B737 CREW DEVIATES FROM SPD RESTR DURING CLB IN LAX AIRSPACE.

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NORMAL OP OUT OF LAX. FLYING THE GORMAN 1 DEP. FIRST CTLR RESTR US TO 250 KTS AND GAVE US A COUPLE VECTOR HDGS. LEVEL AT 9000 FT AT 250 KTS. SECOND CTLR GAVE US ANOTHER VECTOR TO RESUME THE GORMAN 1 AND 13000 FT. (VERY BUSY RADIO WORK.) THROUGH 10000 FT THE CAPT ASKED THAT I PUT MAX SPDS IN THE FMC FOR CLB. I THOUGHT HE WAS IN LEVEL CHANGE MODE WITH THE MCP SO I DID SO. ABOUT THIS TIME WE WERE SWITCHED TO ZLA. A COUPLE MINS LATER CTR ASKED WHAT OUR SPD WAS AND WE WERE AT 310 KTS. HE TOLD US NOT TO EXCEED PRESENT SPD AND THEN ASKED IF WE HAD BEEN RESTR TO 250 KTS. WE SAID WE HAD. IN THE BUSY LOAD OF FLYING AND RADIO WORK; BOTH THE CAPT AND I DIDN'T REALIZE THAT WE DIDN'T HAVE THE LEVEL CHANGE ENGAGED SO WHEN I PUT IN THE MAX SPDS FOR THE CLB THE ACFT ACCELERATED TO THOSE SPDS. WHILE I FEEL THERE IS ALWAYS ROOM FOR IMPROVEMENT I FEEL THAT A PERSON TENDS TO FORGET ABOUT A RESTR LIKE THIS AS YOU GO THROUGH 3 OR 4 CTLRS. ON OUR PART I FEEL AS THOUGH WE COULD COME UP WITH SMALL XCHKS OF THE MCP MUCH LIKE WE DO WITH ALT AWARENESS PROGRAMS.