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Definite vs indefinite integral: is there a connection? The Area function. Given a continuous function f on [a,b], define its area function A(x) ...
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If F and f are two functions on [a, b] such that d dx F^ (x) =^ F^
′(x) = f (x), we say f is the derivative of F , and F is an antiderivative of f.
If F and f are two functions on [a, b] such that d dx F^ (x) =^ F^
′(x) = f (x), we say f is the derivative of F , and F is an antiderivative of f. For instance, we know that d dx sin^ x^ = cos^ x,^
d dx x
(^3) = 3x (^2).
We say that cos x and 3x^2 are the derivatives of sin x and x^3 respectively, or sin x is an antiderivative of cos x, x^3 is an antiderivative of 3x^2.
The derivative identities d dx
[ (^) xa+ a + 1
= xa^ for a 6 = − 1 , d dx ln^ x^ =
x , may be rephrased as ∫ xa^ dx =
xa+ a + 1 +^ C^ if^ a^6 =^ −^1 , ln x + C if a = − 1.
eax^ dx = e
ax ∫^ a^ +^ C^ , sin(ax) dx = − (^1) a cos(ax) + C , ∫ cos(ax) dx =^1 a sin(ax) + C , ∫ sec^2 ax dx =^1 a tan(ax) + C , ∫ sec(ax) tan(ax) dx =^1 a sec(ax) + C.
Given a continuous function f on [a, b], define its area function A(x) as the definite integral A(x) =
∫ (^) x a^ f^ (t)^ dt,^ a^ ≤^ x^ ≤^ b. In other words, A(x) is a function on [a, b] whose value at x is the signed area under the curve y = f (t) between t = a and t = x.
Given a constant function f , its area function is
A. constant B. linear C. quadratic D. always positive
For a linear function f (t) = mt + c, its area function on [0, b] is
A. constant B. linear C. quadratic D. always increasing
For a linear function f (t) = mt + c, its area function on [0, b] is
A. constant B. linear C. quadratic D. always increasing
Question: What is the derivative of the area function of a linear function?
