Practice exam 1

Mathematics: Analysis · DLBDSMFC01

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Practice exam 1

Question 1Not yet answeredMarked out of 3.00Geometric sequences

Which numbers represent the beginning of a geometric sequence?

5,\ 10,\ 15,\ 20,\ \ldots

5,\ 10,\ 20,\ 40,\ \ldots

5,\ 10,\ 15,\ 25,\ \ldots

5,\ 10,\ 30,\ 120,\ \ldots

Question 2Not yet answeredMarked out of 3.00Taylor polynomials

We want to approximate the number

e^{5}

using a Taylor polynomial of

n

th degree for the exponential function. Which of the following degrees likely gives the best approximation?

n = 55

n = 10

n = 50

n = 100

Question 3Not yet answeredMarked out of 3.00Differentiation

Let

f

be a real-valued function. Which of the following tasks does not involve differentiation of

f

Determining the extreme values of the functionCalculating the slope of

f

at the point

x = x_{0}

Calculating the area under the graph of

f

in the interval

[a, b]

Determining the Taylor polynomial of degree

10

of the function

Question 4Not yet answeredMarked out of 3.00Higher derivatives

Let

f(x) = -2x^{3} + 5x^{2} + 1

. Which statement is correct?

f^{\prime\prime\prime}(x) = 6x + 4

f^{\prime\prime\prime}(x) = -6

f^{\prime\prime}(x) = 3 + 4x + 5x^{2}

f^{(4)}(x) = 0

Question 5Not yet answeredMarked out of 3.00Definite integrals

What is the result of

\displaystyle\int_{0}^{\pi} \sin(x)\, dx

2

1

-2

-1

Question 6Not yet answeredMarked out of 3.00Differential equations

Which differential equation is satisfied by the function

y(x) = 3e^{-2x}

y^{\prime}(x) = -3y(x)

y^{\prime}(x) = 2y(x)

y^{\prime}(x) = 3y(x)

y^{\prime}(x) = -2y(x)

Question 7Not yet answeredMarked out of 3.00Sequence limits

Which of the following sequences converges to a finite value for

n \to \infty

a_{n} = (-1)^{n+1}\,\dfrac{n^{3}}{n^{2}}

a_{n} = \dfrac{n^{3}}{n^{2} + 1}

a_{n} = \dfrac{n^{2}}{n^{3} + 1}

a_{n} = (-1)^{n+1}\,\dfrac{n^{2}}{n^{3}}

Question 8Not yet answeredMarked out of 3.00Bijective functions

Which of the following functions is bijective?

f : \mathbb{R} \to \mathbb{R},\ f(x) = \sin(x)

f : \mathbb{R} \to \mathbb{R},\ f(x) = e^{x}

f : \mathbb{R} \to \mathbb{R}^{+},\ f(x) = x^{4}

f : \mathbb{R} \to \mathbb{R},\ f(x) = x^{3}

Question 9Not yet answeredMarked out of 3.00Inverse functions

Suppose the graph of the function

\ln(x)

is reflected along the line of the identity function. Which function describes the reflected graph?

\ln(x)

e^{-x}

-\ln(x)

e^{x}

Question 10Not yet answeredMarked out of 3.00Indefinite integrals

What is the result of

\displaystyle\int \dfrac{4}{4x - 3}\, dx

-\dfrac{32}{(4x - 3)^{2}} + C

-\dfrac{8}{(4x - 3)^{2}} + C

2\ln|4x - 3| + C

8\ln|4x - 3| + C

Question 11Not yet answeredMarked out of 3.00Limits

Determine the value of the following limit:

\displaystyle\lim_{n \to \infty} \left(1 + \dfrac{1}{n}\right)^{n}

e^{0.5}

e^{-1}

e

e^{2}

Question 12Not yet answeredMarked out of 3.00Trigonometric identities

Which of the following equations does not hold for all

x \in \mathbb{R}

\sin(2x) = 2\sin(x)

\sin(x + 2\pi) = \sin(x)

\sin\!\left(x + \dfrac{\pi}{2}\right) = \cos(x)

\cos^{2}(x) + \sin^{2}(x) = 1

Question 13Not yet answeredMarked out of 3.00Antiderivatives

Which of the following functions

F(x)

is an anti-derivative of the function

f(x) = \sin^{2}(x)

F(x) = 2\sin(x)\cos(x)

F(x) = \dfrac{x - \sin(x)\cos(x)}{2}

F(x) = -\dfrac{1}{3}\cos^{3}(x)

F(x) = \dfrac{1}{3}\sin^{3}(x)

Question 14Not yet answeredMarked out of 3.00Classification of differential equations

Which statement is correct?

y^{(4)} - 2y^{\prime\prime} + y(x) = 3\cos(x)

is a differential equation of second order.

y^{\prime\prime}(x)\,y(x) = 1

is a linear differential equation.

y^{\prime\prime}(x) + 2\bigl(y^{\prime}(x)\bigr)^{2} = x

is a linear differential equation.

y^{\prime}(x) = \bigl(y^{\prime\prime}(x) - 1\bigr)^{2}

is a differential equation of second order.

Question 15Not yet answeredMarked out of 6.00Inverse and composite functions

Consider the functions

f : \mathbb{R} \to \mathbb{R}

f(x) = 2x - 1

, and

g : \mathbb{R}^{+} \to \mathbb{R}^{+}

g(x) = x^{2}

. a) Calculate the inverse function of

f

. b) Justify why the function

g

is not invertible. c) Determine the functional equation of the composition

f \circ g

Question 16Not yet answeredMarked out of 6.00Differentiation rules

Calculate the derivatives of the following functions: a)

f(x) = \dfrac{x + 1}{x^{2} + 1}

g(x) = x\sin(x^{2})

Question 17Not yet answeredMarked out of 8.00Local extrema

Determine all local minima and all local maxima of the function

f(x) = 2x^{3} - 3x^{2} - 12x

. Justify your answer.

Question 18Not yet answeredMarked out of 8.00Sequences and partial sums

Consider the sequence

a_{n} = (-1)^{n-1}\,\dfrac{n^{2}}{n^{2} + 1}

and the corresponding series of partial sums

s_{n}

, where

n = 1, 2, 3, \ldots

a) Calculate the values

a_{1}

and

a_{2}

of the sequence. b) Calculate the value

s_{2}

of the series. c) Determine the limit

\displaystyle\lim_{n \to \infty} a_{n}

. Justify your answer.

Question 19Not yet answeredMarked out of 10.00Solids of revolution

Determine the volume of the solid of revolution generated by the function

f(x) = \sqrt{x}\,e^{x}

in the interval

[0, 3]

Question 20Not yet answeredMarked out of 10.00Limited growth model

A landfill is initially populated by

500

rats. Based on food supply and predators, the site can nourish a population of up to

3000

rats. Per day, the population increases at a rate that is equal to

2\%

of the difference between the maximum number and the current number of animals. a) Set up a differential equation for the number

y(t)

of rats as a function of time

t

, measured in days. b) State the general solution of this differential equation as well as the particular solution for the given initial condition. c) Calculate after how many days

t

the population has grown to

2000

rats.

You are allowed to use that.

Unit 1Sequences and Series

Each term adds a constant common difference $d$ to the previous one.

$a_1$ is the first term; $n$ is the position.

Sum of the first $n$ terms: average of the ends times the count.

The defining limit for Euler's number $e$ .

Generalises the Euler limit; the source of continuous growth.

The constant factor between consecutive terms of a geometric sequence.

Each term multiplies the previous one by the common ratio $q$ .

Sum of the first $n$ terms of a geometric sequence.

Converges only when $|q| < 1$ ; otherwise the sum diverges.

A minus sign flips the Euler limit to $1/e$ .

The root grows fast enough to flatten $n$ down to 1.

Any fixed positive base under an nth root tends to 1.

A fixed numerator over a growing denominator goes to zero.

Repeatedly multiplying by a number smaller than 1 collapses to zero.

Unit 2Functions and Inverse Functions

Cosine is sine shifted by a quarter turn; it starts at 1.

Reciprocal of sine; its value never lands inside $(-1, 1)$ .

Undefined where sine is zero; ranges over all reals.

Reciprocal of cosine; its value never lands inside $(-1, 1)$ .

Defined for every real number; output never leaves $[-1, 1]$ .

Undefined where cosine is zero; ranges over all reals.

The quarter-turn values; sine starts at 0 and peaks at $\tfrac{\pi}{2}$ .

Unit 3Differential Calculus

Write it as $x^{-n}$ and apply the power rule.

Seen in Exam formula sheet.

A constant does not change, so its rate of change is zero.

Seen in Exam formula sheet.

A general base picks up a factor of $\ln a$ .

Seen in Exam formula sheet.

Same as arcsin but negated.

Seen in Exam formula sheet.

The arctan derivative, negated.

Seen in Exam formula sheet.

Only defined for $|x| > 1$ , where the root is real.

Seen in Exam formula sheet.

Restricted to $|x| > 1$ .

Seen in Exam formula sheet.

Valid on $(-1, 1)$ , where the square root stays real.

Seen in Exam formula sheet.

Defined for all reals; no square root to worry about.

Seen in Exam formula sheet.

A plus sign under the root, so it is defined for all reals.

Seen in Exam formula sheet.

Restricted to $|x| < 1$ .

Seen in Exam formula sheet.

The minus sign is the easy one to drop on an exam.

Seen in Exam formula sheet.

No minus sign here, unlike $\cos$ .

Seen in Exam formula sheet.

The cotangent mirror of the tangent derivative, with a minus sign.

Seen in Exam formula sheet.

Valid away from $x = 0$ , where $\sinh$ vanishes.

Seen in Exam formula sheet.

The exponential is its own derivative.

Seen in Exam formula sheet.

The natural log has the simplest derivative of the logarithms.

Seen in Exam formula sheet.

A change-of-base factor $1/\ln a$ over the natural log.

Seen in Exam formula sheet.

Differentiating cycles sin -> cos -> -sin -> -cos.

Seen in Exam formula sheet.

Hyperbolic functions differentiate without the sign flips of trig.

Seen in Exam formula sheet.

Two equivalent forms; pick whichever the next step needs.

Seen in Exam formula sheet.

Note the minus in $1 - \tanh^2 x$ , versus $1 + \tan^2 x$ for trig.

Seen in Exam formula sheet.

A special case of the power rule with $n = \tfrac{1}{2}$ .

Seen in Exam formula sheet.

Bring the exponent down, then subtract one from it.

Seen in Exam formula sheet.

Unit 4Integral Calculus

The absolute value covers negative $x$ as well.

Seen in Exam formula sheet.

A constant integrates to a straight line; never forget the $+C$ .

Seen in Exam formula sheet.

Dividing by $\ln a$ undoes the factor that differentiation adds.

Seen in Exam formula sheet.

Comes from $u = \sin x$ substitution; no minus sign this time.

Seen in Exam formula sheet.

A linear inner function divides by its coefficient $a$ .

Seen in Exam formula sheet.

The exponential integrates to itself.

Seen in Exam formula sheet.

Comes from $u = \cos x$ substitution.

Seen in Exam formula sheet.

Add one to the exponent, then divide by the new exponent.

The case $a = -1$ is the logarithm, handled separately.

Seen in Exam formula sheet.

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