Respuesta :
Question 1:
For this case we must find the derivative of the following function:
[tex]f (x) = \frac {7} {x}[/tex] evaluated at [tex]x = 1[/tex]
We have by definition:
[tex]\frac {d} {dx} [x ^ n] = nx ^ {n-1}[/tex]
So:
[tex]\frac {df (x)} {dx} = - 1 * 7 * x ^ {- 1-1} = - 7x ^ {- 2} = - \frac {7} {x ^ 2}[/tex]
We evaluate in [tex]x = 1[/tex]
[tex]- \frac {7} {x ^ 2} = - \frac {7} {1 ^ 2} = - 7[/tex]
ANswer:
Option A
Question 2:
For this we must find the derivative of the following function:
[tex]f (x) = 4x + 7\ evaluated\ at\ x = 5[/tex]
We have by definition:
[tex]\frac {d} {dx} [x ^ n] = nx ^ {n-1}[/tex]
The derivative of a constant is 0
So:
[tex]\frac {df (x)} {dx} = 1 * 4 * x ^ {1-1} + 0 = 4 * x ^ 0 = 4[/tex]
Thus, the value of the derivative is 4.
Answer:
Option A
Question 3:
For this we must find the derivative of the following function:
[tex]f (x) = 12x ^ 2 + 8x\ evaluated\ at\ x = 9[/tex]
We have by definition:
[tex]\frac {d} {dx} [x ^ n] = nx ^ {n-1}[/tex]
So:
[tex]\frac {df (x)} {dx} = 2 * 12 * x ^ {2-1} + 1 * 8 * x ^ {1-1} = 24x + 8 * x ^ 0 = 24x + 8[/tex]
We evaluate for [tex]x = 9[/tex]we have:
[tex]24 (9) + 8 = 224[/tex]
Answer:
Option D
Question 4:
For this we must find the derivative of the following function:
[tex]f (x) = - \frac {11} {x}\ evaluated\ at\ x = 9[/tex]
We have by definition:
[tex]\frac {d} {dx} [x ^ n] = nx ^ {n-1}[/tex]
So:
[tex]\frac {df (x)} {dx} = - (- 1 * 11 * x ^ {- 1-1}) = 11x ^ {- 2} = \frac {11} {x ^ 2}[/tex]
We evaluate for [tex]x = 9[/tex] and we have:
[tex]\frac {11} {9 ^ 2} = \frac {11} {81}[/tex]
ANswer:
Option D
Question 5:
For this case we have by definition, that the derivative of the position is the velocity. That is to say:
[tex]\frac {d (s (t))} {dt} = v (t)[/tex]
Where:
s: It's the position
v: It's the velocity
t: It's time
We have the position is:
[tex]s (t) = 1-10t[/tex]
We derive:
[tex]\frac {d (s (t))} {dt} = 0- (1 * 10 * t ^ {1-1}) = - 10 * t ^ 0 = -10[/tex]
So, the instantaneous velocity is -10
Answer:
-10
