MATHEMATICS - MIDTERM I, 17.09.

2023

Name, Family name, Student's ID:

Write legibly, show your work and justify your answers. Problems are worth 8 points each, 24 points
in total.

Problem 1. Consider a function √

f (x) = x ln(x).

(a) (2 points) Find the rst derivative of f .

(b) (2 points) Find the intervals on which f is increasing or decreasing.
(c) (2 points) Find the second derivative of f .
(d) (1 point) Find the intervals of convexity (=upward concavity), concavity (=down-
ward concavity) and (1 point) the inection points of f .

Problem 2. Consider a function

x+1
f (x) = √ .
x2 − 2x + 1

(a) (2 points) Determine the domain of f .

(b) (6 points) Detemine all asymptotes of f .

Problem 3. Given a function

2 +2x+3
f (x) = e−x ,

(a) (6 points) nd the minimum and maximum values attained by f over an interval
[0, 3],
(b) (2 points) nd the equation of the tangent line to the graph of f at a point (0, f (0)).

Ocial formula sheet for Midterm 1

 ♡ n 1
ˆ lim 1+ = e♡ . ˆ lim (1 + an ) an = e, for an → 0.
n→+∞ n n→+∞

f (x)
ˆ Slant asymptote y = ax + b, where lim =a and lim[f (x) − ax] = b.
x
ˆ eln(♡) = ♡ = ln e♡ ˆ (arctan(x))′ = 1


. .
1+x2

ˆ (sin(x))′ = cos(x). ˆ (arcsin(x))′ = √ 1 .

1−x2
ˆ (cos(x))′ = − sin(x).
′
ˆ f (g(x) = f ′ g(x) · g ′ (x).

ˆ Approximate value of a function at x close to p is f (x) ≈ f (p) + f ′ (p)(x − p).

ˆ Examples of convex functions: x2 and ex , examples of concave functions: −x2 and ln(x).
 MATHEMATICS - MIDTERM I, 17.09.2023

Name, Family name, Student's ID:

Write legibly, show your work and justify your answers. Problems are worth 8 points each, 24 points
in total.

Problem 1. Consider a function √

f (x) = 3
x ln(x).

(a) (2 points) Find the rst derivative of f .

(b) (2 points) Find the intervals on which f is increasing or decreasing.
(c) (2 points) Find the second derivative of f .
(d) (1 point) Find the intervals of convexity (=upward concavity), concavity (=down-
ward concavity) and (1 point) the inection points of f .

Problem 2. Consider a function

x−1
f (x) = √ .
x2 + 2x + 1

(a) (2 points) Determine the domain of f .

(b) (6 points) Detemine all asymptotes of f .

Problem 3. Given a function

2 +4x
f (x) = e−x ,

(a) (6 points) nd the minimum and maximum values attained by f over an interval
[0, 3],
(b) (2 points) nd the equation of the tangent line to the graph of f at a point (0, f (0)).

Ocial formula sheet for Midterm 1

 ♡ n 1
ˆ lim 1+ = e♡ . ˆ lim (1 + an ) an = e, for an → 0.
n→+∞ n n→+∞

f (x)
ˆ Slant asymptote y = ax + b, where lim =a and lim[f (x) − ax] = b.
x
ˆ eln(♡) = ♡ = ln e♡ ˆ (arctan(x))′ = 1


. .
1+x2

ˆ (sin(x))′ = cos(x). ˆ (arcsin(x))′ = √ 1 .

1−x2
ˆ (cos(x))′ = − sin(x).
′
ˆ f (g(x) = f ′ g(x) · g ′ (x).

ˆ Approximate value of a function at x close to p is f (x) ≈ f (p) + f ′ (p)(x − p).

ˆ Examples of convex functions: x2 and ex , examples of concave functions: −x2 and ln(x).
 MATHEMATICS - MIDTERM I, 17.09.2023

Name, Family name, Student's ID:

Write legibly, show your work and justify your answers. Problems are worth 8 points each, 24 points
in total.

Problem 1. Consider a function √ x+5

f (x) = (x + 5) ex+5 = (x + 5)e 2 .

(a) (2 points) Find the rst derivative of f .

(b) (2 points) Find the intervals on which f is increasing or decreasing.
(c) (2 points) Find the second derivative of f .
(d) (1 point) Find the intervals of convexity (=upward concavity), concavity (=down-
ward concavity) and (1 point) the inection points of f .

Problem 2. Consider a function √

x2 − 2x + 1
f (x) = .
x+1
(a) (2 points) Determine the domain of f .
(b) (6 points) Detemine all asymptotes of f .

Problem 3. Given a function

f (x) = ln(x2 − 4x + 5),

(a) (6 points) nd the minimum and maximum values attained by f over an interval
[0, 3],
(b) (2 points) nd the equation of the tangent line to the graph of f at a point (0, f (0)).

Ocial formula sheet for Midterm 1

 ♡ n 1
ˆ lim 1+ = e♡ . ˆ lim (1 + an ) an = e, for an → 0.
n→+∞ n n→+∞

f (x)
ˆ Slant asymptote y = ax + b, where lim =a and lim[f (x) − ax] = b.
x
ˆ eln(♡) = ♡ = ln e♡ ˆ (arctan(x))′ = 1


. .
1+x2

ˆ (sin(x))′ = cos(x). ˆ (arcsin(x))′ = √ 1 .

1−x2
ˆ (cos(x))′ = − sin(x).
′
ˆ f (g(x) = f ′ g(x) · g ′ (x).

ˆ Approximate value of a function at x close to p is f (x) ≈ f (p) + f ′ (p)(x − p).

ˆ Examples of convex functions: x2 and ex , examples of concave functions: −x2 and ln(x).
 MATHEMATICS - MIDTERM I, 17.09.2023

Name, Family name, Student's ID:

Write legibly, show your work and justify your answers. Problems are worth 8 points each, 24 points
in total.

Problem 1. Consider a function √ x+7

f (x) = (x + 7) ex+7 = (x + 7)e 3 .
3

(a) (2 points) Find the rst derivative of f .

(b) (2 points) Find the intervals on which f is increasing or decreasing.
(c) (2 points) Find the second derivative of f .
(d) (1 point) Find the intervals of convexity (=upward concavity), concavity (=down-
ward concavity) and (1 point) the inection points of f .

Problem 2. Consider a function √

x2 + 2x + 1
f (x) = .
x−1
(a) (2 points) Determine the domain of f .
(b) (6 points) Detemine all asymptotes of f .

Problem 3. Given a function

f (x) = ln(2x2 − 4x + 3),

(a) (6 points) nd the minimum and maximum values attained by f over an interval
[0, 3],
(b) (2 points) nd the equation of the tangent line to the graph of f at a point (0, f (0)).

Ocial formula sheet for Midterm 1

 ♡ n 1
ˆ lim 1+ = e♡ . ˆ lim (1 + an ) an = e, for an → 0.
n→+∞ n n→+∞

f (x)
ˆ Slant asymptote y = ax + b, where lim =a and lim[f (x) − ax] = b.
x
ˆ eln(♡) = ♡ = ln e♡ ˆ (arctan(x))′ = 1


. .
1+x2

ˆ (sin(x))′ = cos(x). ˆ (arcsin(x))′ = √ 1 .

1−x2
ˆ (cos(x))′ = − sin(x).
′
ˆ f (g(x) = f ′ g(x) · g ′ (x).

ˆ Approximate value of a function at x close to p is f (x) ≈ f (p) + f ′ (p)(x − p).

ˆ Examples of convex functions: x2 and ex , examples of concave functions: −x2 and ln(x).