Let y(x)=∫0^x f(t) d t, where f is the function whose graph...

Let $y(x)=\int_{0}^{x} f(t) d t,$ where $f$ is the function whose graph is (a) Evaluate g(0) and g(6). (b) Estimate $g(x)$ for $x=1,2,3,4,$ and 5 (c) On what interval is $g$ increasing? (d) Where does $g$ have a maximum value? (e) Sketch a rough graph of $g$. (f) Use the graph in part (e) to sketch the graph of $g^{\prime}(x)$ Compare with the graph of $f$ (graph cannot copy)

Let $y(x)=\int_{0}^{x} f(t) d t,$ where $f$ is the function whose graph is
(a) Evaluate g(0) and g(6).
(b) Estimate $g(x)$ for $x=1,2,3,4,$ and 5
(c) On what interval is $g$ increasing?
(d) Where does $g$ have a maximum value?
(e) Sketch a rough graph of $g$.
(f) Use the graph in part (e) to sketch the graph of $g^{\prime}(x)$ Compare with the graph of $f$
(graph cannot copy)

Key Concepts

Accumulation Function

An accumulation function is defined by an integral such as y(x) = ??? f(t) dt. It represents the total accumulated area under the curve of a function f(t) from a fixed starting point up to x, and it provides insight into the overall behavior of the function over an interval.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration. It states that if y(x) = ??? f(t) dt, then the derivative y'(x) is equal to f(x). This connection allows one to recover the original function from its integral and to understand that the slope of the accumulation function at any point is given by the value of the integrand f at that point.

Monotonicity and the Sign of the Derivative

The behavior of the accumulation function, such as whether it is increasing or decreasing, depends on the sign of its derivative. Since the derivative y'(x) equals f(x), if f(x) is positive over an interval, then y(x) is increasing on that interval, whereas if f(x) is negative, y(x) is decreasing.

Graphical Estimation of Definite Integrals

When provided with the graph of a function f, one can visually estimate the value of the integral by approximating the area under the curve. This estimation technique is useful in situations where an exact analytical evaluation is difficult or when a rough sketch of the accumulation function is required.

Relationship Between a Function and Its Derivative

Since the derivative of the accumulation function y(x) is the original function f(x), the graph of y'(x) directly corresponds to the graph of f(x). Understanding this relationship is key when comparing the behavior of y and f, such as identifying intervals where y is increasing or decreasing, and locating the maximum or minimum values of y.

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