Script called by AJAX from WordPress plugin isn’t getting $wpdb

Solution:

You need to use ajax in WordPress way.

From WordPress Doc:

First add some javascript that will trigger the AJAX request:

<?php
add_action( 'admin_footer', 'my_action_javascript' );

function my_action_javascript() {
?>
<script type="text/javascript" >
jQuery(document).ready(function($) {

    var data = {
        action: 'my_action',
        whatever: 1234
    };

    // since 2.8 ajaxurl is always defined in the admin header and points to admin-ajax.php
    $.post(ajaxurl, data, function(response) {
        alert('Got this from the server: ' + response);
    });
});
</script>
<?php
}

Then, set up a PHP function that will handle that request:

add_action('wp_ajax_my_action', 'my_action_callback');

function my_action_callback() {
    global $wpdb; // this is how you get access to the database

    $whatever = intval( $_POST['whatever'] );

    $whatever += 10;

        echo $whatever;

    die(); // this is required to return a proper result
}

Reference:
1. http://codex.wordpress.org/AJAX_in_Plugins
2. http://wp.tutsplus.com/articles/getting-started-with-ajax-wordpress-pagination/