Calculus Basics Practice Test

Review the following concepts if needed:

Calculus Basics Practice Test

Each of the following is a rational number. Write it as the quotient of two integers.
Plot the numbers on a real number line. Label each plotted point.
Sketch each of the following sets on a separate real number line.

(a) S = { x : ∣ x − 2∣ < 4}

(b) T = { t : t ² + 1 = 5}

(c) U = { s : 2 s − 5 ≤ 3}

(d) V = { y : ∣6 y + 1∣ > 2}

(e) S = { x : x ² + 3 < 6}

(f) T = { s : ∣ s ∣ = ∣ s + 1∣}
Plot each of the points on a pair of cartesian coordinate axes. Label each point.
Plot each of these planar loci on a separate set of axes.

(a) {( x , y ): y = 2 x ² −3}

(b) ( x , y ): x ² + y ² = 9}

(c) y = x ³ + x

(d) x = y ³ + y

(e) x = y ² − y ³

(f) x ² + y ⁴ = 3
Plot each of these regions in the plane.

(a) {( x , y ): x ² + y ² < 4}

(b) {( x , y ): y > x ² }

(c) {( x , y ): y < x ³ }

(d) {( x , y ): x ≥ 2 y + 3}

(e) {( x , y ): y ≤ x + 1}

(f) {( x , y ): 2 x + y ≥ 1}
Calculate the slope of each of the following lines:

(a) The line through the points (−5, 6) and (2, 4)

(b) The line perpendicular to the line through (1, 2) and (3, 4)

(c) The line 2 y + 3 x = 6

(d) The line

(e) The line through the points (1, 1) and (−8, 9)

(f) The line x − y = 4
Write the equation of each of the following lines.

(a) The line parallel to 3 x + 8 y = −9 and passing through the point (4, −9).

(b) The line perpendicular to x + y = 2 and passing through the point (−4, −8).

(c) The line passing through the point (4, 6) and having slope −8.

(d) The line passing through (−6, 4) and (2, 3).

(e) The line passing through the origin and having slope 6.

(f) The line perpendicular to x = 3 y − 7 and passing through (−4, 7).
Graph each of the lines in Exercise 8 on its own set of axes. Label your graphs.
Which of the following is a function and which is not? Give a reason in each case.

(a) f assigns to each person his biological father

(b) g assigns to each man his dog

(c) h assigns to each real number its square root

(d) f assigns to each positive integer its cube

(e) g assigns to each car its driver

(f) h assigns to each toe its foot

(g) f assigns to each rational number the greatest integer that does not exceed it

(h) g assigns to each integer the next integer

(i) h assigns to each real number its square plus six
Graph each of these functions on a separate set of axes. Label your graph.
Calculate each of the following trigonometric quantities.

(a) sin(8π/3)

(b) tan(−5π/6)

(c) sec(7π/4)

(d) csc(13π/4)

(e) cot(−15π/4)

(f) cos(−3π/4)
Calculate the left and right sides of the twelve fundamental trigonometric identities for the values θ = /3 and ψ = − /6, thus confirming the identities for these particular values.
Sketch the graphs of each of the following trigonometric functions.

(a) f ( x ) = sin 2 x

(b) g ( x ) = cos( x + π/2)

(c) h ( x ) = tan(− x + π )

(d) f ( x ) = cot(3 x + π)

(e) g ( x ) = sin( x /3)

(f) h ( x ) = cos(−π + [ x /2])
Convert each of the following angles from radian measure to degree measure .

(a) θ = π/24

(b) θ = − π/3

(c) θ = 27π/12

(d) θ = 9π/16

(e) θ = 3

(f) θ = −5
Convert each of the following angles from degree measure to radian measure .

(a) θ = 65°

(b) θ = 10°

(c) θ = −75°

(d) θ = −120°

(e) θ = π°

(f) θ = 3.14°
For each of the following pairs of functions, calculate f ο g and g ο f .
Consider each of the following as functions from to and say whether the function is invertible. If it is, find the inverse with an explicit formula.

(a) f ( x ) = x ³ + 5

(b) g ( x ) = x ² − x

(c) , where sgn x is +1 if x is positive, −1 if x is negative, 0 if x is 0.

(d) f ( x ) = x ⁵ + 8

(e) g ( x ) = e ^{−3 x}

(f) h ( x ) = sin x

(g) f ( x ) = tan x

(h) g ( x ) = (sgn x ) · x ² , where sgn x is +1 if x is positive, −1 if x is negative, 0 if x is 0.
For each of the functions in Exercise 18, graph both the function and its inverse in the same set of axes.
Determine whether each of the following functions, on the given domain S , is invertible. If it is, then find the inverse explicitly.

(a) f ( x ) = x ² , S = [2, 7]

(b) g ( x ) = In x , S = [1, ∞)

(c) h ( x ) = sin x , S = [0, π/2]

(d) f ( x ) = cos x , S = [0, π]

(e) g ( x ) = tan x , S = (−π/2, π/2)

(f) h ( x ) = x ² , S = [−2, 5]

(g) f ( x ) = x ² − 3 x , S = [4, 7]

Answers:

In Fig. S1.2 , set A = 3.4, B = −π/2, C = 2π, , , F = 9/2, G = −29/10.

Fig. S1.2
Fig. S1.3(a)

Fig. S1.3(b)

Fig. S1.3(c)

Fig. S1.3(d)

Fig. S1.3(e)

Fig. S1.3(f)
Let A = (2, −4), B = (−6, 3), C = (π, π ² ), , F = (1/3, −19/4).
Fig. S1.5(a)

Fig. S1.5(b)

Fig. S1.5(c)

Fig. S1.5(d)

Fig. S1.5(e)

Fig. S1.5(f)
Fig. S1.6(a)

Fig. S1.6(b)

Fig. S1.6(c)

Fig. S1.6(d)

Fig. S1.6(e)

Fig. S1.6(f)
(a) Slope is −3/8 hence line is y − (−9) = (−3/8) · ( x − 4)

(b) Slope is 1 hence line is y − (−8) = 1 · ( x − (−4))

(c) y − 6 = (−8) ( x − 4)

(d) Slope is hence line is y − 3 = (−1/8)( x − 2)

(e) y = 6 x

(f) Slope is −3 hence line is y − 7 = (−3)( x −(−4))
(a) Each person has one and only one father. This is a function.

(b) Some men have more than one dog, others have none. This is not a function.

(c) Some real numbers have two square roots while others have none. This is not a function.

(d) Each positive integer has one and only one cube. This is a function.

(e) Some cars have several drivers. In a one-car family, everyone drives the same car. So this is not a function.

(f) Each toe is attached to one and only one foot. This is a function.

(g) Each rational number succeeds one and only one integer. This is a function.

(h) Each integer has one and only one successor. This is a function.

(i) Each real number has a well defined square, and adding six is a well defined operation. This is a function.
We check the first six identities.
We shall do (a), (c), (e).

Fig. S1.14(a)

Fig. S1.14(c)

Fig. S1.14(e)
(a) θ = (15/2)°

(b) θ = −60°

(c) θ = 405°

(d) θ = (405/4)°

(e) θ = (540/π)°

(f) θ = (−900/π)°
(a) θ = 13π/36 radians

(b) θ = π/18 radians

(c) θ = −5π/12 radians

(d) θ = −2π/3 radians

(e) θ = π ² /180 radians

(e) θ = 157π/9000 radians
(a) f ο g ( x ) = [( x − 1) ² ] ² + 2[( x − 1) ² ] + 3; g ο f ( x ) = ([ x ² + 2 x + 3] − 1) ² .

(b)

(c) f ο g ( x ) = sin([cos( x ² − x )] + 3[cos( x ² − x )] ² ); g ο f ( x ) = cos([sin( x + 3 x ² )] ² − [sin( x + 3 x ² ]).

(d) f ο g ( x ) = e ^{ln( x − 5)+2} ; g ο f ( x ) = ln( e ^{x +2} − 5).

(e) f ο g ( x ) = sin([ln( x ² − x )] ² + [ln( x ² − x )]); g ο f ( x ) = ln([sin( x ² + x )] ² − [sin( x ² + x )]).

(f) f ο g ( x ) = e ^{[ e ^{− x} ² ] ²} ; f ο g ( x ) = e ^{−[ e ^x ² ] ²} .

(g) f ο g ( x ) = [(2 x − 3)( x + 4)] · [(2 x − 3)( x + 4) + 1] · [(2 x − 3)( x + 4) + 2]; g ο f ( x ) = (2[ x ( x + 1)( x + 2)]−3)([ x ( x + 1)( x + 2)] + 4).
(a) f is invertible, with f ⁻¹ ( t ) = ( t − 5) ^1/3 .

(b) g is not invertible since g (0) = g (1) = 0.

(c) h is invertible, with h ⁻¹ ( t ) = sgn t · t ² .

(d) f is invertible, with f ⁻¹ ( t ) = ( t − 8) ^1/5 .

(e) g is invertible, with g ⁻¹ ( t ) = −[ln t ]/3.

(f) h is not invertible, since .

(g) f is not invertible, since tan π/4 = tan 9π/4 = 1.

(h) g is invertible, with .
We will do (a), (c), (e), and (g).

Fig. S1.19(a)

Fig. S1.19(c)

Fig. S1.19(e)

(g) Not invertible.