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# x (b) g ( x )=2 √ f ( x )= x − 3 x 2 1.(5points)Diﬀerentiatefromtheﬁrstprinciplesthefollowingfunctions:(a) BatoryAAHLShortTest8May15,2020Name:

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(1)

Batory AA HL Short Test 8 May 15, 2020

Name:

1. (5 points) Differentiate from the first principles the following functions:

(a) f (x) = x2 − 3x

(b) g(x) = 2

√x

(2)

Batory AA HL Short Test 8, page 2 of 4 May 15, 2020

2. (4 points) Consider the following function:

f (x) =

x3 f or x < 1 ax + b f or x ­ 1

Find the values of a and b so that f is differentiable at x = 1.

(3)

Batory AA HL Short Test 8, page 3 of 4 May 15, 2020

3. (4 points) Solve the following equation:

cos x + cosx

2 + 1 = 0 for 0 ¬ x ¬ 3π.

(4)

Batory AA HL Short Test 8, page 4 of 4 May 15, 2020

4. (7 points) Consider the function

f (x) = x2 − 4 x − 1

(a) Write down the equations of the asymptotes of the graph of y = f (x).

(b) Show that the range of values of f (x) is all real numbers.

(c) Sketch the graph of g(x) = f (|x|) and hence state the set of all possible values of parameter k, such that the equation:

g(x) = k

has four solutions.

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