Category: Differential Calculus, Algebra
"Published in Vacaville, California, USA"
Evaluate
Solution:
To get the value of a given function, let's substitute the value of n to the above equation, we have
Since
the answer is -∞/∞, then it is an Indeterminate Form which is not
accepted as a final answer in Mathematics. We have to do something first
in the given equation so that the final answer will be a real number,
rational, or irrational number.
Method 1:
Since the answer is Indeterminate Form, then we have to divide both sides of the fraction by 2n and simplify the given equation as follows
Substitute the value of n to the above equation, we have
Therefore,
Method 2:
Another
method of solving Indeterminate Form is by using L'Hopital's Rule. This
is the better method especially if the rational functions cannot be
factored. L'Hopitals Rule is applicable if the Indeterminate Form is
either 0/0 or ∞/∞. Let's apply the L'Hopital's Rule to the given
function by taking the derivative of numerator and denominator with
respect to n as follows
Again, apply the L'Hopital's Rule to the above equation, we have
Since the resulting equation is the same as the given equation, then we cannot use the L'Hopital's Rule because of the repetitive solutions and results. There's no end in this process. In this case, we have to consider the Method 1 in solving the given limits.
Category: Differential Calculus, Algebra
"Published in Vacaville, California, USA"
Evaluate:
Solution:
To get the value of a given function, let's substitute the value of x to the above equation, we have
Since the answer is 0/0, then it is an Indeterminate Form which is not accepted as a final answer in Mathematics. We have to do something first in the given equation so that the final answer will be a real number, rational, or irrational number.
Method 1:
Since the answer is Indeterminate Form, then we have to inspect the given function, factor, and then simplify if possible as follows
Substitute the value of x to the above equation, we have
Therefore,
Method 2:
Another method of solving Indeterminate Form is by using L'Hopital's Rule. This is the better method especially if the rational functions cannot be factored. L'Hopitals Rule is applicable if the Indeterminate Form is either 0/0 or ∞/∞. Let's apply the L'Hopital's Rule to the given function by taking the derivative of numerator and denominator with respect to x as follows
Substitute the value of x to the above equation, we have
Therefore,
Category: Differential Equations, Integral Calculus, Analytic Geometry, Algebra
"Published in Newark, California, USA"
Find the equation of a curve having the given slope that passes through the indicated point:
Solution:
The slope of a curve is equal to the first derivative of a curve with respect to x. In this case, y' = dy/dx. Let's consider the given slope of a curve
Multiply both sides of the equation by dx, we have
Integrate on both sides of the equation, we have
In order to get the value of arbitrary constant, substitute the value of the given point which is P(5, 4) to the above equation, we have
Therefore, the equation of a curve is
Category: Differential Calculus, Algebra
"Published in Newark, California, USA"
If y = f(u) and u = Ï•(x), show that
Solution:
Consider the given two equations above
Since y is a function of u and u is a function of x, then we can apply the derivative of y with respect to x by Chain Rule Method.
Take the derivative of the first equation with respect to u, we have
Take the derivative of the second equation with respect to x, we have
Hence, by Chain Rule Method,
Take the derivative of the above equation with respect to x, we have
but
Hence, the above equation becomes
Therefore,
where
