36 Questions
What is the numerator of the first term in the given expansion?
n
What is the denominator of the first term in the given expansion?
2!
What is the power of x in the second term of the expansion?
n-1
What is the coefficient of the second term in the expansion?
n(n-1)
What is the general term in the expansion?
$\frac{n(n-1)...(n-k+1)x^{n-k}}{k!}$
What is the value of the limit as Δx approaches 0?
nx
What is the expression for Δf/Δx?
nx + xΔx + ...
What is the purpose of taking the limit as Δx approaches 0?
To find the derivative of the function
What is an implicit function of x?
A function where one of the variables x or y is not explicitly solved for in terms of the other variable
What is an example of an implicit function?
The equation of a circle given by x^2 + y^2 - c^2 = 0
What is the purpose of implicit differentiation?
To calculate the derivative term when y cannot be explicitly solved for in terms of x
What is the assumption made when differentiating an implicit function?
That the implicit function defines y as a function of x
What is the result of differentiating an implicit function with respect to x?
An equation that can be solved for the derivative term dy/dx
When can an implicit function be converted into an explicit form?
Sometimes
What is the purpose of using the 2nd differential in finding stationary points?
To determine the type of stationary points
What is the value of $f''(x)$ when $x = 0$ in the given function $f(x) = 6x - 6$?
-6
What is the stationary point on the curve $y = x^3$?
(0, 0)
What is the type of stationary point at $x = 0$ in the given function $f(x) = 6x - 6$?
Maximum
What is the purpose of using the 3rd differential in finding the type of stationary point?
To decide whether it is a maximum, minimum or point of inflexion
What is the value of $d^2y/dx^2$ when $x = 0$ in the curve $y = x^3$?
0
What is the type of stationary point at $x = 2$ in the given function $f(x) = 6x - 6$?
Minimum
What do practical problems involving maximum and minimum values often require?
Both of the above
What condition should be satisfied for Theorem II to be applicable?
f(x) and g(x) are differentiable in the open interval (a, b) except possibly at x0
What is the purpose of Theorem II?
To evaluate a limit of a function at a point where the function is not defined
What is the technique used to find the limit in the example?
Algebraic technique of factorization
What is the value of the limit in the example?
3/5
What is the expression of the function in the example?
(x^2 - 9)/(x - 3)(x + 2)
What is the important condition for Theorem II to hold?
g'(x) ≠ ∞ for x ≠ x0
What is the limit of sin(x) / x as x approaches 0?
1
Which series expansion is used to evaluate the limits in the given example?
Maclaurin's series
What is the coefficient of the x^3 term in the Maclaurin's series expansion of sin(x)?
-1/3!
What is the limit of (cos(x) - 1) / x as x approaches 0?
0
What is the coefficient of the x^2 term in the Maclaurin's series expansion of cos(x)?
-1/2!
What is the purpose of using the Maclaurin's series expansion in the given example?
To evaluate the limit of (cos(x) - 1) / x as x approaches 0
What is the general formula for the Maclaurin's series expansion of sin(x)?
x - x^3 / 3! + x^5 / 5! - ...
What is the general formula for the Maclaurin's series expansion of cos(x)?
1 - x^2 / 2! + x^4 / 4! - ...
Study Notes
Differentiation
- The formula for implicit differentiation is derived using the limit definition of a derivative:
- Δf = nx + xΔx + ... + (Δx)^n
- Δf/Δx = nx + x + ... + (Δx)^(n-1)
- The formula is used to find the derivative of an implicit function f(x, y) = 0.
Implicit Functions
- An implicit function is defined as an equation of the form f(x, y) = 0, where one of the variables x or y is not explicitly solved for in terms of the other variable.
- Example: The equation of a circle given by f(x, y) = x^2 + y^2 - c^2 = 0, where c is a constant.
- Implicit functions can be converted into explicit forms, but not always.
Stationary Points
- Stationary points can be found using the first and second derivatives.
- The first derivative is used to find the x-coordinate of the stationary point, and the second derivative is used to determine the type of stationary point (maximum, minimum, or point of inflection).
- Example: Find the stationary points on the curve y = x^3 and determine the type.
Practical Problems
- Many practical problems in science and engineering involve finding maximum and minimum values.
- These problems often require rearranging equations to contain only one variable.
- Example: Evaluate the limit of a function using the series expansion.
Series Expansion
- The series expansion of a function can be used to evaluate limits.
- The Maclaurin's series is used to expand the sine and cosine functions.
- Examples: Show that lim (sin(x)/x) = 1 and lim (cos(x) - 1)/x = 0 as x approaches 0.
Theorem II
- If f(x) and g(x) are differentiable in the open interval (a, b) except possibly at a point x0 in this interval, and if f(x0) = g(x0) = ∞, g'(x) ≠ ∞ for x ≠ x0, then:
- lim (f(x)/g(x)) = lim (f'(x)/g'(x)) as x approaches x0
- Provided the limit on the right-hand side exists.
Algebraic Techniques
- Factorization can be used to find limits of functions.
- Example: Use the algebraic technique of factorization to find the limit of (x^2 - 9)/(x^3 - x^2 - 6x) as x approaches 3.
This quiz covers the concept of forward difference formula in mathematics, which is used to approximate the value of a function at a given point.
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