2021-013

Choose the option that contains the derivative of \( f(x) = g(x) h(x) \), where \( g(x) = x^3 \) and \( h(x) = (1 + \ln x)^2 \).

1. \( 3x^2 + 2(1 + \ln x)/x \)
2. \( x^2 (1 + \ln x) (5 + \ln x) \)
3. \( x^2 (1 + \ln x) (5 + 3 \ln x) \)
4. \( x^2 (1 + \ln x) (3 + 3 \ln x + 2x) \)
5. None of the above

Original idea by: Maria Tejada Begazo

2021-012

Find the derivative of the following function:

$$ f(x) = 2\sqrt{x} - 6\sqrt[3]{x^2} + 5x^2 - 20 $$

1. \( x^{1/2} - 6x^{1/3} + 10x \)
2. \( x^{1/2} - 4x^{1/3} + 10x \)
3. \( x^{-1/2} - 4x^{-1/3} + 10x \)
4. \( x^{-1/2} - 6x^{-1/3} + 10x \)
5. none of the above

Original idea by: Angelica Oliveira

2021-011

Given the linear differential equation below:

$$ y' = 4y + 16 $$

which alternative corresponds to a possible solution?

1. \( y(t) = e^{4t - 4} - 16 \)
2. \( y(t) = 13e^{4t - 4} - 4 \)
3. \( y(t) = e^{t} - 16 \)
4. \( y(t) = 25e^{t} - 256 \)
5. None of the above.

Original Idea by: José Nascimento

2021-010

A crack appeared in a container and the liquid in it began to leak. At instant \( t \ge 0 \), a total of \( 10t - \sqrt{t} \) units of the liquid will have leaked. What is the flow rate \( r \) of the liquid through the crack at instant \( t = 100 \) ?

1. \( r = 99/2 \)
2. \( r = 99/20 \)
3. \( r = 199/2 \)
4. \( r = 199/20 \)
5. None of the above

Original idea by: Bruno Almêda

2021-009

Find the derivative of $$ f(t) = \frac{1}{t} − \frac{1}{2t^3} + \frac{1}{2t^5} $$.

1. \( f'(t) = \dfrac{1}{t} − \dfrac{1}{2t^3} + \dfrac{1}{2t^5} \)
2. \( f'(t) = {t^2} − {2t^4} + \dfrac{1}{2t^5} \)
3. \( f'(t) = -\dfrac{1}{t^2} + \dfrac{1}{2t^4} + \dfrac{t^5}{2} \)
4. \( f'(t) = {t} - {2t^3} + {2t^5} \)
5. None of the above

Original idea by: André Regino

2021-008

The space exploration agency AES is testing a new type of submarine, called XY, which can fly out from under the ocean and reach space. In its first test, there were failures that caused the first XY prototype to have a total loss of its engines, stopping its ascent at a certain time and starting to fall afterwards. The trajectory of the prototype in the time interval \( 0 \leq t \leq 20 \) is given by \( f(t) = 56 + 5t - (t-8)^{2} \).

What is the exact time \( t \) when the XY prototype stopped going up towards space?

1. 20/2.
2. 21/2.
3. 2/21.
4. -21/2.
5. None of the above.

Original idea by: Adolfo Schneider

2021-007

Choose the incorrect alternative below:

1. If \( f(x) = 2x^{−2} \), then \( f'(x) = −4x^{−1} \)
2. If \( f(x) = xe^x \), then \( f'(x) = (x+1)e^x \)
3. If \( f(x) = \sqrt{x^2+1} \), then \( f'(x) = x / \sqrt{x^2+1} \)
4. If \( f(x) = x^{−1} \), then \( \int 𝑓 = \ln x + C \)
   
5. none of the above

Original idea by: Mauricio Schiezaro

2021-006

 Choose the alternative that represents the derivative of the following function

 \( f(x) = (2x-3)(x^2– 5x) \)

1. \( -6 \)
2. \( 2x^2 - 10 \)
3. \( 6x^2 + 26x - 15 \)
4. \( 6x^2 - 26x + 15 \)
5. None of the above

Original idea by: Hismael Costa

2021-005

Given that \( \sin(xyz) = x + 2z + y \), find the correct answer for the partial derivative of \( z \) with respect to \( y \), that is \( \partial z / \partial y \).

1. \( [1 - yz\cos(xyz)] / [xy\cos(xyz) - 2] \)
2. \( [1 + yz\cos(xyz)] / [xy\cos(xyz) - 2] \)
3. \( [1 + xz\cos(xyz)] / [xy\cos(xyz) - 2] \)
4. \( [1 - xz\cos(xyz)] / [xy\cos(xyz) - 2] \)
5. none of the above

Original idea by: Soroor Salavati

2021-004

A builder wants to build a rectangle with a perimeter of 137 cm but wants to do so while maximizing the area of this geometric shape. Help this person and check the alternative that shows the largest area that can be built in this setting.

A. 18769/16 cm²

B. 18753/16 cm²

C. 18833/16 cm²

D. 18945/16 cm²

E. none of the above

Original idea by: Ítalo Fernandes Gonçalves

2021-003

An interesting thing regarding derivatives and real world objects is that the derivative of a sphere's volume with respect to the radius corresponds to its total area. 

Considering that the volume of a sphere V can be computed as:

    V = 4/3 πr³,

where r is the radius of the sphere, we may conclude that the total area of V is given by:

A. 4πr²

B. 2πr

C. 4πr

D. 3/4 πr²

E. none of the above.

 

Original idea by: Leandro Ferlin

2021-002

 The derivative of which function below results in \( x^{-1} \):

1. \( \log_{10} x \)
2. \( \ln x \)
3. \( x^0 \)
4. \( x^1 \)
5. none of the above

 

Original idea by: Margarita Lacuana

2021-001

Choose the incorrect alternative below. Note: \( (f(x))' \) indicates the derivative of \( f(x) \).

1. \( (x^4)' = 4 x^3 \)
2. \( (x^{10})' = 10x^9 \)
3. \( (x^{-2})' = -2x^{-1} \)
4. \( (x^{-3.5})' = -3.5x^{-4.5} \)
5. none of the above

 

Original idea by: Joao Meidanis