All Calculus 1 Resources
Example Questions
Example Question #2885 : Calculus
The velocity of a ball is described by the function . How long does it take the ball to hit the ground?
We need to find when
. To find the position equation, we need to integrate the velocity equation. Integrating the velocity equation gives
To find the constant , we use the inital conditon
Substituting in the constant
To find when
, we will use the quadratic equation
In this equation, ,
,
, and
. Substituting these variables into quadratic equation gives
or
The first answer gives us , which is when the ball is thrown. The second answer is when the ball hits the ground,
Example Question #21 : Solving For Time
The position of an car is described by the function . How long does it take the car to travel
?
We need to find when
.
To solve the above equation, use the quadratic equation
In this equation, ,
,
, and
. Substituting these variables into quadratic equation gives
or
Since time cannot be negative, the answer is
Example Question #22 : Solving For Time
Given the parametric equation for an object
Determine at what the object reaches its maximum point. Assume
.
The maximum point is reached when the derivative of with respect to time
is
.
Using the power rule we know that:
, where
and
are constants.
Therefore,
Solving this for when
The solutions are:
and
Since the function is only defined for when , it hits its maximum at
.
Since the problem asks for the value, we plug in
for that parametric equation.
Example Question #2886 : Calculus
The position of an airplane is described by the function . How long does it take the airplane to travel
?
We need to find when
.
To solve the above equation, use the quadratic equation
In this equation, ,
,
, and
. Substituting these variables into quadratic equation gives
or
Since time cannot be negative, the answer is
Example Question #2887 : Calculus
The position of a person running is described by the function . How long does it take the person to run
miles?
We need to find when
miles.
To solve the above equation, use the quadratic equation
In this equation, ,
,
, and
. Substituting these variables into quadratic equation gives
or
Time cannot be negative, so the answer is
Example Question #1861 : Functions
The velocity of a rocket shot into the air is described by the function . How long does it take the rocket to reach its highest point?
The rocket has reached it's highest point when . Substituting this into the equation gives
Solving for , gives
Example Question #21 : Solving For Time
The position of a particle is described by the function . How long does it take the particle to reach a speed of
?
We need to find when
. The velocity equation is the first derivative of the position equation. Taking the first derivative of the position equation gives
Substituting gives
Example Question #66 : Rate
The velocity of a satellite launched into space is described by the function . How long does it take the satellite to reach its highest point?
The rocket has reached it's highest point when . Substituting this into the equation gives
Solving for , gives
Example Question #2891 : Calculus
The position of a bike is described by the function . How long does it take the bike to reach a speed of
?
We need to find when
. The velocity equation is the first derivative of the position equation. Taking the first derivative of the position equation gives
Substituting gives
Example Question #29 : Solving For Time
The velocity of a missle shot into the air is described by the function . How long does it take the missile to hit the ground if it is launched from the ground?
We need to find when
. To find the position equation, we need to integrate the velocity equation. Integrating the velocity equation gives
To find the constant , we use the inital conditon
Substituting in the constant
To find when
, we will use the quadratic equation
In this equation, ,
,
, and
. Substituting these variables into quadratic equation gives
or
The first answer gives us , which is when the rocket is launched. The second answer is when the missile hits the ground,
All Calculus 1 Resources
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