Hrmm derivative problems (concept?)

A(x)=B(x) for all real numbers x except x=a, thenlimA(x)=limB(x) both as x->a (i.e. as x approaches a) and as x->a+ (i.e. as x approaches a from the right) and as x->a- (i.e. as x approaches a from the left).in short, limits don't care about what's happening at the point in question, only what's happening very close to that point. So, in summary, we discussed finding derivatives at a given point and the concept of limits. We also saw how to apply these concepts to solve for derivatives of different functions, and
  • #1
VikingStorm
I've been trying to do these concept-based questions, (but I think my concept isn't that sound).

"Suppose f'(2)=4, g'(2)=3, f(2)=-1 and g(2)=1. Find the derivative at 2 of each of the following functions
a. s(x)=f(x)+g(x)
b. p(x)=f(x)g(x)
c. q(x)=f(x)/g(x)"
I began doing this, without reading the find the derivative part. What order would I exactly solve it in? Or does it work straight in by plugging in the derivatives? (too simple, so must not be it)

"If f(x)=x, find f'(137)"
This is a pure concept question I'm sure...

"Explain what is wrong with the equation (x^2-1)/(x-1)=x+1, and why lim(x^2)/(x-1)=lim(x+1) both x->1"
The top factors out and supposedly cancels, though I'm not sure why I can't do that.
 
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  • #2
Hello, VikingStorm!

"Suppose f'(2)=4, g'(2)=3, f(2)=-1 and g(2)=1. Find the derivative at 2 of each of the following functions
a. s(x) = f(x) + g(x)
b. p(x) = f(x)*g(x)
c. q(x) = f(x)/g(x)"

Yes , you're right ...
After finding the derivative, just plug in the given values.

(a) s'(x) = f '(x) + g'(x)
Hence: s'(2) = f'(2) + g'(2) = 4 + 3 = 7

(b) p'(x) = f(x)*g'(x) + g(x)*f '(x)
Hence: p'(2) = f(2)*g'(2) + g(2)*f '(2) = (-1)(3) + (1)(4) = 1

(c) q'(x) = [g(x)*f '(x) - f(x)*g'(x)]/[g(x)]^2
Hence: q'(2) = [(1)(4) - (-1)(3)][1^2] = 7
 
  • #3
"If f(x)=x, find f'(137)"
This is a pure concept question I'm sure...

"Explain what is wrong with the equation (x^2-1)/(x-1)=x+1, and why lim(x^2)/(x-1)=lim(x+1) both x->1"
The top factors out and supposedly cancels, though I'm not sure why I can't do that.
for the first one, note that f'(x)=1 for all x, so f'(137)=1. another way to look at is is that for y=x, y=x is a tangent line at all points. the slope of the tangent line is 1 everywhere, so since f'(x) is the slope of the tangent line at (x,f(x)), f'(137)=1.

for the second question, the main thing is what is meant by the equality sign. suppose A(x) and B(x) are two algebraic expressions defined for some set such as the set of real numbers. then we say that A(x)=B(x) if and only if A(x) equals B(x) for all real numbers x. such equations like A(x)=B(x) that are true "everywhere" are called identities.

(x^2-1)/(x-1)=x+1 is *not* an identity because the equation isn't always true: it fails when x=1.

if you let A(x)=(x^2-1)/(x-1) and B(x)=x+1, note that A(x)=B(x) for all real numbers except x=1. when you take the limit as x approaches 1, x is never allowed to actually equal 1, so
limA(x)=limB(x).
 

1. What is a derivative in the context of Hrmm derivative problems?

In mathematics, a derivative is a way to measure the instantaneous rate of change of a function with respect to one of its variables. In the context of Hrmm derivative problems, it refers to the application of this concept to solve problems related to Hrmm, a programming language.

2. How is the derivative calculated in Hrmm derivative problems?

In Hrmm derivative problems, the derivative is calculated using the concept of finite differences. This involves taking small intervals of change in the function and calculating the slope of the secant line connecting two points on the curve. As the interval becomes smaller and smaller, the slope of the secant line approaches the slope of the tangent line at a specific point, which is the derivative of the function at that point.

3. What are some real-world applications of Hrmm derivative problems?

Hrmm derivative problems can be applied in various fields such as economics, engineering, physics, and computer science. In economics, derivatives are used to analyze the rate of change in market trends. In engineering, they are used in designing and optimizing structures. In physics, derivatives are used to understand the motion of objects. In computer science, they are used in programming languages to optimize code and improve efficiency.

4. What is the difference between the derivative and the anti-derivative in Hrmm derivative problems?

While the derivative measures the rate of change of a function at a specific point, the anti-derivative is the inverse operation of the derivative. It allows us to find the original function when given its derivative. In Hrmm derivative problems, finding the anti-derivative can be helpful in solving optimization and integration problems.

5. How can I improve my understanding and skills in solving Hrmm derivative problems?

To improve your understanding and skills in solving Hrmm derivative problems, it is important to practice regularly and familiarize yourself with the concepts and techniques involved. Additionally, seeking guidance from a teacher or tutor, or participating in a study group can also be helpful. There are also various online resources and practice problems available to aid in your learning and improvement.

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