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What is x when its the subject of the equation c minus dx equals ex plus f?
ex+f = c -dx ex+dx = c -f x(e+d) = c -f x = c -f/(e+d)
What are the rules of differentiation?
While no set of rules can handle differentiating every expression, the following should help. For all of the following, assume c and n are constants, f(x) and g(x) are functions of x, and f'(x) and g'(x) mean the derivative of f and g respectively. Constant derivative rule:d/dx(c)=0 Constant multiple rule:d/dx(c*f(x))=c*f'(x) Sum and Difference Rule:d/dx(f(x)±g(x))=f'(x)±g'(x) Power rule:d/dx(xn)=n*xn-1 Product rule:d/dx(f(x)*g(x))=f'(x)*g(x) + g'(x)*f(x) Quotient rule:d/dx(f(x)/g(x))=(f'(x)*g(x)-g'(x)*f(x))/f(x)² Chain rule:d/dx(f(g(x))= f'(g(x))*g'(x)
What is the integral of f prime divided by the quantity f times the square root of the quantity f squared minus a squared with respect to x where f is a function of x and a is a constant?
∫ f'(x)/[f(x)√(f(x)2 - a2)] dx = (1/a)arcses(f(x)/a) + C C is the constant of integration.
What polynomial function below find F(-4) f(x)2x2-x plus 9?
f(x) = 2 * 2 - x + 9 f(-4) = 2 * 2 -(-4) + 9 f(-4) = 4 + 4 + 9 = 17
How do you integrate piecewise continuous functions?
Simply integrate all the pieces apart, en add them up. This is allowed, because int_a^c f(x)dx = int_a^b f(x)dx + int_b^c f(x)dx for all a,b,c in dom(f).
Definition of Differentiable function?
Let f be a function with domain D in R, the real numbers, and D is an open set in R. Then the derivative of f at the point c is defined as: f'(c) =lim as x-> c of the difference quotient [f(x)-f(c)]/[x-c] If that limit exits, the function is called differentiable at c. If f is differentiable at every point in D then f is called differentiable in D.
What is the definition of continuity in calculus?
A function f is continuous at c if:f(c) is defined.lim "as x approaches c" f(x) exists.lim "as x approaches c" f(x) = f(c).
What is the integral of f divided by the quantity f plus b times f plus c with respect to x where f is a function of x and b and c are constants?
∫ f(x)/[(f(x) + b)(f(x) + c)] dx = [b/(b - c)] ∫ 1/(f(x) + b) dx - [c/(b - c)] ∫ 1/(f(x) + c) dx b ≠c
What is the difference between Taylor series and Maclaurin series?
A Maclaurin series is centered about zero, while a Taylor series is centered about any point c. M(x) = [f(0)/0!] + [f'(0)/1!]x +[f''(0)/2!](x^2) + [f'''(0)/3!](x^3) + . . . for f(x). T(x) = [f(c)/0!] + [f'(c)/1!](x-c) +[f''(c)/2!]((x-c)^2) + [f'''(c)/3!]((x-c)^3) + . . . for f(x).
Are vertical and opposite angles the same?
Yes, vertical angles are formed by an x. Opposite angles are the angles opposite on the x. For instance...AB X CDAnd and d are opposite angles, so are B and C. This whole figure is called a vertical angle.
What is x when its the subject of the equation c minus dx equals ex plus f?
ex+f = c -dx ex+dx = c -f x(e+d) = c -f x = c -f/(e+d)
What is a critical point?
Critical point (in one variable): A point on the graph y = f(x) at which f is differentiable and f'(x) = 0. The term is also used for the number c such that f'(c) = 0. The corresponding value f(c) is a critical value. A critical point c can be classified depending upon the behavior of fin the neighborhood of c, as one of following:1. a local minimum, if f'(x) > 0 to the left of c and f'(x)< 0 to the right of c.2. a local maximum, if f'(x) < 0 to the left of c and f'(x)> 0 to the right of c.3. neither local maximum nor local minimum:a) if f'(x) has the same sign to the left and to the right of c, in which case c is a horizontal point of inflection.b) if there is an interval at every point of which f'(x) = 0 and c is an endpoint or interior point of this interval.(This is called the first derivative test).The second derivative test (is also a test for maximum and minimum values. It is a consequence of the concavity test):Suppose f is continuous near c.1. If f'(c) = 0 and f''(c) > 0, then f has a local minimum at c.2. If f'(c) = 0 and f''(c) < 0, then f has a local maximum at c.Example: f(x) = x^3 - 12x + 1(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f.(c) Find the intervals of concavity and the inflection points.Solution:(a) f(x) = x^3 - 12x + 1f'(x) = 3x^2 - 12f'(x) = 3(x +2)(x - 2)Interval: x < -2; -2 x < 2; x > 2x + 2: - ; +; +x - 2: - ; - ; +f'(x): + ; - ; +f: increasing on (-∞, -2); decreasing on (-2, 2); increasing on (2, ∞) So f is increasing on (-∞, -2) and (2, ∞) and f is decreasing on (-2, 2).(b) f changes from increasing to decreasing at x = -2 and from decreasing to increasing at x = 2. Thus f(-2) = 17 is a local maximum value and f(2) = -15 is a local minimum value.(c) f''(x) = 6xf''(x) > 0 ↔ x > 0 and f''(x) < 0 ↔ x < 0. Thus f is concave upward on (0, ∞) and concave downward on (-∞, 0). There is an inflection point where the concavity changes, at (0, f(0)) = (0, 1).
How can you find the point at which the function is continuous?
Using calculus to see if the function f(x) is continuous at a point (point c) involves three steps. These three conditions must be met: 1. f(c) exists, is defined 2. lim f(x) exists x-->c 3. f(c)= lim f(x) x-->c
How many Celsius in 1 F?
-17.22 C °C x 9/5 + 32 = °F (°F - 32) x 5/9 = °C
What are the slopes of vertical lines?
Undefined. Proof, let a function f have a vertical line on x = c. (Notice: By definition of functions, it is not even a function, which means that we do not even need to discuss differentiability. We assume it is a function) Now suppose f'(c) exist, then f'(c) = lim x --> c (f(x) - f(c))/(x - c), the limit exist but since it's a straight line, assume non trivial (a point), we have automatically x = c. But since it's non-trivial, hence f(x) != f(c), let f(x) - f(c) = r for some real number r != 0. we get f'(c) = lim x --> c (f(x) - f(c)) / (x - c) = r/0 which is undefined. Contradiction!. Hence f'(c) doesn't exist. Note: If you see a straight line some where on a "function" and they ask for derivative, write:"This is not a function, what kind of question is this! Go back to Calculus class!" If you are discussing a function with a vertical slope, e.g. let f(x) = cubeRoot(x), then it's a different proof.
What is the integral of the derivative with respect to x of a function of x divided by that same function of x with respect to x?
∫ f'(x)/f(x) dx = ln(f(x)) + C C is the constant of integration.
Why its said that derivation is reverse of antiderivatives?
It is said that derivation is reverse of antiderivation because that is the terminology.If you have a function f(x), then the derivative d/dx f(x) is the slope of the function f(x) at each x. This derivative could be called g(x). Sometimes it is called f'(x), but lets call it g(x).Now, start over...If you have a function g(x), then the antiderivative is a function f(x) such that the derivative d/dx f(x) is g(x). This may sound like circular definition, but it is not...If the derivative of f(x) is g(x), then the antiderivative of g(x) is f(x). Technically, the antiderivative of g(x) is f(x) + C, where C is any constant. This is true because the derivative of a constant C is zero.Now - terminology.Taking the derivative is the same thing as derivation.Taking the antiderivative is the same thing as reverse derivation.Taking the deriviative is the same thing as reverse antiderivation.Antiderivation is also called integration and that is the next topic after derivation. They are simply the reverse processes of each other.
