You cannot use Response object when using ASP.NET Ajax.
In addition, you have typo in UpdatedControls
tag - see in the comment.
protected void Page_Load(object sender, EventArgs e)
{
// You cannot have Response.Write here.
}
<telerik:RadAjaxManager ID="RadAjaxManager1" runat="server">
<AjaxSettings>
<telerik:AjaxSetting AjaxControlID="Button1">
<UpdatedControls>
<telerik:AjaxUpdatedControl ControlID="Panel1" />
<telerik:AjaxUpdatedControl ControlID="Label1" />
</UpdatedControls>
</telerik:AjaxSetting>
<telerik:AjaxSetting AjaxControlID="Button2">
<UpdatedControls>
<%-- Must be Panel2 and Label2 --%>
<telerik:AjaxUpdatedControl ControlID="Panel2" />
<telerik:AjaxUpdatedControl ControlID="Label2" />
</UpdatedControls>
</telerik:AjaxSetting>
</AjaxSettings>
</telerik:RadAjaxManager>