Question

True or False If \(b>a,\) then \(\frac{d}{d x} \int_{a}^{b} e^{x^{2}} d x\) is positive. Justify your answer. .

Short answer

The statement is true. The derivative of the integral of \(e^{x^{2}}\) from a to b is \(e^{b^{2}}\), which is always positive as long as b is real.

Step-by-step solution

Understand Expression

Here we have an integral of \(e^{x^{2}}\) from a to b, and we're finding the derivative of this integral with respect to x. By the first part of the Fundamental Theorem of Calculus, if we have a function that is an integral in the form \(F(x) = \int_{a}^{x} f(t) dt\), the derivative of F(x) with respect to x is just the original function f(x).

Set Up Derivative

We set up the derivative \(\frac{d}{d x}\int_{a}^{b} e^{x^{2}} d x\). If b > a, and according to 1st part of Fundamental theorem of calculus, the derivative of integral from a to b of \(e^{x^{2}}\) dx will just be \(e^{x^{2}}\) at x = b.

Evaluate Derivative Value

Evaluate \(e^{b^{2}}\), which is always positive since the exponential function is always positive, no matter what value b holds as long as b is real number. Hence the statement is true.