LemonmathSelf-study sheet

Grade 12 · Lesson 7.7

Revision: trigonometry, exponentials, logarithms and sequences

Name: Date: Time spent: min

Goals

Use the trigonometric identities correctly, solve basic trigonometric equations and count solutions in an interval.
Manipulate powers and logarithms; solve exponential and logarithmic equations and inequalities with their conditions.
Set up and solve exponential growth and decay models (compound interest, bacteria, a drug in the blood).
Use the formulas for arithmetic and geometric sequences, the sum to infinity and basic limits.

Exercises

Exercise 1 Multiple choice

Given $\sin x=\dfrac13$ , the value of $\cos2x$ is

A. $\dfrac79$ . B. $\dfrac13$ . C. $-\dfrac79$ . D. $\dfrac89$ .

Exercise 2 Multiple choice

The solutions of the equation $\tan x=\sqrt3$ are

A. $x=\dfrac{\pi}{3}+k\pi$ . B. $x=\dfrac{\pi}{3}+k2\pi$ . C. $x=\dfrac{\pi}{6}+k\pi$ . D. $x=\pm\dfrac{\pi}{3}+k\pi$ ( $k\in\mathbb{Z}$ ).

Exercise 3 Multiple choice

Let $a=\log_23$ . Then $\log_224$ equals

A. $3a$ . B. $3+a$ . C. $8+a$ . D. $1+3a$ .

Exercise 4 True/false

The geometric sequence $(u_n)$ has $u_1=3$ and common ratio $q=2$ .

a) $u_5=48$ .

b) The sum of the first $5$ terms is $93$ .

c) $u_n=3\cdot2^n$ for every $n\ge1$ .

d) The number $384$ is the $7$ th term of the sequence.

Exercise 5 Short answer

Find the sum of the solutions of the equation $4^x-6\cdot2^x+8=0$ .

Exercise 6 Short answer

How many integer solutions does the inequality $\log_{\frac12}(x-2)\ge-2$ have?

Exercise 7 Short answer

Your friend An deposits $100$ million đồng in a bank at $6\%$ interest per year, with the interest added to the balance every year. After at least how many years is the balance $150$ million đồng or more?

Exercise 8 Short answer

A culture starts with $1000$ bacteria, and the number doubles every $3$ hours, so after $t$ hours there are $N(t)=1000\cdot2^{\frac t3}$ bacteria. After at least how many whole hours does the number exceed $20\,000$ ?

Exercise 9 Written answer

Solve the equation $\sin2x=\cos x$ on $[0, \pi]$ .

Exercise 10 Short answer (challenge)

A ball is dropped from a height of $10$ m onto the floor. Each time it hits the floor, it bounces back up to $\dfrac34$ of its previous height. Assuming it bounces like this for ever, find the total distance travelled by the ball (in m).

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. The period of the function

y=\sin2x

2\pi

.B.

\pi

.C.

\dfrac{\pi}{2}

.D.

4\pi

Question 2. The solution of the equation

\log_3(2x-1)=2

x=\dfrac92

.B.

x=\dfrac72

.C.

x=5

.D.

x=\dfrac32

Question 3. The arithmetic sequence

(u_n)

has

u_1=2

and common difference

d=3

. The term

u_{10}

32

.B.

26

.C.

30

.D.

29

Question 4. The value of

\lim\dfrac{3n-1}{n+2}

3

.B.

-\dfrac12

.C.

1

.D.

+\infty

Part II. True or false

Question 1. Consider the function

y=\log_2x

a) The domain of the function is $(0, +\infty)$ .	T / F
b) The function is increasing on its domain.	T / F
c) The graph passes through the point $(0, 1)$ .	T / F
d) $\log_2x<3\iff x<8$ .	T / F

Part III. Short answer

Question 1. Find the sum to infinity

S=6+2+\dfrac23+\dfrac29+\cdots

Question 2. How many solutions does the equation

\sin x=\dfrac12

have in

[0, 3\pi]

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $\cos2x=1-2\sin^2x$ .

Steps:

Find $\sin^2x$ .
Substitute into the identity.

Exercise 2.

What you need: $\tan x=\tan\alpha\iff x=\alpha+k\pi$ ; $\tan\dfrac{\pi}{3}=\sqrt3$ .

Steps:

Find $\alpha$ with $\tan\alpha=\sqrt3$ .
Write the general solution with period $\pi$ .

Exercise 3.

What you need: $\log_2(bc)=\log_2b+\log_2c$ .

Steps:

Write $24=8\cdot3$ .
Use $\log_28=3$ .

Exercise 4.

What you need: $u_n=u_1q^{n-1}$ ; $S_n=\dfrac{u_1(1-q^n)}{1-q}$ .

Steps:

Compute $u_5$ and $S_5$ .
Solve $3\cdot2^{n-1}=384$ .

Exercise 5.

What you need: $4^x=(2^x)^2$ ; put $t=2^x>0$ .

Steps:

Solve the quadratic equation in $t$ .
Find $x$ from $2^x=t$ .

Exercise 6.

What you need: the base $\dfrac12<1$ , so the direction reverses; the condition is $x-2>0$ .

Steps:

Write $-2=\log_{\frac12}4$ .
Solve $0<x-2\le4$ .

Exercise 7.

What you need: $A=A_0(1+r)^n$ ; $n$ is a positive integer.

Steps:

Solve $100\cdot1.06^n\ge150$ .
Take the least integer that works.

Exercise 8.

What you need: $2^{\frac t3}>20\iff\dfrac t3>\log_220$ .

Steps:

Solve the exponential inequality.
Take the least integer that works, then check.

Exercise 9.

What you need: $\sin2x=2\sin x\cos x$ ; turn it into a product equal to zero.

Steps:

Write $\cos x(2\sin x-1)=0$ .
Solve each factor and keep the solutions in $[0, \pi]$ .

Exercise 10.

What you need: the sum to infinity $S=\dfrac{u_1}{1-q}$ .

Steps:

The distance is the first drop of $10$ m, then for each bounce one trip up and an equal trip down.
The bounce heights $7.5$ , $7.5\cdot\dfrac34$ , … form a convergent geometric series.

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