 1, so y(x − 3) = 2x + 1M1

f−1(x) = (2x + 1)/(x − 3), x ≠ 3A1

The asymptotes of the inverse are swapped: x = 3 and y = 2.

Common mistake

Writing the horizontal asymptote of (2x − 1)/(x + 3) as y = 0. It is the ratio of the x coefficients, y = 2.

Practice exercise 1

1. Write down the asymptotes of y = 5/(x − 4).

2. Find the asymptotes and intercepts of y = (4x + 2)/(2x − 3).

3. Find the range of f(x) = (x + 5)/(x + 1).

4. Show that (3x + 7)/(x + 2) = 3 + 1/(x + 2).

3 Exponential functions

Growth by a constant factor.

An exponential function has the form f(x) = ax, where a > 0 and a ≠ 1. The variable is in the exponent. Equal steps in x multiply the output by the same factor.

If a > 1 the function shows exponential growth; if 0 < a < 1, exponential decay. All graphs y = ax pass through (0, 1), have the horizontal asymptote y = 0, domain x ∈ ℝ and range y > 0. Note that (1/a)x = a−x, the reflection of ax in the y-axis.

Exponential growth eventually outgrows any polynomial: 2x > x100 for large enough x.

Figure 3. Exponential growth and decay all pass through (0, 1).

Practice exercise 2

1. Write down the y-intercept and asymptote of y = 3x.

2. Sketch y = 2−x, stating its range.

3. Which is larger for x = 30: 2x or x5?

4 The number e and natural growth

The base for continuous growth.

Suppose $1 is invested at 100% interest. Compounded n times a year, it grows to (1 + 1/n)n after one year. As n increases this approaches a limit, the number e ≈ 2.718 28. So e arises naturally from continuous growth.

The natural exponential function ex has a remarkable property: its gradient at every point equals its value (5.6). For this reason it is the standard base in calculus and science. Any exponential ax can be written in base e, since a = eln a:

n	(1 + 1/n)n
1	2
10	2.5937
100	2.7048
10 000	2.7181
1 000 000	2.7183

Practice exercise 3

1. Write 5x in the form ekx.

2. Write e0.3x in the form ax.

5 Logarithmic functions

Inverses of exponentials.

The function f(x) = loga x is the inverse of ax. Its graph is the reflection of y = ax in y = x: it passes through (1, 0), has the vertical asymptote x = 0, domain x > 0 and range ℝ. The natural logarithm ln x is the inverse of ex.

Logarithmic functions grow very slowly: ln x only reaches 10 when x ≈ 22 026.

Figure 4. ln x is the reflection of ex in y = x.

6 Transformed exponential and logarithmic graphs

Track the asymptote.

Transformations (2.11) move asymptotes. For y = kex + c, the horizontal asymptote is y = c, and the y-intercept is k + c. For y = ln(x − h) + c, the vertical asymptote is x = h, the domain is x > h, and the x-intercept solves ln(x − h) = −c.

WORKED EXAMPLE 3 Transformed exponential (standard)

For f(x) = 3 − 2e−x, find the asymptote, intercepts and range.

Solution

As x → ∞, e−x → 0: asymptote y = 3A1

y-intercept (0, 1); x-intercept: e−x = 3/2, x = −ln 1.5 = −0.405A1

Range f(x) < 3A1

WORKED EXAMPLE 4 Transformed logarithm (standard)

For g(x) = ln(2x − 4) + 1, find the domain, asymptote and x-intercept.

Solution

2x − 4 > 0: domain x > 2, asymptote x = 2A1

ln(2x − 4) = −1: 2x − 4 = e−1M1

x = 2 + 1/(2e) ≈ 2.18A1

Practice exercise 4

1. State the asymptote and range of y = 4ex − 1.

2. State the domain and asymptote of y = ln(x + 3) − 2, and find its x-intercept.

3. Sketch y = 5 − e2x, labelling the asymptote and intercepts.

7 Solving equations analytically and graphically

Algebra where possible; technology otherwise.

Analytically, use logarithm laws and algebra (1.5). Graphically, find the intersection points of the two sides with the GDC. Some equations mix types, such as ex = 3 − x or ln x = x − 2, and can only be solved with technology.

WORKED EXAMPLE 5 Graphical solution (standard)

Solve ln x = x − 2.

Solution

Graph y = ln x and y = x − 2M1

x = 0.159 or x = 3.15 (3 s.f.)A1 A1

WORKED EXAMPLE 6 Rational equation (standard)

Solve (2x − 1)/(x + 3) = x − 1.

Solution

2x − 1 = (x − 1)(x + 3) = x2 + 2x − 3M1

x2 = 2A1

x = ±√2A1

GDC tip

When graphs approach each other slowly (for example a logarithm and a horizontal line), a standard window may miss an intersection. Think about end behaviour and extend the window if needed.

Practice exercise 5

1. Solve ex = 3 − x.

2. Solve 2x = x2 for x > 0.

3. Solve 3/(x − 1) = x + 1.

4. Solve ex = 5 − x2.

8 Modelling with exponential functions

Growth and decay models.

Exponential models have the form N(t) = N0ekt or N(t) = N0at, where N0 is the initial value, k > 0 for growth and k < 0 for decay. The half-life (time to halve) is ln 2 ÷ |k|; the doubling time for growth is ln 2 ÷ k.

Newton's law of cooling gives T(t) = Troom + (T0 − Troom)e−kt: the temperature approaches room temperature, a horizontal asymptote. Similar 'limited growth' models, such as P = A − Be−kt, describe learning curves and drug absorption.

WORKED EXAMPLE 7 Exponential decay (standard)

A drug's concentration is C(t) = 40e−0.15t mg L−1 after t hours. (a) Find the concentration after 6 hours. (b) Find the half-life.

Solution

(a) 40e−0.9 = 16.3 mg L−1A1

(b) e−0.15t = ½, t = ln 2 ÷ 0.15M1

= 4.62 hoursA1

WORKED EXAMPLE 8 Finding a model (exam level)

A population is 500 at t = 0 and 800 at t = 4. Assuming P = 500ekt, find k and predict P at t = 10.

Solution

800 = 500e4k, k = ln 1.6 ÷ 4 = 0.1175M1 A1

P(10) = 500e1.175 = 1619A1

WORKED EXAMPLE 9 Newton's law of cooling (exam level)

Coffee at 90 °C cools in a 20 °C room: T = 20 + 70e−0.05t, t in minutes. When does it reach 50 °C?

Solution

70e−0.05t = 30M1

t = ln(7/3) ÷ 0.05M1

= 16.9 minutesA1

Practice exercise 6

1. A sample of 200 g decays by 8% per year. Find the mass after 10 years and the half-life.

2. Bacteria double every 3 hours. Find k in N = N0ekt.

3. A car's value is V = 30 000e−0.18t. When is it worth $10 000?

4. T = 25 + 75e−kt. If T = 70 when t = 10, find k.

9 The reciprocal function and its transformations

y = a/(x − h) + k.

y = 1/x has asymptotes x = 0 and y = 0 and is self-inverse. Transformations give y = a/(x − h) + k with asymptotes x = h and y = k; a stretches the curve and, if negative, reflects it.

Figure 5. Transformations of y = 1/x.

WORKED EXAMPLE 10 Features (standard)

For y = 3/(x − 2) + 1, find the asymptotes and intercepts.

Solution

x = 2, y = 1A1

y-intercept (0, −0.5); x-intercept (−1, 0)A1 A1

Practice exercise 7

1. Find the asymptotes and intercepts of y = −2/(x + 1) + 4.

2. Write y = (2x + 5)/(x + 1) in the form a/(x + 1) + k.

10 Rational functions (ax + b)/(cx + d)

Asymptotes from coefficients.

For f(x) = (ax + b)/(cx + d): vertical asymptote x = −d/c; horizontal asymptote y = a/c; x-intercept −b/a; y-intercept b/d. Its inverse is (−dx + b)/(cx − a), with the asymptotes swapped.

WORKED EXAMPLE 11 Inverse (exam level)

Find the inverse of f(x) = (3x − 1)/(x + 2) and its asymptotes.

Solution

f−1(x) = (−2x − 1)/(x − 3)M1 A1

Asymptotes of f−1: x = 3, y = −2A1

Practice exercise 8

1. Find the range of f(x) = (4x + 1)/(2x − 3).

2. Find the intercepts of y = (x − 6)/(2x + 3).

11 Exponential functions and their features

Growth, decay and asymptotes.

f(x) = kax + c (a > 0, a ≠ 1) has horizontal asymptote y = c and y-intercept k + c. The base e gives f(x) = kerx + c, with r > 0 for growth and r < 0 for decay.

Figure 6. Exponential growth and decay with asymptotes.

WORKED EXAMPLE 12 Features (standard)

For f(x) = 5 − 2e0.3x, find the asymptote, intercepts and range.

Solution

Asymptote y = 5 (as x → −∞)A1

f(0) = 3; f(x) = 0 when e0.3x = 2.5: x = 3.05A1 A1

Range f(x) < 5A1

Practice exercise 9

1. Find the asymptote and y-intercept of y = 2(0.5)x − 3.

2. Find the x-intercept of y = 4 − ex.

12 Logarithmic functions and their features

Inverses of exponentials.

f(x) = loga x has vertical asymptote x = 0 and passes through (1, 0) and (a, 1). y = ln(x − h) + k has asymptote x = h. The domain is where the argument is positive.

WORKED EXAMPLE 13 Features (standard)

For y = 2 ln(x + 3) − 1, find the domain, asymptote and intercepts.

Solution

Domain x > −3; asymptote x = −3A1

x-intercept: ln(x + 3) = 0.5, x = e0.5 − 3 = −1.35A1

y-intercept 2 ln 3 − 1 = 1.20A1

Practice exercise 10

1. Find the inverse of y = ln(2x − 1).

2. State the domain of y = log(5 − x).

13 Modelling with exponential functions

From two points to a model.

To find a model y = Aekt from two data points, substitute both and divide to eliminate A. For y = Aekt + c, the asymptote c is often given by the context (for example, room temperature).

WORKED EXAMPLE 14 Model from data (exam level)

A population is 800 at t = 0 and 1250 at t = 5. Find P = Aekt and predict t = 12.

Solution

A = 800; e5k = 1.5625, k = 0.0893M1 A1

P(12) = 800e1.071 = 2335A1

WORKED EXAMPLE 15 Cooling (exam level)

T = 20 + Ae−kt. T(0) = 85, T(10) = 50. Find A, k and T(25).

Solution

A = 65; 30 = 65e−10k, k = 0.0773A1 A1

T(25) = 20 + 65e−1.933 = 29.4 °CA1

Practice exercise 11

1. A substance decays from 120 g to 90 g in 4 hours. Find its half-life.

2. A car's value V = 30 000e−kt halves in 5 years. Find k.

14 Equations with exponentials and logarithms

Isolate, then take logs.

Isolate the exponential term before taking logarithms. For equations mixing exponentials and polynomials, use technology.

WORKED EXAMPLE 16 Exact solution (standard)

Solve 4e2x − 3 = 17 exactly.

Solution

e2x = 5M1

x = ½ ln 5A1

WORKED EXAMPLE 17 Technology (standard)

Solve ex = x2 + 2.

Solution

Graph both sidesM1

x = 1.32A1

Practice exercise 12

1. Solve 3(2x) = 7x−1 exactly.

2. Solve ln(x + 1) = 2 − x.

15 Comparing growth rates

Exponential beats polynomial beats logarithm.

For large x, exponential functions (with base > 1) grow faster than any polynomial, which in turn grows faster than any logarithm. This explains why exponential models eventually predict huge values and must be used with care.

Figure 7. Logarithmic, polynomial and exponential growth.

WORKED EXAMPLE 18 Overtaking (standard)

Use technology to find where 1.2x = x2, and describe which function is larger for large x.

Solution

Intersections x = −0.920, 1.11 and 40.6M1 A1

For x > 40.6, 1.2x > x2 and the gap grows rapidlyR1

Practice exercise 13

1. Which is larger at x = 100: x3 or 1.1x?

16 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 19 Mixed problem (exam level)

f(x) = 2 + 3/(x − 1), g(x) = ex. (a) State the asymptotes of f. (b) Find (f ∘ g)(x) and its asymptotes.

Solution

(a) x = 1, y = 2A1

(b) 2 + 3/(ex − 1)A1

Vertical asymptote x = 0; horizontal y = 2 (x → ∞) and y = −1 (x → −∞)A1 A1

Practice exercise 14

1. Find the inverse of f(x) = 3 + e2x and state its domain.

2. Find the asymptotes of y = (ex + 2)/(ex − 1).

17 Additional exam-style practice

Paper 1 and Paper 2.

WORKED EXAMPLE 20 Paper 2 question (exam level)

The concentration of a drug is C(t) = 10te−0.4t. (a) Find the maximum concentration. (b) Find the time during which C > 5.

Solution

(a) GDC maximum at t = 2.5: C = 9.20A1 A1

(b) C = 5 at t = 0.648 and 6.36: 5.71 hoursM1 A1

Practice exercise 15

1. Find the range of f(x) = 2 − e−x for x ≥ 0.

2. Find where y = ln x and y = 1/x intersect.

18 Graphs of y = f(x) combined with exponentials

Products and quotients.

Functions such as xe−x, x2ex or ln x/x combine features: zeros of each factor, asymptotes from the exponential or logarithm, and a single maximum or minimum. Sketch them with technology and identify the key points.

Figure 8. Combined functions.

WORKED EXAMPLE 21 Features (standard)

Find the maximum of y = (ln x)/x and its asymptotes.

Solution

Maximum (e, 1/e)A1

Vertical asymptote x = 0; horizontal y = 0A1 A1

Practice exercise 16

1. Find the minimum of y = x2ex.

19 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 15 minutes)

f(x) = (3x + 2)/(x − 1), x ≠ 1; g(x) = ln(x + 2), x > −2.

(a) Write down the equations of the asymptotes of f. [2]

(b) Find the intercepts of f. [2]

(d) Find the x-intercept of g and state its range. [2]

Markscheme

Part	Answer	Marks
(a)	x = 1	A1
	y = 3	A1
(b)	(−2/3, 0)	A1
	(0, −2)	A1
(c)	x(y − 1) = 3y + 2	M1
	y(x − 3) = x + 2	A1
	f−1(x) = (x + 2)/(x − 3), x ≠ 3	A1
(d)	(−1, 0)	A1
	g(x) ∈ ℝ	A1

Examiner's comments. Asymptotes must be given as equations of lines.

Question 2 (Paper 2 style) (about 12 minutes)

A cup of tea is made at 95 °C in a room at 22 °C. Its temperature after t minutes is T(t) = 22 + 73e−kt. After 5 minutes it is 75 °C.

(a) Find k. [3]

(b) Find the temperature after 15 minutes. [2]

(d) State the long-term temperature, with a reason. [1]

Markscheme

Part	Answer	Marks
(a)	75 = 22 + 73e−5k	M1
	e−5k = 53/73	A1
	k = 0.0640	A1
(b)	T(15) = 22 + 73e−0.960	M1
	49.9 °C	A1
(c)	e−kt = 18/73	M1
	t = 21.9 minutes	A1
(d)	22 °C, since e−kt → 0	R1

Examiner's comments. Keeping k unrounded in later parts avoids accuracy errors.

20 Common misconceptions and links

Students often think…	Actually…
the horizontal asymptote of (ax + b)/(cx + d) is y = 0	y = a/c
ln x is defined for x ≤ 0	only x > 0
exponential graphs cross the x-axis	they approach their asymptote
e is just a large number	e is the base of continuous growth

How 2.8 connects to the rest of the course

Topic	Connection
1.5 Logarithms	laws and equations
2.5 Inverse functions	log as inverse of exp
2.13 Rational functions (HL)	oblique asymptotes
5.6 Differentiation	derivatives of ex and ln x

21 Check your understanding

Section A: multiple choice

1. The vertical asymptote of 5/(x − 4) is

A. x = 4 B. y = 4 C. x = 5 D. x = −4

2. The horizontal asymptote of (4x + 1)/(2x − 3) is

A. y = 4 B. y = 0 C. y = 2 D. y = −1/3

3. y = 3x passes through

A. (1, 0) B. (0, 1) C. (0, 3) D. (0, 0)

4. The domain of ln(x + 5) is

A. x > 5 B. x ≥ −5 C. x > 0 D. x > −5

5. The half-life for N = N0e−0.2t is

A. 0.2 B. ln 2/0.2 ≈ 3.47 C. 5 D. ln 0.2

Section B: structured

6. Sketch y = 2 + 3/(x + 1), labelling asymptotes and intercepts. [4]

7. Coffee cools: T = 20 + 70e−0.05t. When does it reach 50 °C? [3]

8. Explain why 1/x is self-inverse. [2]

Answers

1. A 2. C 3. B 4. D 5. B

6. Asymptotes x = −1, y = 2; intercepts (−2.5, 0), (0, 5).

7. 70e−0.05t = 30: t = 16.9 min.

8. Swapping x and y in y = 1/x gives y = 1/x; symmetric about y = x.

22 Summary and topic map

Idea	Remember
1/x	asymptotes x = 0, y = 0; self-inverse
(ax + b)/(cx + d)	x = −d/c, y = a/c
Exponential	through (0, 1); asymptote y = 0
e	limit of (1 + 1/n)n; ax = ex ln a
Logarithmic	inverse of exponential; asymptote x = 0
Transformations	kex + c: y = c; ln(x − h): x = h
Models	N0ekt; half-life ln 2/\|k\|; cooling

Before the exam, make sure you…

•write asymptotes as equations
•label intercepts
•use technology for non-algebraic equations
•interpret models in context

Rational, Exponential and Log Functions on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 9. Topic map for Rational, Exponential and Log Functions.

23 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. x = 4, y = 0

2. x = 1.5, y = 2; (−½, 0), (0, −2/3)

3. f(x) ≠ 1

4. 3(x + 2) + 1 = 3x + 7

Practice exercise 2

1. (0, 1); y = 0

2. decay through (0, 1); range y > 0

3. 230 ≈ 1.07 × 109 > 305 ≈ 2.43 × 107

Practice exercise 3

1. ex ln 5, k = 1.61

2. 1.35x

Practice exercise 4

1. y = −1; y > −1

2. x > −3; x = −3; x = e2 − 3 ≈ 4.39

3. asymptote y = 5; (0, 4); (ln 5/2, 0) ≈ (0.805, 0)

Practice exercise 5

1. x = 0.792

2. x = 2 or x = 4

3. x = ±2

4. x = −2.21 or x = 1.24

Practice exercise 6

1. 86.9 g; 8.31 years

2. ln 2/3 = 0.231

3. 6.10 years

4. k = ln(75/45)/10 = 0.0511

Practice exercise 7

1. x = −1, y = 4; (0, 2) and (−0.5, 0)

2. 2 + 3/(x + 1)

Practice exercise 8

1. y ≠ 2

2. (6, 0) and (0, −2)

Practice exercise 9

1. y = −3; (0, −1)

2. ln 4 = 1.39

Practice exercise 10

1. (ex + 1)/2

2. x < 5

Practice exercise 11

1. 9.64 hours

2. 0.139

Practice exercise 12

1. x ln 2 + ln 3 = (x − 1) ln 7: x = ln 21/ln 3.5 = 2.43

2. 1.21

Practice exercise 13

1. x3 = 106; 1.1100 = 13 781: x3 is larger, but 1.1x overtakes later (x ≈ 160)

Practice exercise 14

1. ½ ln(x − 3), x > 3

2. x = 0; y = 1 and y = −2

Practice exercise 15

1. 1 ≤ f(x) < 2

2. x = 1.76

Practice exercise 16

1. (0, 0) (and a local maximum at x = −2)

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