Solving word problems using derivatives

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In summary, the student is trying to find a solution to a problem where they do not understand the material. The student has posted a picture of a problem they are encountering and is looking for advice on how to study for the problem. The student is looking for help with understanding the concept of a position function, velocity function, and acceleration function. The student wants to be able to solve specific statements using all of the functions.
  • #1
Danatron
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Hi Guys,

I am revising for an exam i have this week, the last module on my subject was calculus. I did not understand it entirely.

I have posted a pic below of a typical problem i can expect to encounter, would anybody be able to point me in the right direction to study material that could teach me how to solve problems like this? my lecture is very vague and study material even vaguer.

XoDC2Jj.jpg
 
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  • #2
Applications of derivatives, found in typical first-year undergraduate Calculus textbooks. h(t) takes the shape (as a cartesian graph) of a parabola with vertex as a maximum. The positive root will be the time when the ball reaches the ground. You can use the first derivative of h, equate this derivative to zero, solve this for t, and that is the time when h is the maximum.
 
  • #3
Which part of the problem is most difficult to you? Conceptually understanding what the question is asking? Or is it the math? Or is it the physical intuition?

The best way to study for exams is different for everybody. It's hard to point you in the direction of the study material if I don't know what you want to study. Do you want to study calculus (how to take derivatives, etc.)? Or do you want to study kinematics?

It might be useful to just find some books with a lot of practice problems similar in difficulty to the ones you will encounter. Have you tried working through the problems in your textbook? Usually introductory physics textbooks will have a lot of practice problems you can do.
 
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  • #4
Also note that dh/dt is a velocity function, so you can use it to determine the velocity for when t is the positive root (meaning, the ball hits the ground).
 
  • #5
symbolipoint said:
Also note that dh/dt is a velocity function, so you can use it to determine the velocity for when t is the positive root (meaning, the ball hits the ground).

How to differentiate that h0 term? Chain rule?
 
  • #6
You want to find the relative maximum, a value of t where the derivative of the position function is equal to zero, and the second derivative is negative, then, as mentioned, determine the value of h(t) just before it reaches the positive zero.
 
  • #7
Arka420 said:
How to differentiate that h0 term? Chain rule?

h[itex]_{0}[/itex] and v[itex]_{0}[/itex] are just constants; the independent variable in the equation is t.
 
  • #8
Reember that h (t)= s (t) this is called the position function.

The derivative h`(t)=v (t) this is called the velocity function

The derivative of h'(t) is h''(t) where h''(t)=a (t) this is called the acceleration function.


Use all 3 functions to solve specific given statements. Understand what it means by position, velocity, and acceleration.
 

1. What are derivatives and how are they used to solve word problems?

Derivatives are a mathematical tool used to calculate the rate of change of a function. In word problems, derivatives are used to find the maximum or minimum value of a function, as well as the rate of change at a specific point.

2. How do I know when to use derivatives to solve a word problem?

Typically, word problems involving rates of change, optimization, or related rates can be solved using derivatives. Look for key words or phrases such as "rate of change," "maximum," "minimum," or "optimization" to determine if derivatives should be used.

3. What are the steps to solving a word problem using derivatives?

The first step is to read the problem carefully and identify what information is given and what is being asked. Then, use the given information to create a mathematical equation or function. Take the derivative of this function and set it equal to zero to find critical points. Finally, plug these critical points back into the original function to find the maximum or minimum value.

4. Are there any common mistakes to avoid when solving word problems using derivatives?

One common mistake is forgetting to check the domain of the function. It's important to make sure that the critical points found are within the given domain. Additionally, make sure to double check the units of measurement, as they may need to be converted for the answer to make sense.

5. Can derivatives be used to solve word problems in real-life situations?

Yes, derivatives are used in various fields such as economics, physics, and engineering to solve real-life problems. They are especially useful in situations where there is a changing rate of something, such as the growth of a population or the decay of a radioactive substance.

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