LemonmathSelf-study sheet

Grade 12 · Lesson 7.2

Revision: derivatives and graphs of functions

Name: Date: Time spent: min

Goals

Find where a function increases or decreases and find its turning points from $f(x)$ , from $f'(x)$ or from a table of variations.
Find the maximum and minimum values of a function on a closed interval and on an open interval.
Find the vertical, horizontal and oblique asymptotes of the graph of a rational function.
Read tables of variations and graphs; count the roots of $f(x)=k$ ; identify a function from its graph.

Exercises

Exercise 1 Multiple choice

How many turning points does $y=x^4-2x^2+1$ have? A. $3$ . B. $1$ . C. $2$ . D. $0$ .

Exercise 2 Multiple choice

The vertical and horizontal asymptotes of the graph of $y=\dfrac{3x-1}{x-2}$ are A. $x=2$ and $y=3$ . B. $x=-2$ and $y=3$ . C. $x=2$ and $y=\dfrac12$ . D. $x=3$ and $y=2$ .

Exercise 3 Short answer

Let $M$ , $m$ be the maximum and minimum values of $y=x+\dfrac4x$ on $[1, 5]$ . Find $M+m$ .

Exercise 4 Written answer

Find the asymptotes of the graph of $y=\dfrac{x^2+x+1}{x-1}$ .

Exercise 5 True/False

The function $y=f(x)$ has derivative $f'(x)=x(x-1)^2(x+2)$ for all $x\in\mathbb{R}$ .

a) The function has three turning points.

b) The function has a local minimum at $x=0$ .

c) The function is increasing on $(0, +\infty)$ .

d) $f(-1)>f(0)$ .

Exercise 6 Short answer

The concentration of a drug in the blood $t$ hours after it is taken is $C(t)=\dfrac{5t}{t^2+4}$ (mg/l), $t\ge0$ . What is the highest concentration, in mg/l?

Exercise 7 Multiple choice

The function $y=f(x)$ is continuous on $\mathbb{R}$ with this table of variations:

$x$	$-\infty$		$0$		$2$		$+\infty$
$f'(x)$		$+$	$0$	$-$	$0$	$+$
$f(x)$	$-\infty$	$\nearrow$	$1$	$\searrow$	$-3$	$\nearrow$	$+\infty$

The number of roots of $2f(x)+6=0$ is A. $1$ . B. $2$ . C. $3$ . D. $0$ .

Exercise 8 True/False

Let $y=\dfrac{x+1}{x-1}$ .

a) The domain of the function is $\mathbb{R}\setminus\{1\}$ .

b) The function is decreasing on $\mathbb{R}\setminus\{1\}$ .

c) The asymptotes of the graph meet at $I(1, 1)$ .

d) The maximum value of the function on $[2, 4]$ is $3$ .

Exercise 9 Short answer

Find the minimum value of $y=(x^2-3)e^x$ on $[0, 2]$ . Give your answer to 1 decimal place.

Exercise 10 Challenge

The graph of $y=ax^3+bx^2+cx+d$ has a local maximum at $(0, 2)$ and a local minimum at $(2, -2)$ . Find $a$ , $b$ , $c$ , $d$ .

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 function

y=f(x)

has derivative

f'(x)=(x-1)(x+3)

. On which interval is it increasing?

(1, +\infty)

.B.

(-3, 1)

.C.

(-\infty, 1)

.D.

(-3, +\infty)

Question 2. How many vertical asymptotes does the graph of

y=\dfrac{x+2}{x^2-4}

have?

1

.B.

2

.C.

0

.D.

3

Question 3. The maximum value of

y=-x^3+3x+1

[0, 2]

3

.B.

1

.C.

-1

.D.

2

Question 4.

The function $y=f(x)$ has this table of variations:

$x$	$-\infty$		$-1$		$1$		$+\infty$
$f'(x)$		$+$	$0$	$-$	$0$	$+$
$f(x)$	$-\infty$	$\nearrow$	$4$	$\searrow$	$0$	$\nearrow$	$+\infty$

The local minimum value of the function is

0

.B.

1

.C.

4

.D.

-1

Part II. True or false

Question 1. Let

f(x)=x^4-2x^2+3

a) $f'(x)=4x^3-4x$ .	T / F
b) The function has three turning points.	T / F
c) The local maximum value of the function is $2$ .	T / F
d) The minimum value of the function on $\mathbb{R}$ is $2$ .	T / F

Part III. Short answer

Question 1. The two asymptotes of the graph of

y=\dfrac{2x+1}{x-3}

meet at

I(a, b)

. Find

a+b

Question 2. Let

M

m

be the maximum and minimum values of

y=x+\dfrac1x

\left[\dfrac12, 3\right]

. Find

M+m

, to 1 decimal place.

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

Hints (only when stuck)

Exercise 1.

What you need: a turning point is a root of $y'$ where $y'$ changes sign.

Steps:

Factorise $y'=4x^3-4x$ .
Check the sign change at each root.

Exercise 2.

What you need: $y=\dfrac{ax+b}{cx+d}$ has vertical asymptote $x=-\dfrac dc$ and horizontal asymptote $y=\dfrac ac$ .

Steps:

Find the root of the denominator and check that the numerator is not $0$ there.
Take the ratio of the coefficients of $x$ .

Exercise 3.

What you need: compare the values at the roots of $y'$ in $(1, 5)$ and at the two endpoints.

Steps:

Solve $y'=1-\dfrac{4}{x^2}=0$ .
Compute $y(1)$ , $y(2)$ , $y(5)$ .

Exercise 4.

What you need: divide the numerator by the denominator to write $y=ax+b+\dfrac{r}{x-1}$ .

Steps:

Carry out the polynomial division.
Look at the limits as $x\to1^\pm$ and at $y-(ax+b)$ as $x\to\pm\infty$ .

Exercise 5.

What you need: a repeated root of even multiplicity does not make $f'$ change sign.

Steps:

Make a sign table for $f'$ with roots $-2$ , $0$ , $1$ .
Judge each statement.

Exercise 6.

What you need: the quotient rule $\left(\dfrac uv\right)'=\dfrac{u'v-uv'}{v^2}$ ; the maximum on $[0, +\infty)$ .

Steps:

Compute $C'(t)$ and find its positive root.
Find the sign of $C'$ , noting that $C(0)=0$ and $C(t)\to0$ as $t\to+\infty$ .

Exercise 7.

What you need: the number of roots of $f(x)=k$ is the number of points where the graph meets the line $y=k$ .

Steps:

Rewrite as $f(x)=-3$ .
Read the table: $-3$ is the local minimum value.

Exercise 8.

What you need: $y=\dfrac{ax+b}{cx+d}$ has $y'=\dfrac{ad-bc}{(cx+d)^2}$ ; monotonicity is stated on each interval separately.

Steps:

Compute $y'$ .
Find the two asymptotes and compare $y(2)$ , $y(4)$ .

Exercise 9.

What you need: $(uv)'=u'v+uv'$ and $(e^x)'=e^x$ .

Steps:

Compute $y'=(x^2+2x-3)e^x$ and solve $y'=0$ on $(0, 2)$ .
Compare with $y(0)$ , $y(2)$ .

Exercise 10.

What you need: at a turning point $x_0$ , $y'(x_0)=0$ ; turning points lie on the graph.

Steps:

Write four equations: $y(0)=2$ , $y'(0)=0$ , $y(2)=-2$ , $y'(2)=0$ .
Solve, then check the type of each turning point.

Self-check

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

Redo the ones you missed after: ☐ 1 day ☐ 3 days ☐ 7 days

Scan for step-by-step solutions

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐