Differentiation / Integration Help

In summary, the gradient function has a turning point at (0.5, -4) and a constant q can be found by setting the gradient expression to 0 and solving for q.
  • #1
Joe20
53
1
The curve has a gradient function dy/dx = 2 +q/(5x^2) where q is a constant, and a turning point at (0.5, -4). Find the value of q.

option 1 : 2.5
option 2: -2.5
option 3: 0
Option 4: -3

I couldn't find the answer and will need assistance to how the answer can be obtained.

I have substituted x = 0.5 into dy/dx to get the gradient expression of 2 + 4q/5 and integrated to get y = 2x - q/(5x) + c.
It seems impossible for me to get the value of q since c could not be found. I am not sure if the question has some missing information to continue. Your help will be greatly appreciated. Thanks.
 
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  • #2
Since the given point is a turning point, we must have:

\(\displaystyle \left.\d{y}{x}\right|_{x=\frac{1}{2}}=2+\frac{q}{5\left(\frac{1}{2}\right)^2}=2+\frac{4q}{5}=0\)

Now you just need to solve for $q$.
 

1. What is the difference between differentiation and integration?

Differentiation is a mathematical process of finding the rate of change of a function at a specific point, while integration is the reverse process of finding the area under a curve. In simple terms, differentiation helps us determine the slope of a curve, while integration helps us find the total value or quantity represented by the curve.

2. Why are differentiation and integration important in science?

Differentiation and integration are important in science because they help us understand and analyze natural phenomena, such as the speed and acceleration of objects in motion, the growth rate of populations, and the rate of change in chemical reactions. They are also used in many scientific fields, including physics, biology, and economics.

3. What are the basic rules of differentiation and integration?

The basic rules of differentiation include the power rule, product rule, quotient rule, and chain rule. The basic rules of integration include the power rule, constant multiple rule, sum rule, and substitution rule. These rules help us find the derivatives and integrals of more complex functions by breaking them down into simpler parts.

4. How is differentiation and integration used in real-life applications?

Differentiation and integration are used in many real-life applications, such as predicting the stock market, designing bridges and buildings, and modeling the spread of diseases. They are also used in engineering, economics, and statistics to analyze data and make predictions.

5. What are some common mistakes to avoid when differentiating or integrating?

Some common mistakes to avoid when differentiating include forgetting to apply the chain rule, not simplifying the final answer, and not checking for extraneous solutions. When integrating, common mistakes include forgetting to use the constant of integration, not using the correct integration rules, and not checking for any mistakes in the calculations.

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